SOME QUESTIONS IN THE THEORY OF ELECTRICAL FLUCTUATIONS

V. L. Ginzburg

INTRODUCTION

Electrical fluctuations are known to be of great scientific and practical interest and, at the same time, still cannot be regarded as comprehensively studied. It is therefore understandable that the theory of electrical fluctuations continues to attract attention, and this also applies to such a fundamental result in the field under consideration as the Nyquist formula. Recently the author has encountered three questions belonging here: the peculiarities of electrical fluctuations near points of a second-order phase transition (Curie points), the application of thermodynamics to electrical fluctuations, and the quantum generalization of the Nyquist formula. All these questions appear not to be without interest, and at the same time their understanding can, as it seems to us, be brought to a certain clarity. The present article is devoted to the consideration of these three questions and is mainly methodological in character. For the latter reason the author has sought to avoid excessive brevity of exposition.

1. CLASSICAL NYQUIST FORMULA

For the purposes of the further exposition it is necessary first to dwell on the definition of the basic quantities used in the theory of electrical fluctuations and on the derivation of the Nyquist formula. Here we shall be concerned with that well-known form of the Nyquist formula which is valid only when quantum effects are neglected.

Thermal motion in a certain electrical circuit leads to the fact that, even in the absence of an external electromotive force, a current \(I(t)\) flows in the circuit. The instantaneous value of this current changes rapidly both in magnitude and in direction, owing to which interest

is represented not by the quantity \(I(t)\), but by its mean square \(\overline{I^2}\) and the spectral density \(|I_\omega|^2\), defined by the relations:

\[ \overline{I^2}=\lim_{T\to\infty}\frac{1}{T}\int_0^T I^2\,dt =4\pi\int_0^\infty |I_\omega|^2\,d\omega, \tag{1,1} \]

\[ |I_\omega|^2=\lim_{T\to\infty}\frac{1}{4\pi^2T} \left|\int_0^T I(t)e^{-i\omega t}\,dt\right|^2 . \tag{1,2} \]

It is necessary to define the quantity \(|I_\omega|^2\) as a certain limit because \(I(t)\) cannot be expanded in a Fourier integral (for more detail see, for example\(^1\), § 117, where the definition of quantities of the type \(|I_\omega|^2\) differs from (1,2) by the factor \(4\pi^2\)).

In a number of cases it proves convenient to define the density \(|I_\omega|^2\) not directly, but by expressing it through the density \(|\mathcal{E}_\omega|^2\) of the fluctuating electromotive force \(\mathcal{E}(t)\):

\[ |\mathcal{E}_\omega|^2=\lim_{T\to\infty}\frac{1}{4\pi^2T} \left|\int_0^T \mathcal{E}(t)e^{-i\omega t}\,dt\right|^2, \]

\[ \overline{\mathcal{E}^2} =4\pi\int_0^\infty|\mathcal{E}_\omega|^2\,d\omega =\int_0^\infty w(\omega)\,d\omega =\int_0^\infty w(\nu)\,d\nu, \tag{1,3} \]

\[ w(\omega)=\frac{w(\nu)}{2\pi}=4\pi|\mathcal{E}_\omega|^2, \]

where the frequency \(\nu=\dfrac{\omega}{2\pi}\) is introduced because, in practice, when discussing the questions under consideration, it is used more often than \(\omega\). The fluctuating e.m.f. \(\mathcal{E}(t)\), by definition, is the e.m.f. necessary in order to produce in the circuit the fluctuating current \(I(t)\). But within the framework of the quasistationary approximation to which we confine ourselves, the current \(I(t)\) does not differ in principle from the current arising in the circuit under the influence of some external e.m.f. Therefore, if, as will be assumed, the system (circuit) is linear, i.e. in the presence of an e.m.f. \(\mathcal{E}(\omega)\) varying according to the law \(e^{i\omega t}\), the current in the system is equal to \(I(\omega)=\dfrac{\mathcal{E}(\omega)}{Z(\omega)}\), then an analogous relation also connects \(\mathcal{E}_\omega\) and \(I_\omega\):

\[ |I_\omega|^2=\frac{|\mathcal{E}_\omega|^2}{|Z_\omega|^2} =\frac{4\pi w(\omega)}{|Z(\omega)|^2}. \tag{1,4} \]

In the simplest case, when the system under consideration is a circuit with resistance \(R\), capacitance \(C\), and self-inductance \(L\), the impedance is

\[ Z=R+i\left(L\omega-\frac{1}{\omega C}\right) \]

(for definiteness, below we shall often be speaking precisely of such a circuit). In the general case, below \(R\) should be understood as \(\operatorname{Re} Z\). If we are interested not in the current \(I\), but in the charge on the capacitor entering the circuit, then in the quasistationary case under consideration

\[ |q_\omega|^2=\frac{|I_\omega|^2}{\omega^2},\qquad \overline{q^2}=4\pi\int_0^\infty |q_\omega|^2\,d\omega, \tag{1.5} \]

where \(\overline{q^2}\) is the mean square of the charge on the capacitor. The impedance of the system \(Z(\omega)\) may be regarded as known, or in any case its determination has no bearing on the question of fluctuations. Thus, as is clear from (1.4) and (1.5), the problem of the theory of electrical fluctuations in linear systems reduces to finding the function \(w(\omega)\).

The spectral density of the fluctuation emf, \(w(\omega)\), is determined by the character of the thermal motion in the system and, in the most general case, of course, cannot be determined. However, the function \(w(\omega)\) can be found\(^2\) in the most important particular case, when the system is in a state of thermodynamic equilibrium\(^*\). For this purpose it is most convenient first to establish one general property of the function \(w(\omega)\) in the equilibrium state. Namely, it turns out that in this case the quantity \(w(\omega)\) for a system with impedance \(Z(\omega)\) is the product of \(R(\omega)=\operatorname{Re} Z\) and a certain universal (i.e. independent of the properties of the circuit) function of the frequency \(\omega\) and the temperature of the system \(T\):

\[ w(\omega)=R(\omega)\cdot f(\omega,T). \tag{1.6} \]

To prove formula (1.6), one considers a closed circuit consisting of two two-terminal networks located in thermostats with the same temperature \(T\). In a state of thermodynamic equilibrium, the mean power \(P_{12}\) delivered by the first two-terminal network to the second must be equal to the mean power \(P_{21}\) delivered by the second two-terminal network to the first. On the other hand, for example, the mean power

\[ P_{12}=\overline{I_1(t)V_1(t)}, \]

where the bar denotes averaging over time, \(I_1\) is the current in the circuit produced by the fluctuation emf \(\mathcal{E}_1\) acting in the first two-terminal network, and \(V_1\) is the potential difference on the second corresponding to this current—

\(^*\) Of course, nonequilibrium fluctuations, as well as equilibrium fluctuations in nonlinear systems, are also of interest. However, there are either no results in this field at all, or in any case they are unknown to the author. (After the present article had already been written, several works appeared devoted to fluctuations in nonlinear systems; see J. Appl. Phys. 22, 1143, 1153, 1211 (1951).)

two-terminal network. Therefore, as is easy to see, using the expressions introduced above:

\[ P_{12}=\int_0^\infty \frac{R_2 w_1(\omega)\,d\omega}{|Z|^2},\qquad P_{21}=\int_0^\infty \frac{R_1 w_2(\omega)\,d\omega}{|Z|^2}, \tag{1,7} \]

where \(Z_1\) and \(Z_2\) are the impedances of two-terminal networks 1 and 2, \(R_1=\operatorname{Re} Z_1\), \(R_2=\operatorname{Re} Z_2\), \(Z=Z_1+Z_2\) is the impedance of the whole circuit, and \(w_{1,2}\) are the spectral densities of the fluctuation e.m.f. respectively in two-terminal networks 1 and 2. Equating the expressions \(P_{12}\) and \(P_{21}\) and taking into account that this equality must hold for any two-terminal networks, we arrive at (1,6), i.e. at the conclusion that for any two-terminal networks \(1,2,\ldots,k\)

\[ \frac{w_1}{R_1}=\frac{w_2}{R_2}=\cdots=\frac{w_i}{R_i}=f(\omega,T), \]

where \(f\) is a universal function of \(\omega\) and \(T\). Nyquist himself based his derivation of the formula for \(w(\omega)\) on relation (1,6); it is proved in more detail and very clearly in § 4 of the paper by G. S. Gorelik\(^3\).

The universality of the function \(f(\omega,T)\) makes it possible, in order to find it, to consider some single simplest (“test”) system. For this purpose Nyquist considered a two-wire line. Sometimes, as a “test” system, an antenna situated in a field of thermal radiation is chosen. In our opinion, it is simplest and most convenient to take, as the “test” system, an ordinary electrical \(RCL\)-circuit with weak damping.

The mean electrical energy \(\overline{U}\) in the capacitor belonging to the \(RCL\)-circuit is equal to

\[ \overline{U}=\frac{\overline{q^2}}{2C} =\frac{2\pi}{C}\int_0^\infty |q_\omega|^2\,d\omega =\frac{2\pi}{C}\int_0^\infty \frac{|\varepsilon_\omega|^2\,d\omega}{\omega^2|Z(\omega)|^2} = \]

\[ =\frac12\int_0^\infty \frac{Cw(\omega)\,d\omega}{R^2C^2\omega^2+(LC\omega^2-1)^2}. \tag{1,8} \]

In the state of thermodynamic equilibrium, on the one hand, \(w=R(\omega)\cdot f(\omega,T)\), and, on the other hand, within the framework of classical theory the mean energy in the capacitor, entirely analogous to the mean potential energy of an oscillator, is equal to \(\frac{kT}{2}\), where \(k=1.38\cdot10^{-16}\) erg/degree. Further, for a sufficiently weakly damped circuit the function

\[ \frac{CR}{R^2C^2\omega^2+(LC\omega^2-1)^2}, \]

standing under the integral sign in (1,8) after substitution of (1,6), is a very sharp delta-shaped function with a maximum at \(\omega=\omega_0=\frac{1}{\sqrt{LC}}\) (\(\omega_0\) is the natural frequency of the circuit). Therefore one may put \(R(\omega)=R(\omega_0)\) and \(f=f(\omega_0,T)\), after which the integral proves to be

equal to \(\pi/4\)1. As a result we obtain \(\dfrac{kT}{2}=\dfrac{\pi f(\omega_0,T)}{4}\), and, since the frequency \(\omega_0\) may be arbitrary, \(f(\omega,T)=\dfrac{2kT}{\pi}\), and

\[ w(\omega)=R(\omega)\cdot f(\omega,T)=\frac{2}{\pi}R(\omega)\cdot kT \tag{1,9} \]

or

\[ w(\nu)=2\pi w(\omega)=4R(\nu)\cdot kT. \tag{1,9'} \]

Expression (1,9) is the Nyquist formula, applicable within the limits of validity of the classical theory. For this latter reason, formula (1,9′), which until now alone has been used in practice, we shall call the classical Nyquist formula. The fact that the spectral density of the fluctuating e.m.f. \(w(\omega)\), even in a state of thermodynamic equilibrium, depends on the resistance \(R\), which characterizes the rate of dissipation of the energy of the system under consideration when it has been brought out of equilibrium, is physically quite understandable. Indeed, if, for example, in a metal, owing to a fluctuation in the distribution of the electrons with respect to velocities, some fluctuating current has arisen, then, since this current is in principle no different from any other cur-

\[ \frac{1}{2}\int_{0}^{\infty} \frac{RC\,d\omega}{R^{2}C^{2}\omega^{2}+(LC\omega^{2}-1)^{2}} = \frac{1}{2}\int_{0}^{\infty} \frac{\alpha\,d\eta}{\alpha^{2}\eta^{2}+(\eta^{2}-1)^{2}} = \frac{\pi}{4} \]

(here \(\eta=\sqrt{LC}\cdot\omega\)). Instead of considering the electric energy \(\overline{U}=\dfrac{\overline{q^{2}}}{2C}\), in order to determine the form of the function \(f(\omega,T)\) one may calculate the mean magnetic (kinetic) energy

\[ \overline{K}=\frac{L\overline{I^{2}}}{2} = \frac{kT}{2} = \frac{1}{2}\int \frac{C^{2}L\omega^{2}w(\omega)\,d\omega} {R^{2}C^{2}\omega^{2}+(LC\omega^{2}-1)^{2}}. \]

The integral

\[ \frac{1}{2}\int \frac{C^{2}LR\omega^{2}\,d\omega} {R^{2}C^{2}\omega^{2}+(LC\omega^{2}-1)^{2}} = \frac{1}{2}\int_{0}^{\infty} \frac{\alpha\eta^{2}\,d\eta} {\alpha^{2}\eta^{2}+(\eta^{2}-1)^{2}} \]

is also equal to \(\dfrac{\pi}{4}\) for any value of \(\alpha\). The indicated integrals are easiest to compute by using the residue theorem. In considering a weakly damped circuit, as was done in the text, the integrals are evaluated at once, since in this case

\[ \frac{1}{2}\int_{0}^{\infty} \frac{\alpha\,d\eta} {\alpha^{2}\eta^{2}+(\eta^{2}-1)^{2}} \approx \frac{1}{2}\int_{0}^{\infty} \frac{\alpha\,d\eta} {\alpha^{2}+4(\eta-1)^{2}} \approx \frac{\pi}{4}. \]

...so that its damping rate, which determines the form of the functions \(|I_\omega|^2\) and \(w(\omega)\), must depend precisely on the resistance \(R(\omega)\).

The electrical fluctuations under consideration are analogous to the Brownian motion of particles in a gas or liquid. In particular, fluctuations in an oscillatory circuit described by the equation

\[ L\ddot q+R\dot q+\frac{q}{C}=\mathcal{E}, \tag{1,10} \]

are completely analogous to the Brownian motion of an oscillator, for which

\[ m\ddot x+r\dot x+kx=F, \tag{1,11} \]

where \(F\) is a random force \((\overline F=0)\). Hence it is clear that results obtained for the electrical system can at once be transferred to the mechanical system, and conversely.

However, in the mechanical case the thermodynamic considerations that led to the important formula (1,6) are never applied, since here they would be, at the very least, highly artificial. Therefore the calculation of all quantities analogous to \(|I_\omega|^2\), \(w(\omega)\), etc., in the theory of Brownian motion is carried out by other methods.

Without dwelling on the theory of Brownian motion (see \(^{4,5,6}\)), let us give directly the corresponding derivation of Nyquist’s formula for the system (1,10). To this end let us assume that the fluctuating e.m.f. has the character of random, instantaneous kicks uncorrelated with one another. In this case the spectrum of the e.m.f. \(\mathcal{E}(t)\) will be continuous and independent of frequency [\(\mathcal{E}(t)\) is represented as a sum of \(\delta\)-functions, and the spectrum of a \(\delta\)-function does not depend on \(\omega\)]. Thus it is assumed that \(w(\omega,R,T,\ldots)=w(R,T,\ldots)\).

Further, the mean electric energy \(\overline U\) and the mean magnetic energy \(\overline K\), corresponding to the mean potential and mean kinetic energy of the oscillator, are, on the one hand, equal to \(\dfrac{kT}{2}\), and, on the other hand, are expressed through the function \(w\):

\[ \overline U=\frac{\overline{q^2}}{2C} =\frac12\int_0^\infty \frac{Cw\,d\omega}{R^2C^2\omega^2+(LC\omega^2-1)^2} =\frac{kT}{2} =\overline K =\frac{L\overline{I^2}}{2} = \]

\[ =\frac12\int_0^\infty \frac{LC^2\omega^2 w\,d\omega}{R^2C^2\omega^2+(LC\omega^2-1)^2}. \tag{1,12} \]

If the function \(w\) does not depend on \(\omega\), then, under the assumption that the quantities \(L\), \(C\), and \(R\) also do not depend on \(\omega\), the integrals in (1,12) are evaluated (see the note to p. 352; to find \(w\) it is sufficient, of course, to evaluate one of these integrals) and are equal ...

\[ \frac{\pi w}{4R}=\frac{kT}{2}, \]

whence for \(w\) one obtains Nyquist’s formula \((1.9)^{*}\). More precisely, one obtains Nyquist’s formula with \(R(\omega)=R=\mathrm{const}\), whereas in (1.9) the resistance \(R\) may depend on the frequency. This limitation is very substantial. For this reason alone the first derivation of Nyquist’s formula should be preferred to the second.

In conclusion of this section let us make one more remark. Using Nyquist’s formula (1.9), we can, with the aid of the expressions for \(\overline U\) and \(\overline K\) [see (1.12)], calculate the mean electric (potential) and magnetic (kinetic) energies of a circuit with arbitrary parameters \(L\), \(C\), and \(R\), independent of \(\omega\). As a result, for any values of these parameters we obtain

\[ \overline U=\overline K=\frac{kT}{2}. \tag{1.13} \]

At first sight it may seem that this conclusion is completely obvious by virtue of the theorem, valid in classical statistics, on the uniform distribution of energy over degrees of freedom (see, for example, \({}^{6}\S 15\) or \({}^{1}\S 44\)). This, however, is not quite so. The point is that the canonical distribution for a “system in a thermostat,” used in proving the law of equipartition, is valid, generally speaking, only if the “system” interacts weakly with the thermostat, so that the corresponding interaction energy may be neglected. In our case the “system” is characterized by some macroscopic coordinate \(q\) or \(x\) [see (1.10) and (1.11)], and its interaction with the thermostat is determined by the value of the resistance \(R\) or of the coefficient \(r\). The term “system” is put in quotation marks here because earlier the thermostat also was included in the system (without quotation marks). Therefore, to avoid confusion, instead of “system” we shall speak of a subsystem—the macroscopic part of the whole system, that part whose energy changes only as a result of interaction with other parts of the system. For example, in the case of a pendulum moving in a gas, the coordinate \(x\) is the coordinate of the center of gravity of the pendulum bob (the mass of the bob is \(m\)), measured from its equilibrium position. The role of the thermostat is played by the gas surrounding the pendulum, while the friction is due to the interaction of the subsystem—the bob—with the thermostat—the gas—which reduces to collisions of gas molecules with the bob. In the case of a circuit, the subsystem is the fields in the capacitor and in the self-induction coil; the role of \(x\) is played by the charge on the capacitor \(q\); the thermostat may be regarded as the material of the conductors themselves, while the resis-

*) Let us note that the equality used in (1.12), \(\overline U=\overline K=\dfrac{kT}{2}\), for a strongly damped circuit is by no means self-evident; but under the condition \(R=\mathrm{const}\) it in fact holds (see the end of this section and Section 3).

resistance leads to the conversion of electromagnetic energy into thermal motion in the conductors.

If the resistance of the circuit is so small that during the time of an oscillation its energy changes little, i.e. if

\[ \frac{R}{L} \ll \frac{1}{\sqrt{LC}}=\omega_0, \tag{1,14} \]

then there can be no objections to applying the formulas of statistics and, consequently, the equalities (1,13)*). But when the inequality (1,14) is not satisfied, when the energy of the subsystem changes strongly during one period, the possibility of neglecting the interaction energy is a priori by no means clear. Therefore, assuming, for example, that when the coordinate of the center of gravity of a pendulum is changed by an amount \(x\), the potential—or, more precisely, the free—energy of the entire pendulum and of the surrounding medium of arbitrary viscosity changes only by the amount \(U=\dfrac{kx^2}{2}\), we are making an additional assumption; the same applies to the electrical case. The assumption in question means, in particular, that the subsystem as such remains essentially unchanged irrespective of the magnitude of the damping, i.e. of the rate at which energy is transferred from the subsystem to the thermostat.

Within the framework of classical theory, neglect of the interaction energy at large damping is in principle possible and, as experience shows, is often practically admissible. The interaction energy may be neglected if the interaction forces of the subsystem with its surroundings have the character of instantaneous impacts. For example, in a gas of hard spheres the mean interaction energy (mean potential energy) is equal to zero, despite the fact that energy is transferred in collisions. This assertion that the mean potential energy is zero in the case of instantaneous collisions, i.e. when the interaction energy between particles depends very sharply on their mutual distance, follows from the virial theorem (see \(^1\), p. 111). It is also quite clear from simple physical considerations: at the moment of collision, when, say, both particles have stopped, the instantaneous potential energy is finite and equal to the kinetic energy of the particles before impact, whence it follows that, as the collision time tends to zero, the time-averaged potential energy also tends to zero.

*) Therefore the use, in deriving formula (1,9), of the relation

\[ \overline{U}=\frac{\overline{q^2}}{2C}=\frac{kT}{2} \quad \text{or} \quad \overline{K}=\frac{L\overline{I^2}}{2}=\frac{kT}{2} \]

for a weakly damped circuit is certainly lawful.

Thus, in the case of instantaneous forces, relation (1.13) must be valid. This conclusion is in complete agreement with that made above on the basis of Nyquist’s formula and the assumption that the resistance \(R\) is independent of frequency, since it is precisely for instantaneous impulses that the resistance \(R\) cannot depend on \(\omega\) (for instantaneous impulses, as indicated above, the quantity \(\mathfrak{w}(\omega)=\dfrac{2}{\pi}R(\omega)\,kT\) does not depend on \(\omega\), whence it follows that in this case \(R(\omega)=\mathrm{const}\)).

If, however, \(R=R(\omega)\), then from Nyquist’s formula and the expressions for \(\overline{K}\) and \(\overline{U}\) in terms of \(C\), \(L\), and \(R\) [see (1.12)], the relations (1.13) are not obtained. In this case the whole situation is in general more complicated, since the equation of motion for the charge \(q\) can no longer be written in the form (1.10). Instead of this equation, assuming for simplicity that \(q(t)\) is expanded in a Fourier integral
\[ q(t)=\int_{-\infty}^{+\infty} q_{\omega}e^{i\omega t}\,d\omega, \]
we have:
\[ \left. \begin{aligned} L\ddot q+\frac{q}{C}+\mathscr{E}(R)&=\mathscr{E},\\[6pt] \mathscr{E}(R)&=\int_{-\infty}^{+\infty} i\omega R(\omega)q_{\omega}e^{i\omega t}\,d\omega=\\[6pt] &=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \frac{i\omega}{2\pi}R(\omega)q(t-\tau)e^{i\omega\tau}\,d\omega\,d\tau, \end{aligned} \right\} \tag{1.15} \]
where \(\mathscr{E}\) is the fluctuation electromotive force.

The integro-differential equation (1.15), for \(R\ne\mathrm{const}\), obviously does not reduce to the mechanical equation for a particle with one degree of freedom, and therefore the question of applying statistics to the quantities \(\overline{K}\) and \(\overline{U}\), associated with the “quasi-degree of freedom” \(q\), requires special consideration*). Another aspect of the same question is the fact that the resistance \(R\) depends on \(\omega\) for non-instantaneous impulses, and in this case, generally speaking, one cannot neglect the energy of interaction of the subsystem with its surroundings; therefore one also cannot, without further ado, make use of all the re-

*) In this connection, the fact that in the mechanical case, when \(r=r(\omega)\), the quantity \(\overline{K}\), computed by means of formula (1.9), is not equal to \(\dfrac{kT}{2}\), does not contradict the rigorously following conclusion from classical statistics that the kinetic energy is equal to
\[ \overline{K}=\frac{m\overline{v^{2}}}{2} \]
and has the value \(\dfrac{kT}{2}\), independently of the nature of the interaction of the particle of mass \(m\) under consideration with its surroundings.

results of statistical mechanics. We shall not dwell here in greater detail on this circle of questions, not devoid of interest, since for what follows it was important only to clarify the character and meaning of the conditions under which, for an electric circuit, the relation (1.13) is valid:

\[ \overline{U}=\overline{K}=\frac{kT}{2}. \]

2. THE QUANTUM NYQUIST FORMULA

The classical Nyquist formula (1.9) is valid only when quantum effects may be neglected, i.e., if for the frequency \(\omega\) under consideration the condition

\[ \hbar\omega \ll kT, \tag{2.1} \]

is satisfied, or

\[ \omega \ll 1.3\cdot 10^{11}T. \tag{2.1'} \]

Even for the frequency \(\omega=2\cdot 10^{12}\) \(\left(\lambda=\dfrac{2\pi c}{\omega}\simeq 1\ \text{mm}\right)\), this condition means that \(T\gg 15^\circ\mathrm{K}\). If, for example, \(\omega=6\cdot 10^{10}\) \((\lambda\simeq 3\ \text{cm})\), then for the application of the classical formulas it is necessary that \(T\gg 0.5^\circ\mathrm{K}\).

Thus, in the radio range, quantum effects can strongly affect the character of electrical fluctuations in conductors only at low temperatures. At room temperature, \(T\simeq 300^\circ\mathrm{K}\), quantum corrections need be taken into account only on passing to waves shorter than \(1\ \text{mm}\), or if there arises the question of observing very small deviations from the classical Nyquist formula.

Despite what has been said, there can hardly be any doubt as to the desirability of obtaining the quantum Nyquist formula, i.e., a formula suitable for any frequencies \(\omega\) and temperatures \(T\) and reducing to the classical formula (1.9) under condition (2.1). It is clear that only by having such a quantum formula can one, for given \(\omega\) and \(T\), quantitatively determine the accuracy of formula (1.9), quite apart from the fact that the region of low temperatures and very short waves, where the classical formula cannot be used at all, is also of definite interest. The quantum Nyquist formula can be obtained immediately by proceeding exactly as in the derivation of the classical formula, but taking into account that the mean potential and mean kinetic energy of a weakly damped circuit (oscillator) in the quantum case is equal not to \(kT/2\), but to the known expression

\[ \overline{U}=\overline{K}=\frac{1}{2}\left(\frac{\hbar\omega_0}{2}+\frac{\hbar\omega_0}{e^{\frac{\hbar\omega_0}{kT}}-1}\right), \tag{2.2} \]

where \(\omega_0=\dfrac{1}{\sqrt{LC}}\) is the natural frequency of the circuit.

Thus, for a sufficiently weakly damped circuit, from (1.8) we obtain

\[ \frac{\pi}{4} f(\omega_0,T)=\frac{1}{2}\left(\frac{\hbar\omega_0}{2}+ \frac{\hbar\omega_0}{e^{\frac{\hbar\omega_0}{kT}}-1}\right) \]

and, taking into account the possibility of choosing a circuit with arbitrary frequency \(\omega_0\) and the universality of the function \(f\), we find the quantum Nyquist formula, valid for any \(R(\omega)\):

\[ w(\omega)=R(\omega)\cdot f(\omega,T)= \frac{2}{\pi}R(\omega)\left\{\frac{\hbar\omega}{2}+ \frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}\right\}. \tag{2.3} \]

Under condition (2.1) this formula passes over into (1.9), as it should. Formula (2.3), however, without the zero-point energy \(\hbar\omega/2\), was obtained earlier by Nyquist\(^2\) also essentially by replacing, in (1.9), the mean energy of the oscillator \(\bar E=kT\) by the quantum expression

\[ \bar E=\bar K+\bar U=\frac{\hbar\omega_0}{2}+ \frac{\hbar\omega_0}{e^{\frac{\hbar\omega_0}{kT}}-1}. \]

The derivation given, however, cannot be considered satisfactory without further explanations. The point is that, in addition to the classical expression for the mean energy of the oscillator, in deriving the classical Nyquist formula (1.9) the concept of impedance \(Z(\omega)\) [see (1.4)] and the concrete expression of this impedance through the circuit parameters \(L\), \(C\), and \(R\) were used. At the same time, impedance is usually introduced on the basis of the equations of motion (1.10), which have a classical character. In other words, the question arises whether the classical element in the derivation of Nyquist’s formula given in Section 1 is limited only by the assumption that the mean energy of the circuit is equal to the classical value \(kT\).

To resolve this question, let us clarify how a certain electric circuit (system) is to be considered in quantum theory.

Already in the classical domain, the introduction of resistance, i.e. of the dissipative force \(R\dot q\) in the equation of motion (1.10), is the result of a certain averaging of a microscopic picture in which there is no resistance, and all forces are conservative in character. Carrying out such an averaging and introducing the force \(R\dot q\) is possible only on the condition that the system (oscillatory circuit, pendulum + gas) has a very large number of degrees of freedom, i.e. is a macroscopic system.

in the sense given to this concept in statistical physics. From the point of view of quantum theory this means that the energy levels of the system are arranged very densely, very close to one another*). If the Hamiltonian operator for such a system in the absence of an external perturbation is \(H_0(x,p)\), then the energy levels of the system \(E_n\) are determined from the equation

\[ H_0 \Psi_n(x) = E_n \Psi_n(x), \tag{2,4} \]

where \(x,\ p=-i\hbar \dfrac{\partial}{\partial x}\) are the set of all coordinates and momenta of the particles entering the system.

Let now an external voltage \(\mathcal{E}=\mathcal{E}_0\sin\omega t\) be applied to a conductor of length \(l\) that is part of the system. Then the electric-field strength in the conductor is \(E=\dfrac{\mathcal{E}}{l}\), and the perturbation energy is equal to

\[ V=\sum_k e_k x_k E=\mathcal{E}Q, \]

where \(e_k\) and \(x_k\) are the charge and coordinate of the \(k\)-th particle, and \(Q=\sum_k \dfrac{e_k x_k}{l}\).

The Hamiltonian of the system in the presence of the perturbation is \(H_0+V\), and its behavior in time is determined by the Schrödinger equation:

\[ i\hbar \frac{\partial \Psi}{\partial t} = (H_0+V)\Psi = H_0\Psi+\mathcal{E}_0\sin\omega t\cdot Q\Psi. \tag{2,5} \]

To solve this equation one uses nonstationary perturbation theory, presented in all courses of quantum mechanics (see, for example, \(^{7}\), § 40). For this reason we shall not dwell on the details; nevertheless we shall give all the basic expressions necessary for understanding what follows. The starting point of nonstationary perturbation theory is the representation of the function \(\Psi(x,t)\) in the form of an expansion in a series in the functions of the unperturbed problem \(\Psi_k^0\):

\[ \Psi=\sum_k a_k(t)\Psi_k(x)e^{\frac{iE_k t}{\hbar}}. \tag{2,6} \]

*) To avoid misunderstandings, let us emphasize that, as was already indicated in Section 1, the system includes not only that part of it characterized by some coordinate \(q\) or \(x\) (this part we call the “subsystem”), but also other parts (the gas in the case of a pendulum, the conductors in the case of a circuit), interacting with the “subsystem” and bringing about the damping of its oscillations. The levels of the subsystem, if one abstracts from its interaction with the other parts of the system, may be situated at any distance from one another.

Substituting (2.6) into (2.5), we obtain a system of equations for \(a_k(t)\):

\[ \left. \begin{gathered} i\hbar\,\frac{d a_k}{dt}=\sum_n V_{kn}a_n e^{\frac{i}{\hbar}(E_k-E_n)t},\\ V_{kn}=\int \Psi_k^*(x)V(x,t)\Psi_n(x)\,dx . \end{gathered} \right\} \tag{2.7} \]

If at the instant \(t=0\) the system is in the state \(n\), i.e. at \(t=0\)

\[ a_k=a_k^{(0)}=\delta_{kn}, \]

then in the first approximation

\[ i\hbar\,\frac{d a_k^{(1)}}{dt} = V_{kn}e^{\frac{i}{\hbar}(E_k-E_n)t} \quad\text{and}\quad a_k^{(1)} = -\frac{i}{\hbar}\int_0^t V_{kn}e^{i\omega_{kn}t'}\,dt', \]

where

\[ \omega_{kn}=\frac{E_k-E_n}{\hbar}. \]

In the case (2.5), when

\[ V=\xi_0 Q\sin\omega t, \]

we have:

\[ \left. \begin{gathered} a_k^{(1)} = -\frac{\xi_0 Q_{kn}}{2i} \left[ \frac{e^{i(\omega_{kn}-\omega)t}-1}{\hbar(\omega_{kn}-\omega)} + \frac{e^{i(\omega_{kn}+\omega)t}-1}{\hbar(\omega_{kn}+\omega)} \right], \\ Q_{kn}=\int \Psi_k^*Q\Psi_n\,dx,\qquad \omega_{kn}=\frac{E_k-E_n}{\hbar}. \end{gathered} \right\} \tag{2.8} \]

Under the influence of the perturbation the system passes from the initial state \(n\) into other states, primarily into those for which the frequency \(\omega\) is close to \(\pm\omega_{kn}\). In the case of sufficiently dense, practically continuous levels, only transitions to levels for which the resonance condition \(\omega=\pm\omega_{kn}\) is satisfied with high accuracy play a role; moreover, the probability of the transition of the system from the level \(n\) to these levels is proportional to the time \(t\), and the transition probability per unit time is equal to

\[ W_n= \frac{\pi}{2\hbar}\xi_0^2 \left\{ \left|\langle E_n+\hbar\omega|Q|E_n\rangle\right|^2 \rho(E_n+\hbar\omega) + \left|\langle E_n-\hbar\omega|Q|E_n\rangle\right|^2 \rho(E_n-\hbar\omega) \right\}. \tag{2.9} \]

Here \(\rho(E)dE\) is the density of energy levels in the interval \(E, E+dE\), and the matrix elements \(Q_{kn}\) are written in a more convenient form, so that \(\langle E_n\pm\hbar\omega|Q|E_n\rangle\equiv Q_{kn}\) and \(E_k=E_n\pm\hbar\omega\).

In transitions from the state \(E_n\) to the state \(E_k=E_n+\hbar\omega\), the system absorbs a quantum \(\hbar\omega\); in a transition to the state \(E_k=E_n-\hbar\omega\), the system loses the energy \(\hbar\omega\). Therefore the total absorbed power is equal to

\[ P_n(\omega)=\frac{\pi}{2}\mathcal{E}_0^2\omega\left\{ |\langle E_n+\hbar\omega|Q|E_n\rangle|^2\rho(E_n+\hbar\omega)- |\langle E_n-\hbar\omega|Q|E_n\rangle|^2\rho(E_n-\hbar\omega) \right\}. \tag{2,10} \]

A real system, even before the perturbation \(V\) is switched on, at a temperature different from zero is never in any one state \(n\). Therefore, in order to find the power absorbed by the system, it is necessary to average expression (2,10) over the initial states.

Denoting the statistical weight of a state with energy \(E_n\) by \(f(E_n)\), so that \(\sum_n f(E_n)=1\), for the absorbed power we obtain:

\[ \begin{aligned} P(\omega) &=\frac{\pi}{2}\mathcal{E}_0^2\omega\sum_n f(E_n) \left\{ |\langle E_n+\hbar\omega|Q|E_n\rangle|^2\rho(E_n+\hbar\omega) -\right.\\ &\qquad\qquad\qquad\left. |\langle E_n-\hbar\omega|Q|E_n\rangle|^2\rho(E_n-\hbar\omega) \right\}\\ &=\frac{\pi}{2}\mathcal{E}_0^2\omega \int_0^\infty \rho(E)f(E) \left\{ |\langle E+\hbar\omega|Q|E\rangle|^2\rho(E+\hbar\omega) -\right.\\ &\qquad\qquad\qquad\left. |\langle E-\hbar\omega|Q|E\rangle|^2\rho(E-\hbar\omega) \right\}\,dE, \end{aligned} \tag{2,11} \]

where it has been taken into account that in the interval \(dE\) the number of levels is equal to \(\rho(E)\,dE\), and the transition from summation to integration is possible because of the large density of levels.

The result (2,11) is in full agreement with the well-known conclusion of the theory of alternating currents, according to which the average power absorbed by a linear system with impedance \(Z(\omega)\) under the action of the voltage \(\mathcal{E}=\mathcal{E}_0\sin\omega t\) is proportional to \(\mathcal{E}_0^2\) and is equal to

\[ P(\omega)=\frac{\mathcal{E}_0^2}{2}\,\frac{R(\omega)}{|Z(\omega)|^2}, \tag{2,12} \]

where \(R=\operatorname{Re} Z\), and the impedance \(Z\) is determined from the relation

\[ \mathcal{E}=Z(\omega)I \tag{2,13} \]

under the condition that \(\mathcal{E}\) depends on time according to the law \(e^{i\omega t}\). Hence it is clear that the linearity of the system and, consequently, the existence of the impedance \(Z(\omega)\) hold for any dissipative system (a system with dense levels) under the sole condition that the perturbation be sufficiently small to allow one to restrict oneself to the first

approximation of perturbation theory. This conclusion can, of course, also be reached without computing the power absorbed by the system, but by directly establishing the relation between the current and the voltage.

In the classical theory the total current in the conductor under consideration is

\[ I=\sum_k \frac{e_k \dot{x}_k}{l}=\dot{Q}\equiv \frac{dQ}{dt} \]

(for example, if the charges and velocities of all particles are identical and equal to \(e\) and \(\dot{x}\), then \(\sum e_k\dot{x}_k=eN\dot{x}\), where \(N\) is the total number of particles in a wire of length \(l\); the current strength is then equal to \(en\dot{x}\), where \(n=\frac{N}{l}\) is the number of particles per unit length). In the quantum theory the current strength is given by the same expression, but \(\dot{x}_k\) must be understood as the mean velocity

\[ \frac{d\bar{x}_k}{dt} = \frac{d}{dt}\int \Psi^* x_k \Psi\, dx = \frac{i}{\hbar}\int \Psi^*(H_0x_k-x_kH_0)\Psi\, dx = \int \Psi^* \frac{p_k}{m_k}\Psi\, dx, \]

where \(m_k\) is the mass and \(p_k\) the momentum of the \(k\)-th particle. If, before the perturbation was switched on, the system was in the state \(\Psi_n\), in which the current is zero, then after the perturbation is switched on, in the first approximation of perturbation theory

\[ \Psi=\Psi_n+\Psi^{(1)} \]

and the current is

\[ I=\dot{Q}=\frac{d}{dt}\int \Psi^* Q\Psi\, dx = \frac{d}{dt}\int\left(\Psi_n^*Q\Psi^{(1)}+\Psi^{*(1)}Q\Psi_n\right)\, dx, \]

where

\[ \Psi^{(1)}=\sum_k a_k^{(1)}\Psi_k e^{\frac{i}{\hbar}E_kt} \]

[see (2.6) and (2.8)]. Since the coefficients \(a_k^{(1)}\), according to (2.8), are proportional to \(\mathcal{E}_0\), the same is true of the current \(I\). In complex notation this proportionality can be written in the form (2.13), which is what we wanted to show*). It is not necessary to write out explicitly the cumbersome general expression for \(Z(\omega)\) here, since it will not be encountered below. As for the quantity entering into all expressions

*) For simplicity, the expression for the current \(I\) has been written under the assumption that at \(t=0\) the system is in the state described by the function \(\Psi_n\). But the conclusion about the existence of impedance is, of course, very general and remains valid when the system in the initial state is described not by a single function \(\Psi\), but by a certain statistical matrix (see 1).

of the quantity \(\dfrac{R(\omega)}{|Z(\omega)|^2}\), then, comparing (2,11) and (2,12), we obtain:

\[ \frac{R(\omega)}{|Z(\omega)|^2} = \pi\omega \int_0^\infty \rho(E) f(E) \left\{ |\langle E+\hbar\omega|Q|E\rangle|^2 \rho(E+\hbar\omega) - |\langle E-\hbar\omega|Q|E\rangle|^2 \rho(E-\hbar\omega) \right\} \, dE . \tag{2,14} \]

If the external field acting on the system is not sinusoidal, then all the results presented remain valid for the Fourier components of the quantities \(\mathcal E\) and \(\dot Q=I\). In this case, instead of the quantity \(\frac{1}{2}\mathcal E_0^2\), there appears the spectral density of the voltage, \(w(\omega)\),

\[ \left[ \text{if } \mathcal E(t)=\mathcal E_0 \sin \omega t,\ \text{then }\ w(\omega')=\frac{\mathcal E_0^2}{2}\delta(\omega-\omega') \right], \]

and the total absorbed power is equal to

\[ P=\int \frac{R(\omega) w(\omega)}{|Z|^2}\, d\omega, \]

where the quantity \(\dfrac{R(\omega)}{|Z(\omega)|^2}\) is determined by formula (2,14).

The possibility thus established of introducing the concept of impedance within the framework of quantum theory is, to a certain extent, trivial, since the quantum approach is the most general one; and if it is known from experiment that, for a given system, the relation (2,13) holds, then there is no reason to doubt the fundamental possibility of obtaining this relation also as a result of a detailed quantum-mechanical analysis of the behavior of the system. Nevertheless, the discussion presented is useful for a better understanding of the whole question and, in particular, from the point of view of clarifying the conditions of applicability of the concept of a linear dissipative system.

In view of what has been said, it is clear that the important theorem (1,6), in the derivation of which no specific properties of the impedance were used, remains valid also in the quantum domain. Hence it further follows that, in order to obtain the quantum Nyquist formula, one may consider any, very simple, “test” linear dissipative system. As such a system, as in the classical case, we shall choose an oscillatory circuit (oscillator) with arbitrarily small damping.

In the absence of damping, the Hamiltonian operator for an oscillator acted upon by a coordinate-independent force \(F(t)\) has the form

\[ H_0=\frac{p^2}{2m}+\frac{kx^2}{2}-F(t)x, \]

where \(p=-i\hbar\dfrac{\partial}{\partial x}\). Further, for the mean value

\[ x=\int \Psi^* x \Psi\, dx \]

for a system with any potential energy \(V(x)\), the following relation holds:

equation

\[ m\ddot{x}=-\frac{\partial V(x)}{\partial x} \]

(this statement is sometimes called Ehrenfest’s theorem). In the case when the function \(V(x)\) is quadratic with respect to \(x\), it is obvious that

\[ \frac{\partial \bar V}{\partial x} = \frac{\partial V(\bar x)}{\partial x} \equiv \left[\frac{\partial V(x)}{\partial x}\right]_{x=\bar x} \]

and for \(\bar x\) the classical equation of motion holds, i.e. in our case the equation \(m\ddot{x}+kx=F(t)\). Hence, if \(F=F_0 e^{i\omega t}\), then

\[ \dot{x}= \frac{F}{i\left(\omega m-\dfrac{k}{\omega}\right)}, \]

and since by definition \(F=Z(\omega)\dot{x}\), then for the impedance we obtain the expression

\[ Z(\omega)=i\left(\omega m-\frac{k}{\omega}\right). \]

In the electrical case the role of \(F\) is played by the voltage \(\mathcal E\), and one must replace \(m\) by \(L\) and \(k\) by \(\dfrac{1}{C}\), i.e.

\[ Z(\omega)=i\left(\omega L-\frac{1}{\omega C}\right). \]

The case of the mechanical oscillator was considered first because here everything is especially transparent. In passing to the electrical circuit, if one does not confine oneself only to the customary replacement of \(m\) and \(k\) by \(L\) and \(\dfrac{1}{C}\), it is also necessary to trace how, for the energy in the circuit, one obtains the expression

\[ H_0=\frac{p^2}{2L}+\frac{q^2}{2C}-q\mathcal E . \]

However, in this point there is nothing specifically quantum, and it would be inexpedient to dwell on it here*).

Thus, in the absence of damping the impedance of the oscillator and of the oscillatory circuit has in quantum theory exactly the same expression as in the classical theory (this circumstance is widely used in the theory of dispersion). In the presence of arbitrarily weak damping the imaginary part of \(Z\) may be regarded as unchanged, while the real part is some function \(R(\omega)\), i.e.

\[ Z(\omega)=R(\omega)+i\left(\omega L-\frac{1}{\omega C}\right). \]

But precisely such an expression was used in (1.8) in deriving Nyquist’s formula, and, as indicated at the beginning of this section, to obtain the quantum formula (2.3) the expression (1.8) must be

*) Let us note that under conditions when the entire charge may be regarded as concentrated on the plates of the capacitor to which the voltage \(\mathcal E\) is applied, the quantity introduced above

\[ Q=\frac{\sum e_k x_k}{l} \]

is just equal to the charge of one of the plates of the capacitor \(q\), since in this case \(l\) is the distance between the plates of the capacitor and \(\sum e_k x_k=q(x_2-x_1)\), where \(q\) is the sum of \(|e_k|\), taken over one plate, while \(x_2\) and \(x_1\) are the coordinates of the plates charged, respectively, with positive and negative charges.

assign the quantum expression (2.2) for the mean potential energy of the oscillator (circuit).

The totality of the considerations set forth may be regarded as a quite rigorous justification of Nyquist’s quantum formula (2.3). It is interesting that this formula was recently derived in another way, without using the auxiliary theorem (1.6). The corresponding proof, due to Callen and Welton\(^8\), also appears quite convincing and, moreover, valuable from the methodological point of view. It is therefore given at the end of the article (see the supplement).

Let us proceed to a discussion of some results that can be obtained on the basis of Nyquist’s quantum formula. From the classical Nyquist formula (1.9) it follows that the mean square of the fluctuational e.m.f. \(\mathcal{E}^2\) is equal to

\[ \mathcal{E}^2=\int_{0}^{\infty} w(\omega)\,d\omega = \frac{2}{\pi}\,kT\int_{0}^{\infty} R(\omega)\,d\omega . \tag{2.15} \]

If \(R=\mathrm{const}\), then \(\mathcal{E}^2\to\infty\). This result is quite understandable, since, as indicated in Section 1, the assumption of constancy of \(R\) is equivalent to the assertion that the voltage impulses are instantaneous; and in the spectrum of instantaneous impulses arbitrarily high frequencies are present with the same weight as low frequencies. In fact, of course, at sufficiently high frequencies no impulses can be regarded as instantaneous; for example, if the question concerns impacts of molecules on the load of a pendulum, then the duration of the corresponding impulses is

\[ \Delta\tau \simeq \frac{a}{v}, \quad \text{where } \quad v \simeq \sqrt{\frac{kT}{m}} \]

is the thermal velocity of the molecules and \(a\simeq 10^{-8}\) is the size of a molecule (for air molecules at \(T\simeq 300^\circ\mathrm{K}\), \(\Delta\tau\simeq 10^{-13}\) sec). Hence it is clear that for high frequencies \(\omega \gtrsim \Delta\tau\) the resistance \(R\) will necessarily depend on the frequency, and as a result the integral in (2.15) will converge\(^*\). In this case the use of expression (2.15) is lawful only when, for the maximum frequency essential for the value of the integral

\[ \int_{0}^{\infty} R(\omega)\,d\omega, \]

the condition of classicality (2.1) is still satisfied. Otherwise, for the calculation one must use the quantum formula (2.3), which ensures the finiteness of the temperature-dependent

\(^*\) We shall disregard, here and below, other factors that may hinder the use of all the formulas obtained at high frequencies (first of all, it must be remembered that above we everywhere assumed the condition of quasistationarity to be fulfilled, which is not valid for sufficiently high frequencies).

temperature part \(\overline{\mathcal{E}^{2}}\) even when \(R=\mathrm{const}\). According to (2,3)

\[ \begin{aligned} \overline{\mathcal{E}^{2}} &=\overline{\mathcal{E}^{2}(0)}+\overline{\mathcal{E}^{2}(T)} =\frac{2}{\pi}\int_{0}^{\infty} R(\omega) \left\{ \frac{\hbar\omega}{2} + \frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1} \right\}\,d\omega,\\ \overline{\mathcal{E}^{2}(T)} &=\frac{2}{\pi}\int_{0}^{\infty} \frac{R(\omega)\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}\,d\omega . \end{aligned} \tag{2,16} \]

The first part of \(\overline{\mathcal{E}^{2}}\), i.e. the quantity

\[ \overline{\mathcal{E}^{2}(0)} = \frac{\hbar}{\pi}\int_{0}^{\infty} R(\omega)\,\omega\,d\omega, \]

is purely quantum in character (it is a consequence of the existence of zero-point oscillations), does not depend on temperature, and therefore is not a source of “thermal noise,” with which one is usually concerned in the study of electrical fluctuations. As for the mean square of the thermal fluctuational e.m.f. \(\mathcal{E}^{2}(T)\), then, for \(R=\mathrm{const}\),

\[ \overline{\mathcal{E}^{2}(T)} = \frac{\pi R}{3\hbar}(kT)^{2}, \qquad \text{since } \int_{0}^{\infty}\frac{x\,dx}{e^{x}-1}=\frac{\pi^{2}}{6}. \]

In a circuit consisting only of two conductors, each of which has a frequency-independent resistance \(R\), the “fluctuational power” delivered by one conductor is equal to

\[ P=\int_{0}^{\infty}\frac{Rw\,d\omega}{|Z|^{2}} = \frac{1}{4R}\int_{0}^{\infty}w\,d\omega = \frac{\overline{\mathcal{E}^{2}(T)}}{4R} = \frac{\pi}{12\hbar}(kT)^{2} = 4.7\cdot 10^{-6}T^{2}; \]

for \(T\simeq 300^\circ K\), \(P\simeq 0.5\) erg.

Let us now write down the quantum expressions for the mean electric and magnetic energies in the circuit [see (1,8) and (2,3)]:

\[ \begin{aligned} \overline{U} &= \frac{\overline{q^{2}}}{2C} = \frac{1}{2}\int_{0}^{\infty} \frac{CR f(\omega,T)\,d\omega} {R^{2}C^{2}\omega^{2}+(LC\omega^{2}-1)^{2}} \\ &= \frac{1}{\pi}\int_{0}^{\infty} \frac{ a\left\{ \frac{\hbar\eta}{2\sqrt{LC}} + \frac{\frac{\hbar\eta}{\sqrt{LC}}} {\exp\!\left(\frac{\hbar\eta}{\sqrt{LC}\,kT}\right)-1} \right\} } {a^{2}\eta^{2}+(\eta^{2}-1)^{2}} \,d\eta, \\[1ex] \overline{K} &= \frac{L\overline{I^{2}}}{2} = \frac{1}{2}\int_{0}^{\infty} \frac{CRLC\omega^{2} f(\omega,T)\,d\omega} {R^{2}C^{2}\omega^{2}+(LC\omega^{2}-1)^{2}} \\ &= \frac{1}{\pi}\int_{0}^{\infty} \frac{ a\eta^{2}\left\{ \frac{\hbar\eta}{2\sqrt{LC}} + \frac{\frac{\hbar\eta}{\sqrt{LC}}} {\exp\!\left(\frac{\hbar\eta}{\sqrt{LC}\,kT}\right)-1} \right\} } {a^{2}\eta^{2}+(\eta^{2}-1)^{2}} \,d\eta . \end{aligned} \tag{2,17} \]

\[ \alpha=\frac{CR}{\sqrt{LC}},\quad \eta=\sqrt{LC}\,\omega,\quad \overline U=\overline U(0)+\overline U(T);\quad \overline K=\overline K(0)+\overline K(T). \]

If the circuit is sharp, i.e. condition (1.14) is fulfilled,

\[ \frac{R(\omega_0)}{L}\ll \omega_0=\frac{1}{\sqrt{LC}}, \]

then for \(\overline U\) and \(\overline K\) one obtains expression (2.2) for the mean energy of a harmonic oscillator—this circumstance was used in the very derivation of Nyquist’s quantum formula (2.3). Such a weakly damped circuit is classical if

\[ \hbar\omega_0=\frac{\hbar}{\sqrt{LC}}\ll kT. \tag{2.18} \]

Under this condition

\[ \overline U=\overline K=\frac{kT}{2}. \]

For nonvanishing damping, any concrete conclusions from formula (2.17) can, of course, be drawn only by specifying the function \(R(\omega)\) (\(L\) and \(C\) are always regarded as constants). Therefore below only the most important and at the same time the simplest case will be considered, when \(R=\mathrm{const}\), i.e. all parameters of the circuit are independent of frequency.

The temperature-independent parts of \(\overline U\) and \(\overline K\), i.e. the quantities \(\overline U(0)\) and \(\overline K(0)\), can be calculated explicitly for arbitrary parameter values. Thus, for \(\alpha<2\),

\[ \overline U(0)=\frac{\hbar}{2\pi\sqrt{LC}}\int_0^\infty \frac{\alpha\eta\,d\eta}{\alpha^2\eta^2+(\eta^2-1)^2} = \]

\[ =\frac{\hbar}{2\pi\sqrt{LC}}\cdot \frac{1}{\sqrt{4-\alpha^2}} \left[ \frac{\pi}{2}-\operatorname{arctg} \frac{\alpha^2-2}{\alpha\sqrt{4-\alpha^2}} \right] \]

and for \(\alpha=2\)

\[ \overline U(0)=\frac{\hbar}{2\pi\sqrt{LC}}. \]

The value of \(\overline K(0)\) for \(\alpha\ne0\) and \(R=\mathrm{const}\) is infinite, since

\[ \overline K(0)= \frac{\hbar\alpha}{4\pi\sqrt{LC}} \left[ \ln\left|\eta^4+(\alpha^2-2)\eta^2+1\right| \right]_0^\infty +\frac{2-\alpha^2}{2}\,\overline U(0). \]

Taking into account the dependence of \(R\) on \(\omega\) at high frequencies will, of course, lead to the finiteness of \(\overline K(0)\). For

\[ \alpha=\frac{CR}{\sqrt{LC}}\to0, \]

in agreement with (2.2),

\[ \overline U(0)=\overline K(0)=\frac{\hbar}{4\sqrt{LC}}=\frac{\hbar\omega_0}{4}.\,^{*}) \]

For \(\alpha\ne0\) the energies \(\overline U(0)\) and \(\overline K(0)\)

\[ ^{*})\ \text{If, for example, one assumes that } R(\omega)=R \text{ for } \omega<\omega_m=\frac{\eta_m}{\sqrt{LC}} \text{ and} \]

\[ R=0 \text{ for } \omega>\omega_m,\text{ then } \overline K(0)=\frac{\hbar\alpha}{4\pi\sqrt{LC}} \ln\left[ \eta_m^4+(\alpha^2-2)\eta_m^2+1 \right] + \]

\[ +\frac{2-\alpha^2}{2}\,\overline U(0), \]

where the value of the first part of \(\overline K(0)\) is determined by high frequencies; as \(\alpha\to0\) this part tends to zero.

are not equal to each other and depend on both parameters \(\alpha = \dfrac{CR}{\sqrt{LC}}\) and \(\dfrac{\hbar}{\sqrt{LC}}\). An analogous situation also occurs for the parts \(\overline{U}\) and \(\overline{K}\), which are of principal interest and depend on temperature, i.e. for the quantities \(\overline{U}(T)\) and \(\overline{K}(T)\):

\[ \left. \begin{aligned} \overline{U}(T) &= \frac{kT}{\pi}\int_{0}^{\infty} \frac{\alpha\beta\eta\,d\eta} {\left(e^{\beta\eta}-1\right)\left[\alpha^{2}\eta^{2}+(\eta^{2}-1)^{2}\right]} \\ &= T\Phi_{u}\left(\sqrt{LC}\,T,\frac{CR}{\sqrt{LC}}\right), \\[1em] \overline{K}(T) &= \frac{kT}{\pi}\int_{0}^{\infty} \frac{\alpha\beta\eta^{3}\,d\eta} {\left(e^{\beta\eta}-1\right)\left[\alpha^{2}\eta^{2}+(\eta^{2}-1)^{2}\right]} \\ &= T\Phi_{k}\left(\sqrt{LC}\,T,\frac{CR}{\sqrt{LC}}\right), \\[1em] \alpha &= \frac{CR}{\sqrt{LC}},\qquad \beta = \frac{\hbar}{kT\sqrt{LC}} . \end{aligned} \right\} \tag{2,19} \]

For arbitrary \(\alpha\) and \(\beta\) it is not possible to express the functions \(\Phi_u\) and \(\Phi_k\) otherwise than through integrals. If the damping is weak, so that

\[ \alpha=\frac{CR}{\sqrt{LC}}\ll 1 \tag{2,20} \]

[this condition is equivalent to condition (1,14): \(\dfrac{R}{L}\ll\dfrac{1}{\sqrt{LC}}\)], then, as is clear from what was said above,

\[ \Phi_u=\Phi_k=\Phi\left(\sqrt{LC}\,T\right) =\frac{k}{2}\cdot \frac{\dfrac{\hbar}{\sqrt{LC}\,kT}} {e^{\dfrac{\hbar}{\sqrt{LC}\,kT}}-1}. \tag{2,21} \]

Under condition (2,20), and also putting \(\beta=\dfrac{\hbar}{\sqrt{LC}\,kT}\ll 1\), for calculating the integrals (2,19) one may use the expansion

\[ \frac{\beta\eta}{e^{\beta\eta}-1} = 1-\frac{\beta\eta}{2}+\frac{(\beta\eta)^2}{12}-\cdots . \]

As a result, with accuracy up to

terms of order \(\beta\), for example for \(\overline{U(T)}\), we have:

\[ \overline{U(T)} = \frac{kT}{2}\left\{1-\frac{\hbar}{\pi\sqrt{LC}\,kT}\, \frac{1}{\sqrt{4-\alpha^2}}\left[ \frac{\pi}{2}-\operatorname{arctg} \frac{\alpha^2-2}{\alpha\sqrt{4-\alpha^2}} \right]\right\} = \]

\[ = \frac{kT}{2}\left\{1-\frac{\hbar}{2\sqrt{LC}\,kT} -\frac{\hbar R}{2\pi L\,kT}-\cdots\right\}. \tag{2,22} \]

When the damping is strong, i.e.

\[ \alpha=\frac{CR}{\sqrt{LC}}\gg 1, \tag{2,23} \]

the relation between the quantities \(\alpha\) and \(\beta\) is also essential. In this case we have (in all cases \(\alpha\gg 1\)):

\[ \frac{\hbar}{RC}\ll kT\;(\beta\ll\alpha): \qquad \overline{U(T)}=\frac{kT}{2}, \tag{2,24a} \]

\[ \frac{\hbar}{RC}\gg kT\;(\beta\gg\alpha): \qquad \overline{U(T)}=\frac{\pi}{6}\left(\frac{kT}{\dfrac{\hbar}{RC}}\right)kT, \tag{2,24b} \]

\[ \frac{\hbar R}{L}\ll kT\;\left(\beta\ll\frac{1}{\alpha}\right): \qquad \overline{K(T)}=\frac{kT}{2}, \tag{2,25a} \]

\[ \frac{\hbar R}{L}\gg kT\;\left(\beta\gg\frac{1}{\alpha}\right): \qquad \overline{K(T)}=\frac{\pi}{6}\left(\frac{kT}{\dfrac{\hbar R}{L}}\right)kT. \tag{2,25b} \]

The meaning of the inequalities given is clear from the fact that, when quantum effects are neglected, the characteristic frequencies determining the values of the integrals in the expressions for \(\overline{U(T)}\) and \(\overline{K(T)}\), for \(\alpha\gg 1\), are respectively

\[ \omega_U=\frac{\eta_U}{\sqrt{LC}}=\frac{1}{RC} \quad\text{and}\quad \omega_K=\frac{\eta_K}{\sqrt{LC}}=\frac{R}{L} \]

(or \(\eta_U=\frac{1}{\alpha}\) and \(\eta_K=\alpha\)); the frequencies \(\omega_U\) and \(\omega_K\) are at the same time the frequencies at which the frictional force in the circuit \(R\dot q\) becomes comparable respectively with the elastic force \(\frac{q}{C}\) or the inertial term \(L\ddot q\). Thus, for example, the condition \(\frac{\hbar}{RC}\ll kT\) [see (2,24a)] is the condition for the classicality of the circuit from the standpoint of calculating the electric energy \(\overline{U(T)}\). This condition does not coincide with the condition of classicality for the magnetic energy \(\overline{K(T)}\), which for \(\alpha\gg 1\) is determined by oscillations with the higher frequency

\[ \omega_K=\frac{R}{L}\gg \omega_U=\frac{1}{RC}. \]

It is interesting that even in the case when, for small \(R\), the circuit is completely nonclassical (i.e. \(\hbar\omega_0=\frac{\hbar}{\sqrt{LC}}\gg kT\)), with a sufficient increase of \(R\) and unchanged \(L\) and \(C\), the mean electric

energy \(\overline{U(T)}\) tends to the classical limit \(kT/2\), while the magnetic energy \(\overline{K(T)}\) tends to zero. And, in general, let us emphasize once again that the mean electric (potential) and magnetic (kinetic) energies in the circuit are not equal to each other and depend substantially on the value of the resistance \(R\), even if this resistance is the same for all frequencies. The dependence of \(\overline{U(T)}\) and \(\overline{K(T)}\) on \(R\) disappears only for \(R \to 0\), or in the classical limiting case when \(R=\operatorname{const}\).

The reason for this situation is that in the quantum case, in contrast to the classical one, even for \(R=\operatorname{const}\) one cannot neglect the change in the energy of the subsystem associated with its damping. This conclusion follows directly from the uncertainty relation for energy, according to which the uncertainty in the energy of the subsystem \(\Delta E \simeq \hbar/\tau\), where \(\tau\) is the damping time (see [7], § 44). A manifestation of this general relation is the fact that, in the presence of damping, the subsystem has no discrete levels and one can speak only of broadened quasistationary levels, whose width is smaller than the distance between levels only for weak damping [in the case of a circuit, this small-damping condition coincides with condition (2,20)]. If, however, the damping is so strong that the width of the levels is greater than the distance between the levels of the unperturbed system, then it is quite impossible to regard the latter as unchanged. For example, the subsystem obtained by introducing strong damping into an oscillator differs radically from a quantum oscillator, whose energy levels are discrete and are determined by the familiar expression \(E_n=\hbar\omega_0\left(n+\dfrac{1}{2}\right)\). The impossibility, under strong damping, of neglecting the change in the energy of the subsystem does not allow one to calculate its mean energy by the formulas of statistics. This also explains the fact that in the quantum case the mean energies \(\overline{U}\) and \(\overline{K}\) for \(R\ne0\) differ from the statistical expressions (2,2), valid for the mean potential and kinetic energies of an oscillator with negligibly small damping.

3. ON THE APPLICATION OF THERMODYNAMICS TO ELECTRICAL FLUCTUATIONS

The results set forth above make it possible to make several remarks concerning the question, recently raised by G. S. Gorelik[^3], of “what can be said about electrical fluctuations on the basis of phenomenological thermodynamics alone, without using statistical considerations.” The legitimacy of such a formulation of the question becomes clear if one takes into account

the known conditionality of the delimitation of phenomena into fluctuational and nonfluctuational ones (see³ § 2 and⁹ § 34). For example, thermal radiation is usually not regarded as a fluctuation phenomenon, although in principle it differs in no way from thermal fluctuations of electromagnetic energy in an electric circuit. Hence it is clear that on the basis of thermodynamics one can indeed draw definite conclusions about the mean energies in a circuit

\[ \overline{U}=\frac{\overline{q^2}}{2C} \]

and

\[ K=\frac{\overline{LI^2}}{2}, \]

just as the well-known thermodynamic Wien equation characterizes the mean spectral density of thermal radiation \(u_\omega\):

\[ u_\omega(T,\omega)=T\Phi\left(\frac{T}{\omega}\right)\omega^2, \tag{3,1} \]

where \(\Phi\) is some unknown function of the ratio \(T/\omega\). Equation (3,1) is obtained thermodynamically by taking into account the Doppler formula for the change in frequency of light reflected from a moving mirror and the relation valid for isotropic radiation,

\[ p=\frac{u}{3}, \]

where \(p\) is the light pressure and

\[ u=\int_0^\infty u_\omega\,d\omega \]

is the total energy density of the radiation*). Both the Doppler formula and the expression

\[ p=\frac{u}{3} \]

are of a very general character—they are obtained both in classical electromagnetic theory and from the standpoint of the concept of light quanta.

Wien’s relation (3,1), which reduces the problem of finding the function \(u_\omega(\omega,T)\) of two variables to the problem of finding a function \(\Phi\) of one variable, played an essential role in the development of the theory of thermal radiation. At the present time, however, there is no special reason to resort to Wien’s equation, since it is simpler to establish at once the complete expression for \(u_\omega\), i.e. Planck’s formula,

\[ u_\omega=\frac{\hbar\omega^3}{\pi^2 c^3\left(e^{\hbar\omega/kT}-1\right)}. \]

Thermal radiation in a certain cavity may be regarded as an ensemble of “field oscillators” with mean energy

*) For a derivation of formula (3,1), see, for example, in¹⁰. This formula is more often written in the equivalent to (3,1) form:

\[ u_\omega=\omega^3\Phi'\left(\frac{\omega}{T}\right), \]

where \(\Phi'\) is an unknown function.

\[ \overline E=\frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}, \]

moreover the number of oscillators in the frequency interval \(d\omega\) is equal to \(\dfrac{\omega^2\,d\omega}{\pi^2 c^3}\). Hence it follows that, as applied to a single oscillator (circuit), thermodynamics must, instead of expression (3,1), lead to the relation

\[ \overline E=T\Phi\left(\frac{T}{\omega}\right), \]

where \(\omega\) is the natural frequency of the oscillator*). At the same time, since we are relying on the example of undamped “field oscillators,” the discussion can concern, at least if one argues by analogy, only an undamped or sufficiently weakly damped oscillator. But for an undamped oscillator the purely mechanical virial theorem is valid, according to which the time-averaged potential and kinetic energies \(\overline U\) and \(\overline K\) are equal to one another. As a result of what has been said we come to the conclusion that, in the case of a sufficiently weakly damped electric circuit with frequency \(\omega_0=\dfrac{1}{\sqrt{LC}}\), the relation equivalent to Wien’s equation (3,1) must have the form:

\[ \overline U=\overline K=T\Phi\left(\sqrt{LC}\,T\right). \tag{3,2} \]

As is shown below, expression (3,2) for a weakly damped electric circuit is obtained directly as well, without resorting to analogy with Wien’s equation. However, in his article G. S. Gorelik did not aim to obtain, for electric fluctuations, an analogue of Wien’s equation, but tried in essence to consider thermodynamically the more general problem of fluctuations in a circuit with any resistance. Moreover, in\(^3\) only the limiting case of an \(RC\)-circuit is analyzed, i.e. a circuit consisting of a capacitor and a resistance, but without self-inductance. For the reader’s convenience we shall briefly present the corresponding arguments of G. S. Gorelik\(^3\).

The mean energy of an \(RC\)-circuit is equal to \(\overline U=\dfrac{\overline{q^2}}{2C}\); the work of the electric forces when the capacitance of the capacitor is changed by \(dC\) is equal to

\[ \overline{\delta A} = -\frac{\partial}{\partial C}\left(\frac{\overline{q^2}}{2C}\right)dC = \frac{\overline{q^2}}{2C^2}\,dC, \]

* ) At first glance any possibility of obtaining thermodynamic results for a single oscillator, i.e. a subsystem with one degree of freedom, may seem strange. The point, however, is that thermodynamics is applied to an ideal gas consisting of oscillators, and the energy of such a gas is equal to the sum of the energies of the individual oscillators. Therefore, in the final analysis, thermodynamic conclusions can also be obtained about the properties of one oscillator, just as is done in the text.

and thus the thermodynamic equality expressing the second law of thermodynamics has the form:

\[ T\,dS=d\overline{U}+\delta\overline{A} =d\left(\frac{\overline{q^2}}{2C}\right) +\frac{\overline{q^2}}{2C^2}\,dC. \tag{3,3} \]

Hence, putting \(\overline{q^2}=\varphi(T,C)\) and equating to one another the derivatives \(\dfrac{\partial^2 S}{\partial T\partial C}\) and \(\dfrac{\partial^2 S}{\partial C\partial T}\), we obtain for \(\varphi\) the equation

\[ T\,\frac{\partial \varphi}{\partial T} = C\,\frac{\partial \varphi}{\partial C}, \]

from which it follows that \(\varphi(T,C)=\varphi(TC)\), and

\[ \overline{U}=\frac{\overline{q^2}}{2C}=T\Phi(TC), \tag{3,4} \]

where \(\Phi=\dfrac{\varphi}{2CT}\) is an unknown function of the product \(CT\). The result (3,4) remains unchanged if one assumes that the energy in the circuit is equal not to \(\dfrac{\overline{q^2}}{2C}\), but to \(\dfrac{\overline{q^2}}{2C}+F(T)\), where \(F\) is an unknown function of \(T\). Since, when the resistance is varied, the system performs no work, the quantities \(\overline{q^2}\) and \(F\) cannot depend on \(R\). This conclusion is general for any system and follows immediately from (3,3), since, for example, in an adiabatic process, if for \(dR\ne0\) one has \(\delta A=0\), then \(\dfrac{\partial U}{\partial R}=0\), and consequently the energy does not depend on \(R\).

Let us now turn to the expression of the energy \(\dfrac{\overline{q^2}}{2C}\) in terms of the circuit parameters and the spectral density \(w(\omega)\) in the state of equilibrium, equal to \(w(\omega)=R(\omega)f(T,\omega)\). Then for an \(RC\)-circuit \((L=0)\), taking into account (1,8) and (3,4), we obtain:

\[ \overline{U}=\frac{\overline{q^2}}{2C} = T\Phi(TC) = \frac{1}{2}\int_{0}^{\infty} \frac{CR\,f(T,\omega)\,d\omega}{R^2C^2\omega^2+1}. \tag{3,5} \]

In the case of a resistance \(R\) independent of frequency, we see that the right-hand side of equality (3,5) depends on \(T\) and the parameter \(CR\), whereas the left-hand side depends on \(T\) and \(CT\). Hence it follows that the function \(\Phi(CT)\) is some universal constant \(\dfrac{a}{2}\), i.e.

\[ \frac{\overline{q^2}}{2C}=\frac{a}{2}\,T. \tag{3,6a} \]

In view of (3,6a), the integral in (3,5) must not depend on \(RC\), whence

it follows that*)

\[ f=\frac{2}{\pi}aT,\qquad a=\mathrm{const}, \tag{3,66} \]

i.e. the classical Nyquist formula (1,9) is obtained, since the universal constant \(a\) can be determined for some one circuit and, obviously, will prove equal to Boltzmann’s constant \(k\).

The results (3,6) presented are, in essence, completely paradoxical, since they are purely classical and contradict the quantum Nyquist formula (2,3), whereas in deriving them, it would seem, no classical assumptions were made. Indeed, thermodynamic relations must be equally valid in both the classical and the quantum regions. As for the use of expressions for the energies

\[ \frac{q^2}{2C}=\frac{1}{2}\int \frac{CR f\,d\omega}{C^2R^2\omega^2+1}, \]

they are likewise applicable far beyond the limits of classical theory (see Section 2).

Thus, there is no doubt that the above derivation of formulas (3,6) is not thermodynamic, but contains in hidden form certain far-reaching assertions equivalent to the use of classical statistics. In light of what was said in Sections 1 and 2, it is not difficult to clarify what the matter is here. From the quantum Nyquist formula it is evident that the mean energy in a circuit, generally speaking, depends on the resistance of this circuit \(R\) (see the end of Section 2). Meanwhile, if thermodynamics is applied to any circuit, in the manner in which this was done above for the \(RC\)-circuit, one always obtains expressions for the energies \(\bar U\) and \(\bar K\) that do not depend on the resistance \(R\). The reason for this contradiction is that in the case of a circuit or any system with nonvanishing resistance, thermodynamic relations, generally speaking, cannot be applied. All thermodynamics, as well as statistics, is constructed for quasi-closed systems, for systems whose interaction energy with the thermostat may be neglected. Only under this assumption, as was already emphasized in Section 1,

*) For a rigorous proof of this assertion see in \(^3\). The naturalness of such a result is clear from the fact that, by virtue of (3,6a), relation (3,5) is written in the form

\[ \frac{aT}{2}=\frac{1}{2}\int_0^\infty \frac{f\!\left(T,\frac{\xi}{CR}\right)}{\xi^2+1}\,d\xi, \]

i.e. its right-hand side must not depend on \(CR\), which is a very stringent condition imposed on the form of the function \(f\!\left(T,\frac{\xi}{CR}\right)\). Setting \(f\!\left(T,\frac{\xi}{CR}\right)=f(T)\), we obtain (3,6b). The uniqueness of this solution follows from the theory of integral equations.

the Gibbs canonical distribution is valid, from which the fundamental thermodynamic equality can also be obtained. An analogous situation also holds in the development of thermodynamics independently of statistics: here, too, one cannot proceed without assuming the additivity of energy (see, for example, §§ 7 and 16). At the same time, damping is precisely the expression of the presence of an interaction leading to an exchange of energy between the system and the thermostat, and the damping is regarded as large in the case when the rate of this exchange is comparable with or greater than the rate of energy exchange between the parts of the system itself [see condition (2.22), which means that \(\dfrac{R}{L} \gg \dfrac{1}{\sqrt{LC}}\)]. In our case the system to which thermodynamics is applied is, using the terminology of Section 1, a subsystem consisting of the oscillator, or of the electromagnetic field in the capacitor and the self-inductance coil, but not including the gas surrounding the oscillator or the material of the wires (see Section 1). If the damping of the system is large, then the energy of the entire system (oscillator + gas, the circuit as a whole), generally speaking, is not equal to the sum of the energies of the isolated subsystem and its surroundings, and thermodynamics cannot be applied to the subsystem.

Thermodynamics can, of course, be applied to the system as a whole, assuming it to be placed in some external thermostat with which it interacts weakly. But in this case the use of the thermodynamic equality cannot lead to any concrete results, since the energy of the system is some unknown function of \(T\) and of the parameters characterizing the state of the system, among which may be the parameters \(C\) and \(L\). (The point here is again that, in the presence of a strong interaction of the subsystem with its surroundings, the energy of the whole system is not equal to the sum of the energy of the subsystem \(\dfrac{q^{2}}{2C} + \dfrac{Li^{2}}{2}\) and a function \(f(T)\) independent of the state of the subsystem.)

Thus, irrespective of whether we apply thermodynamics to the subsystem or to the system as a whole, the results (3.6), obtained for a definitively strongly damped circuit (an \(RC\)-circuit), are indeed not purely thermodynamic, but are obtained only under an additional, very strong assumption that the interaction energy is equal to zero, despite the strong damping of the subsystem. In the quantum domain such an assumption is altogether impossible, since it contradicts the uncertainty relation for energy. In classical theory, neglect of the interaction energy is possible when the action of the surrounding medium on the subsystem occurs as a result of instantaneous collisions. But in that case, as is clear from what was said in Section 1, formulas (3.6a) and (3.6b) must indeed be valid. Thereby the paradox is completely resolved.

The paradoxical nature of G. S. Gorelik’s result (3.6b) can also be seen in the fact that, even under the assumption of the instantaneous character of the impacts, the Nyquist formula was obtained by him without the usually used statistical result

\[ \overline{U}=\overline{K}=\frac{kT}{2}. \]

However, as is indicated in the following note, the theorem on the equipartition of energy over degrees of freedom follows from the assumption that the impacts are instantaneous, i.e., that the function \(\mathfrak w\) is independent of \(\omega\). Therefore there is no contradiction on this point either.

Thus, for a damped circuit no purely thermodynamic conclusions, i.e. conclusions independent of the character of the forces of interaction in the system, its classical nature, etc., can be obtained.*)

*) For a damped circuit with a resistance \(R\) independent of frequency, under the assumption that for any value of \(R\) the energy of interaction of the subsystem under consideration with its surroundings may be neglected, or, in other words, that the energy of the whole system may be represented in the form of the sum

\[ \frac{q^2}{2C}+F(T), \]

thermodynamics leads to formulas (3.6). This is precisely what was shown by G. S. Gorelik for an \(RC\)-circuit (see above). As the author has learned, M. L. Levin obtained the same result also for an arbitrary \(RCL\)-circuit (thus, in this respect, neglect of self-induction, generally speaking, as indicated in the following note, is inadmissible in the consideration of thermal fluctuations, and is not essential). However, this conclusion cannot be regarded as thermodynamic in the usual sense of the word. The last is clear, in particular, from the fact that formulas (3.6) can also be reached without using the second law of thermodynamics and at the same time without additional statistical assumptions. Indeed, neglect of the interaction energy in the case of strong damping is possible if the impacts are instantaneous, and in this case the function \(\mathfrak w=Rf\) is independent of frequency (see Section 1). Therefore, since, by assumption, \(R=\mathrm{const}\), from the expressions for \(\overline{K}\) and \(\overline{U}\) [see (1.12)] we obtain:

\[ \overline{K}=\overline{U}=\frac{\pi}{4}f(T), \]

where, by virtue of (1.6), \(f(T,\omega)=f(T)\) is a universal function of \(T\). Thus a result is obtained which is equivalent to the theorem on the equipartition of energy over degrees of freedom, and it remains only to show that if \(T\) is the absolute thermodynamic temperature, then \(f(T)=\mathrm{const}\cdot T\). For this purpose let us consider an ideal monatomic gas, for which, on the one hand, by virtue of the universality of the function \(f\), also \(K=\frac{\pi}{4}f(T)\), and, on the other hand, by the virial theorem the mean energy in the volume \(V\) is equal to

\[ \overline{E}=N\overline{K}=\frac{3}{2}pV, \]

where \(N\) is the number of particles and \(p\) is the pressure. Further, as follows from experiment, for an ideal gas \(pV=\mathrm{const}\cdot T\), and thus, using only the properties of an ideal gas, we arrive at formula (3.6b) \(f=\mathrm{const}\cdot T\), from which formula (3.6a) also follows.

G. S. Gorelik’s derivation of Nyquist’s formula, if extended to an \(RCL\)-circuit, is a method of obtaining the universal function \(f\) on the basis of considering a “test” system (see Section 1) for which \(R=\mathrm{const}\), and for any \(R\) the energies \(\overline{U}\) and \(\overline{K}\) do not depend on \(R\). Since such a “test” system can in fact be realized only in the classical region, with its aid one can obtain only the classical Nyquist formula.

On the contrary, for a sufficiently weakly damped circuit such conclusions can be drawn; namely, one can obtain relation (3.2).

The work \(\overline{\delta A}\) associated with changing the capacitance of the circuit by \(dC\) and its self-inductance by \(dL\) is equal to

\[ \overline{\delta A} = -\frac{\partial}{\partial C}\left(\frac{q^2}{2C}\right)dC + \frac{\partial}{\partial L}\left(\frac{LI^2}{2}\right) = \frac{\overline{q^2}}{2C^2}\,dC - \frac{\overline{I^2}}{2}\,dL, \]

since the potential energy of the electric field is

\[ U=\frac{q^2}{2C}, \]

and the potential function determining the forces of the magnetic field is equal to

\[ -\frac{LI^2}{2}. \]

The energy in the circuit is

\[ \overline{E}=\overline{K}+\overline{U} = \frac{\overline{q^2}}{2C} + \frac{L\overline{I^2}}{2}. \]

Further, for a sufficiently weakly damped circuit, which alone is being considered, since this permits one, without further assumptions, to use thermodynamics, it follows from the virial theorem, which is of a purely dynamical character, that \(\overline{K}=\overline{U}\)*). Making use of this circumstance from the very beginning, for convenience, and taking into account the expressions written above for \(\overline{\delta A}\) and \(\overline{E}\), we may write the thermodynamic equality in the form:

\[ T\,dS=d\overline{E}+\overline{\delta A} =d(L\varphi)+\frac{L}{2C}\varphi\,dC+\frac{\varphi}{2}\,dL, \tag{3.7} \]

where \(\varphi(T,L,C)=\overline{I^2}\).

Equating now the derivatives

\[ \frac{\partial^2 S}{\partial T\,\partial C} \quad\text{and}\quad \frac{\partial^2 S}{\partial C\,\partial T} \]

and so forth, we obtain:

\[ \left. \begin{aligned} &L\frac{\partial \varphi}{\partial L} - C\frac{\partial \varphi}{\partial C} +\varphi=0,\\[4pt] &T\frac{\partial \varphi}{\partial T} - 2L\frac{\partial \varphi}{\partial L} - 3\varphi=0,\\[4pt] &T\frac{\partial \varphi}{\partial T} - 2C\frac{\partial \varphi}{\partial C} - \varphi=0. \end{aligned} \right\} \tag{3.8} \]

*) This relation, for \(R\to0\), also follows immediately from the integral expressions for \(\overline{K}\) and \(\overline{U}\) in terms of the parameters \(L,C,R\) and the function \(f\) [see (2.17)] and holds both in the classical and in the quantum regions. A similar proof for the case of an oscillator is equivalent to proving the equality of \(\overline{K}\) and \(\overline{U}\) on the basis of a more general virial theorem.

Let us also note that, since in the classical region for \(R=\mathrm{const}\)

\[ \overline{K}=\overline{U}=\frac{kT}{2} \]

for any \(R\), in studying electrical fluctuations one generally must never consider an \(RC\)-circuit, i.e. neglect the self-inductance and the magnetic energy. This is explained by the fact that, as is clear from equation (1.10), the self-inductance may be neglected only if the frequency of the applied electromotive force \(\mathcal{E}\) satisfies the inequality

\[ \omega \ll \frac{R}{L}. \]

At the same time, in the case of thermal fluctuations the spectrum of the electromotive force \(\mathcal{E}\) is not known in advance and in fact extends to very high frequencies that do not satisfy the indicated inequality.

The solution of this system is \(\varphi = I^2 = \dfrac{2\overline U}{L}\Phi(\sqrt{\overline{LC}}\,T)\), where \(\Phi\) is an arbitrary function. Hence for \(\overline C=\overline K=\dfrac{LT^2}{2}\) we obtain formula (3.2), as was to be expected. Since formula (3.2) has now been obtained by an independent method, using it it is easy to arrive at Wien’s equation (3.1).

Taking (3.2) into account, the expression, for example, for the potential energy in a weakly damped circuit can be written in the form [see (2.17)]:

\[ \overline U = T\Phi(\sqrt{LC}\,T) = \frac12\int_0^\infty \frac{CRf(\omega,T)\,d\omega} {R^2C^2\omega^2+(LC\omega^2-1)^2}. \]

Hence, just as was done in Sections 1 and 2 in deriving the Nyquist formula, we immediately find that

\[ f(\sqrt{LC},T)=\frac4\pi\,T\Phi(\sqrt{LC}\,T). \]

or, replacing \(\sqrt{LC}\) by \(1/\omega\), we obtain:

\[ f(\omega,T)=\frac4\pi\,T\Phi\left(\frac{T}{\omega}\right). \tag{3.9} \]

This result is, of course, in complete agreement with the quantum Nyquist formula (2.3), from which, just as from comparison of expressions (3.2) and (2.2), it follows that

\[ T\Phi\left(\frac{T}{\omega}\right) = \frac12\left( \frac{\hbar\omega}{2} + \frac{\hbar\omega}{e^{\hbar\omega/kT}-1} \right). \]

It is not possible to obtain, thermodynamically, without introducing additional assumptions, any further results. As for formulas (3.2) and (3.9), they can have only pedagogical value, since the much more far-reaching quantum Nyquist formula (2.3) can be obtained more simply and in fact was obtained without any use of the indicated thermodynamic relations.

4. ON ELECTRICAL FLUCTUATIONS NEAR SECOND-ORDER PHASE-TRANSITION POINTS IN SEIGNETTE ELECTRICS, FERROMAGNETICS, AND SUPERCONDUCTORS

In connection with the questions touched upon above, we also wish to dwell on the features of electrical fluctuations near second-order phase-transition points, also called Curie points. This case is of interest because near Curie points the fluctuations of a certain parameter determining the character of the transition increase strongly and, formally, even tend to infinity.

For definiteness, let us immediately dwell on a concrete case—the phase transition in ferroelectrics. In this case the thermodynamic potential near the Curie points has the form (see \(^{11}\))

\[ \Phi=\Phi_0+\alpha P^2+\frac{\beta}{2}P^4, \tag{4,1} \]

where \(P\) is the electric polarization, \(\Phi_0,\alpha\), and \(\beta\) are functions of temperature and pressure; the electric field is assumed to be absent, and, for simplicity, a crystal with one ferroelectric axis is considered, for example Rochelle salt. At the Curie point (at \(T=\theta\)) the coefficient \(\alpha_\theta=0\), while \(\beta_\theta>0\); near the Curie point one may put \(\alpha=\alpha'_\theta(T-\theta)\) and \(\beta=\beta_\theta\), where \(\alpha'_\theta=\left(\dfrac{d\alpha}{dT}\right)_\theta>0\), and thus for \(T>\theta\) \(\alpha>0\). For \(T>\theta\), in a state of thermodynamic equilibrium \(P=0\), while for \(T<\theta\), when \(\alpha<0\), in equilibrium \(P^2=P_0^2=-\dfrac{\alpha}{\beta}>0\).

As is known from the theory of thermodynamic fluctuations (see \(^{1}\), ch. 12 and \(^{6}\), § 30), the probability of a fluctuation of some quantity \(\eta\) is proportional to \(e^{-\Delta\Phi/kT}\), where \(\Delta\Phi\) is the change of the thermodynamic potential associated with the change of the quantity \(\eta\) by \(\Delta\eta\) in the volume \(\Delta V\) under consideration (we regard the temperature and pressure of the body as constant). In our case \(\eta=P\) and, say, above the Curie point \(\Delta\eta=P\) and \(\Delta\Phi=\alpha P^2\Delta V\) (the term \(\dfrac{\beta P^4}{2}\) may, in any case so long as the fluctuations are not too large, be neglected; the potential \(\Phi\) is referred to unit volume). Hence we find that the mean square of the polarization fluctuation above the Curie point is

\[ \left. \begin{aligned} \overline{P^2} &=\frac{kT}{2\alpha\Delta V} =\frac{kT}{2\alpha'_\theta(T-\theta)\Delta V},\\ &\qquad T>\theta. \end{aligned} \right\} \tag{4,2} \]

On approaching the Curie point (\(T\to\theta\)), if one uses expression (4,2), the quantity \(\overline{P^2}\) increases without bound. The same occurs for any second-order transitions, where the role of \(P^2\) is played by some other parameter, or by several other parameters, having the meaning of the square of the spontaneous magnetization \(M^2\) (ferromagnets), the concentration of superconducting electrons \(n_s\) (superconductors), the density of the superfluid part of the liquid \(\rho_s\) (helium II), the square of the degree of order \(\eta^2\) (ordering alloys), etc.* The presence of large fluctuations \(\overline{P^2}, M^2, n_s\), etc., above the point

* For definiteness we speak only of second-order transitions. In fact, however, all anomalies (although, in general, to a lesser degree) also occur for first-order transitions close to the so-called \(\lambda\)-point or critical Curie point (see \(^{1}\), § 134 and \(^{11}\), § 2).

It is all the more interesting that, in this region, the existence is impossible not only of stable, but even of metastable nuclei corresponding to a relative minimum of the thermodynamic potential—nuclei of the ordered phase (the phase stable below the Curie point); with this is connected the impossibility of superheating in second-order phase transitions. But, as has been said, fluctuational rapidly varying “nuclei” of the ordered phase can appear even above the Curie point and, on approaching it, become significant.

The increase in fluctuations of the characteristic parameter near Curie points is, to a certain extent, analogous to the increase in density fluctuations near the critical point on the liquid–vapor equilibrium diagram. Just as this latter phenomenon leads to critical opalescence, the growth of fluctuations of order near Curie points leads to anomalous scattering of X-rays[^13].

In the case of second-order transitions connected with ordering, the electrical properties of the substance change only indirectly, and from the point of view of electrical fluctuations these transitions are not of special interest. But in the case of ferroelectrics, ferromagnets, and superconductors, the characteristic and strongly fluctuating quantity is already an “electromagnetic” parameter—polarization \(P\), magnetization \(M\), and, connected with the depth of penetration of the magnetic field, the concentration of superconducting electrons \(n_s\) (see[^13]).

How will these fluctuations manifest themselves in experiment? What are the features of electrical fluctuations near Curie points in ferroelectrics, ferromagnets, and superconductors? These naturally arising questions must be answered. To do this at first sight seems very difficult.

In fact, suppose that we want to calculate the fluctuation of the electric moment of some ferroelectric specimen. The fluctuation of the moment of the specimen as a whole is made up of fluctuations of the moment in its separate parts. Far from the Curie point, these latter fluctuations are determined by expression (4.2), but near Curie points formula (4.2) is no longer suitable, and it is necessary first of all to take into account the correlation of fluctuations in different volumes [for this purpose expression (4.1) must be supplemented by the term \(\gamma(\nabla P)^2\); for more detail see[^1,^12]]. Already from this it is clear that to calculate in this way the fluctuation of the electric moment of the specimen as a whole is a very difficult problem.

Fortunately, this problem need not be solved—or, if one likes, it has already been solved.

Indeed, finding the fluctuations of the electric moment of a ferroelectric, or of the magnetic moment of some ferromagnetic specimen, reduces in the final analysis to determining the fluctuations

charges on a capacitor in which there is a ferroelectric, fluctuations of currents in a coil surrounding a ferromagnet, etc. But the fluctuations of \(q^{2}\) and \(I^{2}\), or of the quantity \(w(\omega)\) connected with them, in a state of thermodynamic equilibrium have already been determined, and moreover in a general form, irrespective of the specific properties of the substances contained in the circuit; thus the electrical fluctuations near Curie points, just as in other cases, are determined by the usual Nyquist formula \(w=\dfrac{2}{\pi} R(\omega)kT\) *). Therefore, if fluctuations near Curie points have any anomalies, they are completely characterized by the corresponding anomaly in the behavior of \(R(\omega)=\operatorname{Re} Z\). It follows from this that from measurements of equilibrium electrical fluctuations near Curie points one cannot obtain any interesting data other than the quantity \(R(\omega)\), which is considerably simpler to measure directly.

For completeness, however, let us indicate the behavior of \(R(\omega)\) near Curie points.

If the question concerns a capacitor with a ferroelectric, then the impedance \(Z\) is easily determined by using the equation of motion for the total polarization \(P\) (see \(^{11}\) § 3; we restrict ourselves to the case \(T>\theta\)):

\[ 2\delta \dot P + 2\alpha P = E_0 e^{i\omega t}=E, \tag{4,3} \]

where \(\alpha\) is the same coefficient as in (4,1), \(E\) is the electric-field strength, \(\delta\) is the coefficient determining the losses, and the term with \(\ddot P\) has been neglected, which can be done at sufficiently low frequency. Further, by definition,

\[ P=\frac{\varepsilon'-1}{4\pi}E \approx \frac{\varepsilon'}{4\pi}E \]

(since \(|\varepsilon'|\gg 1\)), where

\[ \varepsilon'=\varepsilon_1-i\varepsilon_2=\varepsilon-i\frac{4\pi\sigma}{\omega} \]

is the complex dielectric constant (\(\varepsilon\) is the real dielectric constant, \(\sigma\) is the effective conductivity). At the same time, from (4,3) it is clear that

\[ P=\frac{E}{2\delta i\omega+2\alpha}, \]

whence

\[ \varepsilon'=\frac{2\pi}{\alpha+i\delta\omega};\qquad \varepsilon_1\equiv \varepsilon=\frac{2\pi\alpha}{\alpha^2+\delta^2\omega^2};\qquad \varepsilon_2\equiv \frac{4\pi\sigma}{\omega}=\frac{2\pi\delta\omega}{\alpha^2+\delta^2\omega^2}; \]

\[ \operatorname{tg}\delta'=\frac{\varepsilon_2}{\varepsilon_1}=\frac{\delta\omega}{\alpha}. \tag{4,4} \]

We see that the quantities \(\varepsilon_1\), \(\varepsilon_2\), and \(\operatorname{tg}\delta'\) near Curie points depend characteristically on temperature, since \(\delta=\mathrm{const}\), while

*) We restrict ourselves here to the classical case. In addition, it is necessary once more to emphasize that Nyquist’s formula characterizes only fluctuations in a state of thermodynamic equilibrium. Possible, in principle, nonequilibrium electrical fluctuations, which need independent consideration and accounting, may play a role. Attempts to detect such nonequilibrium fluctuations under the specific conditions obtaining near phase-transition points would be of interest.

\[ \alpha=\alpha'_0(T-\theta). \]

However, the real part of the impedance \(R(\omega)\) has no “anomalous” behavior at \(T\simeq\theta\), since the impedance of a capacitor with a ferroelectric is

\[ Z=\frac{1}{i\omega C'}=\frac{4\pi d}{i\omega \varepsilon' S} =\frac{2\delta d}{S}-i\frac{2\alpha d}{\omega S},\quad R=\frac{2\delta d}{S}=\text{const.}, \tag{4.5} \]

where \(S\) is the area of the capacitor plate and \(d\) is the distance between the plates \([\,\text{we also arrive at (4.5) directly from (4.3), since }SP=I,\ \text{emf }{\mathcal E}=Ed\text{ and the resistance }R=2\delta d/S,\ \text{i.e. }2\delta\text{ is the specific resistance}\,]\).

In the case of ferromagnets it is natural to consider a toroidal ferromagnetic specimen placed in the same kind of coil. The impedance of this coil is proportional to the complex magnetic permeability \(\mu'\) of the specimen. The value of \(\mu'\), in turn, is determined by equation (4.3), where \(P\) must be replaced by the magnetization \(M\), and \(E\) by the magnetic field \(H\). As a result, for \(\mu'=\mu_1-i\mu_2\) one obtains formulas analogous to (4.4), while \(\operatorname{Re} Z=R(\omega)\) is proportional to

\[ \mu_2=\frac{2\pi\delta\omega}{\alpha^2+\delta^2\omega^2} \]

and depends on \(T\), this dependence being capable of being very sharp for small \(\delta\).

In the case of superconductors the characteristic parameter—the concentration of superconducting electrons \(n_s\)—determines the penetration depth of the magnetic field into the superconductor and thereby the self-inductance of a coil on a superconducting core. However, measurement of the noise in this coil is not of particular interest, especially since these noises must be very small both because of the smallness of \(T\) and because of the insignificance of losses in superconductors at low frequencies.

Nevertheless, in the case of superconductors the considerations presented concerning fluctuations near Curie points are of interest from an entirely different standpoint. The point is that, in the theoretical analysis of the behavior of superconductors both in a constant and in a high-frequency field, it has always until now been assumed explicitly or implicitly (see \(^{13,14}\)) that above the transition point (in the normal state) the properties of the metal do not differ in any way from the properties of nonsuperconductors. This assumption, which is immaterial for the questions usually considered, is supported by the fact that there is no anomaly of the electrical conductivity in the region above the transition point and, in the case of a constant field, raises no doubts, since the superconducting state cannot exist, even as a metastable (superheated) one, for \(T>T_k\) (\(T_k=T_0\) is the transition point to the superconducting state in the absence of a magnetic field). But fluctuations of the quantity \(n_s\) must be present also above \(T_k\), and thus, as it were, anticipate the superconducting transition. The presence of these fluctuations must somehow affect the properties of superconduc-

of conducting metals above, but near the transition point \(T_k\). First of all, in this respect it is necessary to clarify the influence of fluctuations of \(n_s\) on the complex dielectric constant of the metal \(\varepsilon'\), which for \(T>T_k\) is usually taken to be the same as in nonsuperconducting metals. Of course, in a constant field one should not expect the rapidly varying fluctuations of \(n_s\) to affect \(\varepsilon'\), but in a high-frequency field they could manifest themselves. At the same time it is possible that, in the normal state, the influence of fluctuations of \(n_s\) on the value of \(\varepsilon'\) will be practically completely masked by the main contribution to \(\varepsilon'\), connected with the ordinary conductivity, which in the cases under consideration (pure metals at low temperature) is especially large. In this connection it is interesting to note that one phenomenon of “anticipation” of the superconducting transition, if the corresponding experiments are to be believed, has already been observed, although it has so far received no interpretation: we have in mind the anomalous behavior of the thermoelectric properties of a metal, appearing in an interval of several tenths of a degree above \(T_k\) (see \(^{15,14}\)).

Whether this phenomenon, whose existence still cannot be considered reliably established (in work \(^{16}\) it was not observed), is connected with fluctuations of \(n_s\) at \(T>T_k\) is entirely unclear, just as the whole question of the role of these fluctuations for other effects in superconductors is unclear.

We hope to return to this problem elsewhere; we have mentioned it here in order to demonstrate, by this example, as also by the example of fluctuations in ferroelectrics and ferromagnets, the connection between the questions touched upon in Sections 1–3 concerning electric fluctuations in circuits with a number of quite different physical phenomena.

Appendix. DERIVATION OF NYQUIST’S QUANTUM FORMULA ACCORDING TO CALLEN AND WELTON

The derivation of Nyquist’s quantum formula (2,3), proposed by Callen and Welton \(^{8}\), is based on comparing the general quantum-mechanical expression for

\[ \frac{R(\omega)}{|Z(\omega)|^2}, \]

given in Section 2 [see (2,14)], with the independently calculated value of the mean square of the current force \(\overline{Q}=\overline{I^2}\) in a state of thermodynamic equilibrium and, of course, in the absence of an external voltage.

Let the system be in some stationary state \(n\), in which the energy is equal to \(E_n\) and \(\Psi=\Psi_n\) [see (2,4)]. In such a state the current \(I\), i.e. the mean value of the operator of the current force

\[ \dot Q=\frac{i}{\hbar}(H_0Q-QH_0), \]

which has the form

\[ \langle E_n|\dot Q|E_n\rangle=\int \Psi_n^{*}\dot Q\Psi_n\,dx, \]

is equal to zero*). The mean value of the quantity \(\dot Q^{2}\) is equal to

\[ \langle E_n|\dot Q^{\,2}|E_n\rangle = \sum_m \langle E_n|\dot Q|E_m\rangle\cdot \langle E_m|\dot Q|E_n\rangle = \]

\[ = \frac{1}{\hbar^2}\sum_m \langle E_n|H_0Q-QH_0|E_m\rangle\cdot \langle E_m|H_0Q-QH_0|E_n\rangle = \]

\[ = \frac{1}{\hbar^2}\sum_m (E_n-E_m)^2\,|\langle E_m|Q|E_n\rangle|^2, \tag{D1} \]

where, in the transformations, the usual devices of quantum mechanics have been used

\[ \left( \dot Q\Psi_n=\sum_m \langle E_m|\dot Q|E_n\rangle\Psi_m,\quad \langle E_m|1|E_n\rangle = \int \Psi_m^{*}\Psi_n\,dx=\delta_{nm}; \right. \]

see also the footnote on the preceding page). Introducing now the frequency \(\omega\), defined according to

\[ \hbar\omega=|E_n-E_m|, \tag{D2} \]

and taking into account that we are dealing with a system with densely spaced levels, one may replace in (D1) the summation over \(m\) by integration over \(\omega\):

\[ \langle E_n|\dot Q^{\,2}|E_n\rangle = \]

\[ = \frac{1}{\hbar^2}\int_0^\infty (\hbar\omega)^2\,|\langle E_n+\hbar\omega|Q|E_n\rangle|^2\, \rho(E_n+\hbar\omega)\,\hbar\,d\omega+ \]

\[ + \frac{1}{\hbar^2}\int_0^\infty (\hbar\omega)^2\,|\langle E_n-\hbar\omega|Q|E_n\rangle|^2\, \rho(E_n-\hbar\omega)\,\hbar\,d\omega = \]

\[ = \int_0^\infty \hbar\omega^2\left\{ |\langle E_n+\hbar\omega|Q|E_n\rangle|^2\rho(E_n+\hbar\omega)+ \right. \]

\[ \left. + |\langle E_n-\hbar\omega|Q|E_n\rangle|^2\rho(E_n-\hbar\omega) \right\}\,d\omega, \tag{D3} \]

where the first integral corresponds to the terms of the sum in (D2) with \(E_n<E_m\), and the second to the terms of the sum with \(E_n>E_m\) [on the introduction of the density of levels \(\rho(E)\), see § 2].

The fluctuations observed in a real system are obtained as the result of averaging the fluctuations in the various states \(n\) with the weight factor \(f(E_n)\). Thus, the quantity that interests us—

*) By virtue of the Hermiticity of \(H_0\), for any functions of the class under consideration \(\Psi^{*}\) and \(\varphi\),

\[ \int \Psi^{*}H_0\varphi\,dx=\int \varphi H_0\Psi^{*}\,dx. \]

Therefore

\[ \int \Psi_n^{*}H_0Q\Psi_n\,dx = \int (Q\Psi_n)H_0\Psi_n^{*}\,dx = E_n\int Q\Psi_n\Psi_n^{*}\,dx = E_n\int \Psi_n^{*}Q\Psi_n\,dx. \]

Obviously, the expression

\[ \int \Psi_n^{*}QH_0\Psi_n\,dx \]

is equal to the same value, whence follows the assertion that the quantity

\[ \langle E_n|\dot Q|E_n\rangle \]

is equal to zero.

for \(\overline{\dot Q^{\,2}}=\overline{I^2}\) is equal to

\[ \overline{\dot Q^{\,2}}=\sum_n f(E_n)\langle E_n|\dot Q^{\,2}|E_n\rangle = \]

\[ =\int_0^\infty \hbar\omega^2\left[\int_0^\infty \rho(E)f(E)\left\{|\langle E+\hbar\omega|Q|E\rangle|^2\rho(E+\hbar\omega)+ \right.\right. \]

\[ \left.\left. +|\langle E-\hbar\omega|Q|E\rangle|^2\rho(E-\hbar\omega)\right\}\,dE\right]\,d\omega, \tag{Д4} \]

since the number of energy levels in the interval \(dE\) is equal to \(\rho(E)\,dE\). For the linear system under consideration, when a voltage \(\mathcal E=\mathcal E_0 e^{i\omega t}\) is applied to it, \(\dot Q=I=\dfrac{\mathcal E}{Z(\omega)}\), where \(Z\) is the impedance. Therefore one may regard the fluctuation currents in a system with impedance \(Z(\omega)\) as being caused by some fluctuating emf \(\mathcal E\), and, in accordance with (Д4) (see also Section 1),

\[ \overline{\mathcal E^2}=\int_0^\infty w(\omega)\,d\omega =\int_0^\infty |Z(\omega)|^2|I_\omega|^2\,d\omega = \]

\[ =\int_0^\infty \hbar\omega^2|Z(\omega)|^2\left[\int_0^\infty \rho(E)f(E)\left\{|\langle E+\hbar\omega|Q|E\rangle|^2\rho(E+\hbar\omega)+ \right.\right. \]

\[ \left.\left. +|\langle E-\hbar\omega|Q|E\rangle|^2\rho(E-\hbar\omega)\right\}\,dE\right]\,d\omega. \tag{Д5} \]

To obtain the quantum Nyquist formula it is necessary to relate expression (Д5) to expression (2.14) for \(\dfrac{R}{|Z|^2}\), which we write here once more:

\[ \frac{R}{|Z|^2} = \pi\omega\int_0^\infty \rho(E)f(E)\left\{|\langle E+\hbar\omega|Q|E\rangle|^2\rho(E+\hbar\omega)- \right. \]

\[ \left. -|\langle E-\hbar\omega|Q|E\rangle|^2\rho(E-\hbar\omega)\right\}\,dE. \tag{Д6} \]

Expressions (Д5) and (Д6) are determined by the quantities \(C(\pm)\) entering into them:

\[ \overline{\mathcal E^2} = \int_0^\infty |Z(\omega)|^2\hbar\omega^2 C(+)\,d\omega, \qquad \frac{R(\omega)}{|Z(\omega)|^2}=\pi\omega C(-), \]

\[ C(\pm)=\int_0^\infty \rho(E)f(E)\left\{|\langle E+\hbar\omega|Q|E\rangle|^2\rho(E+\hbar\omega)\pm \right. \]

\[ \left. \pm|\langle E-\hbar\omega|Q|E\rangle|^2\rho(E-\hbar\omega)\right\}\,dE. \tag{Д7} \]

We shall now, in the second integral in (D7), before which there is the sign \(\pm\), make the change of variables \(E \to E+\hbar\omega\) and take into account that the matrix element
\(\langle E-\hbar\omega|Q|E\rangle=\int \Psi^*_{E-\hbar\omega}Q\Psi_E\,dx\)
is equal to zero for \(E<\hbar\omega\) [all energy levels are regarded as positive, which is reflected in (D7), since the integration is carried out from the energy \(E=0\)]. As a result we obtain (recall that, by virtue of the Hermiticity of \(Q\), \(|\langle E|Q|E+\hbar\omega\rangle|^2=|\langle E+\hbar\omega|Q|E\rangle|^2\)):

\[ \begin{aligned} C(\pm)&=\int_0^\infty |\langle E+\hbar\omega|Q|E\rangle|^2 \rho(E+\hbar\omega)\rho(E)f(E)\times \\ &\qquad \times \left[1\pm \frac{f(E+\hbar\omega)}{f(E)}\right]\,dE=\\ &=\left\{1\pm e^{-\frac{\hbar\omega}{kT}}\right\} \int_0^\infty |\langle E+\hbar\omega|Q|E\rangle|^2 \rho(E+\hbar\omega)\rho(E)f(E)\,dE, \end{aligned} \tag{D8} \]

where, in passing to the second expression, it has been taken into account that in a state of thermodynamic equilibrium the ratio of weights is

\[ \frac{f(E+\hbar\omega)}{f(E)}=e^{-\frac{\hbar\omega}{kT}}. \]

From (D8) and (D7) we have:

\[ \left. \begin{aligned} C(+)&=\frac{1+e^{-\frac{\hbar\omega}{kT}}}{1-e^{-\frac{\hbar\omega}{kT}}}\,C(-) =\left(1+\frac{2}{e^{\frac{\hbar\omega}{kT}}-1}\right)C(-)\\ &=\frac{2}{\pi\omega}\left(\frac12+\frac{1}{e^{\frac{\hbar\omega}{kT}}-1}\right) \frac{R(\omega)}{|Z(\omega)|^2},\\[6pt] \overline{\mathscr{E}^{\,2}}&=\int_0^\infty w(\omega)\,d\omega =\int_0^\infty |Z|^2\hbar\omega\,C(+)\,d\omega=\\ &=\frac{2}{\pi}\int_0^\infty R(\omega) \left(\frac{\hbar\omega}{2}+ \frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}\right)\,d\omega, \end{aligned} \right\} \tag{D9} \]

i.e., for \(w\) one obtains the quantum Nyquist formula (2,3):

\[ w(\omega)=\frac{2}{\pi}R(\omega) \left(\frac{\hbar\omega}{2}+ \frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}\right). \tag{D10} \]

The expression appearing in (D10),

\[ \frac{\hbar\omega}{2}+ \frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}, \]

is formally the expression for the mean energy of an oscillator with frequency \(\omega\) at temperature \(T\). Therefore, in order to avoid misunderstandings, it may be worth emphasizing that this result nevertheless is not

is not connected with any special model representations of the character of the linear dissipative system under consideration.

Nyquist’s formula has been discussed in this article almost exclusively for the electrical case. However, as has already been noted, and as is especially clear, in particular, from its derivation given in the present supplement, Nyquist’s formula applies in fact to any linear dissipative system, regardless of whether it is electrical or mechanical. The application of Nyquist’s formula to the theory of Brownian motion has already been discussed in Section 1. This case is treated in more detail, as is the question of using Nyquist’s formula to establish the connection between radiative friction and fluctuations of the electric field in vacuum, or between acoustic radiative friction and pressure fluctuations in a gas, in⁸.

Cited Literature

1. L. Landau and E. Lifshitz, Statistical Physics, Gostekhizdat, 1951.
2. H. Nyquist, Phys. Rev. 32, 110 (1928).
3. G. S. Gorelik, UFN 44, 33 (1951).
4. Collection “Brownian Motion.” Works of A. Einstein and M. Smoluchowski. Supplementary articles by Yu. A. Krutkov and B. I. Davydov, ONTI, 1936.
5. G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930); Rev. Mod. Phys. 17, 323 (1945).
6. M. A. Leontovich, Statistical Physics, Gostekhizdat, 1944.
7. L. Landau and E. Lifshitz, Quantum Mechanics, Part 1, Gostekhizdat, 1948.
8. H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).
9. M. A. Leontovich, Introduction to Thermodynamics, Gostekhizdat, 1951.
10. M. Planck, Theory of Thermal Radiation, ONTI, 1935.
11. V. L. Ginzburg, UFN 38, 490 (1949).
12. L. Landau, Sow. Phys. 12, 123 (1937).
13. V. L. Ginzburg, UFN 42, 169, 333 (1950).
14. V. L. Ginzburg, ZhETF 21, 979 (1951).
15. H. Casimir and A. Rademakers, Physica 13, 33 (1947).
16. M. Steele, Phys. Rev. 81, 262 (1951).

1. With a view to what follows, let us emphasize that for any values of the parameter \(\alpha=\dfrac{RC}{\sqrt{LC}}\) the resulting integral ↩