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LETTERS TO THE EDITOR
ON THE THERMODYNAMIC TREATMENT OF EQUILIBRIUM ELECTRICAL FLUCTUATIONS
The question of applying phenomenological thermodynamics to thermal electrical fluctuations in quasistationary circuits has already been discussed in the pages of this journal. In G. S. Gorelik’s article,^1 with the aid of the second law of thermodynamics and the general equations of the theory of alternating currents, a theorem on the uniform distribution of energy over degrees of freedom in electrodynamics was derived—a result whose paradoxical character was noted by the author himself. Considerable space in V. L. Ginzburg’s article^2 is devoted to explaining this paradox. However, V. L. Ginzburg’s argumentation, containing, along with correct assertions, also erroneous ones, does not on the whole seem convincing to us. Meanwhile, the question of interest to us can be set out with due clarity, which is the aim of the present article.
1. MEAN ENERGY OF ELECTRICAL FLUCTUATIONS IN AN OSCILLATORY CIRCUIT
Let a macroscopic \(LCR\)-circuit be in thermal equilibrium with a thermostat at temperature \(T\). The mean magnetic energy \(W_m\) and the mean electrical energy \(W_e\) of the circuit are respectively equal to
\[ W_m=\frac{1}{2}L\overline{I^2};\qquad W_e=\frac{\overline{q^2}}{2C}, \tag{1.1} \]
where \(I\) is the fluctuation current in the circuit, \(q\) is the fluctuation charge on the plates of the capacitor, and the bar denotes averaging. Since the circuit is in a state of thermodynamic equilibrium, the quantities \(W_m\) and \(W_e\) are certain functions of the temperature and of the circuit parameters
\[ W_m=T\varphi(T,L,C,R);\qquad W_e=T\psi(T,L,C,R). \tag{1.2} \]
Here \(\varphi\) and \(\psi\) are as yet undetermined functions of their arguments, so that the extraction of the factor \(T\), made to simplify the subsequent exposition, does not reduce the generality of the reasoning.
The work done when the parameters of the circuit are changed is equal to
\[ \delta A= \left(\frac{\partial W_m}{\partial L}\right)_{I}dL - \left(\frac{\partial W_e}{\partial C}\right)_{q}dC = \frac{1}{2}\overline{I^2}\,dL + \frac{\overline{q^2}}{2C^2}\,dC = \]
\[ = \frac{W_m}{L}\,dL+\frac{W_e}{C}\,dC. \tag{1.3} \]
(when \(R\) is changed no work is done), or on the basis of (1.2)
\[ \delta A=T\left(\frac{\varphi}{L}\,dL+\frac{\psi}{C}\,dC\right). \tag{1.4} \]
Substituting (1.2) and (1.4) into the equation of the second law of thermodynamics\(^*\)
\[ T\,dS=d(W_m+W_e)+\delta A, \]
we obtain:
\[ dS=d(\varphi+\psi)+\frac{\varphi+\psi}{T}\,dT+\frac{\varphi}{L}\,dL+\frac{\psi}{C}\,dC. \tag{1.5} \]
The first term on the right-hand side of (1.5) is a total differential; in order that the last three terms also constitute a total differential, the functions \(\varphi\) and \(\psi\), first, must not depend on \(R\), and, second, must satisfy the system of equations
\[ T\frac{\partial\varphi}{\partial T} = L\frac{\partial}{\partial L}(\varphi+\psi), \qquad L\frac{\partial\psi}{\partial L} = C\frac{\partial\varphi}{\partial C}, \]
\[ T\frac{\partial\psi}{\partial T} = C\frac{\partial}{\partial C}(\varphi+\psi), \]
or the system equivalent to it
\[ T\frac{\partial\varphi}{\partial T} = L\frac{\partial\varphi}{\partial L} + C\frac{\partial\varphi}{\partial C}, \qquad L\frac{\partial\psi}{\partial L} = C\frac{\partial\varphi}{\partial C}. \tag{1.6} \]
\[ T\frac{\partial\psi}{\partial T} = L\frac{\partial\psi}{\partial L} + C\frac{\partial\psi}{\partial C}, \]
The solution of (1.6) is
\[ \varphi=\varphi(LT,\,CT); \qquad \psi=\psi(LT,\,CT), \tag{1.7} \]
where \(\varphi(x,y)\), \(\psi(x,y)\) are connected by the relation
\[ x\frac{\partial\psi}{\partial x} = y\frac{\partial\varphi}{\partial y}, \tag{1.8} \]
and otherwise are completely arbitrary functions of their arguments.
Thus, from the second law of thermodynamics it follows that the energy of an oscillatory circuit (magnetic, electric, or their sum, the total energy), initially considered as a function of four arguments, has the form of the product of the temperature by a function of two arguments: \(LT\) and \(CT\). In particular, for \(L=0\) we arrive at the result of G. S. Gorelik\(^1\), who from the very beginning considered a degenerate \(CR\)-circuit. If, however, one assumes that the mean magnetic energy is equal to the mean electric energy (the virial theorem), i.e. that \(\varphi=\psi\), then from (1.8) it immediately follows that \(\varphi(x,y)\) will be a function of the product \(xy\). Then
\[ \varphi(LT,\,CT)=\psi(LT,\,CT)=\Phi\left(\frac{\omega_0}{T}\right), \tag{1.9} \]
where \(\omega_0=\dfrac{1}{\sqrt{LC}}\), and \(\Phi\) is, for the time being, an arbitrary function. Formula (1.9) was
\(^*\) The internal energy of the circuit may differ from \(W_m+W_e\) by some function of the resistance and temperature (see\(^1\)). However, by generalizing the arguments of\(^1\), it is easy to show that for an \(LCR\)-circuit as well this function does not depend on \(R\). Therefore we have omitted it at once.
obtained by V. L. Ginzburg,^2 who from the very beginning assumed that the virial theorem is valid for an oscillatory circuit. However, since the latter is not a thermodynamic theorem, the derivation of formula (1.9) based on it also cannot be regarded as strictly thermodynamic (as V. L. Ginzburg himself indicates, it is valid only for \(R \to 0\)).
2. SPECTRAL DENSITY OF THE FLUCTUATING E.M.F.
For a more detailed description of fluctuating currents and charges, let us introduce, as is usually done, the random part of the e.m.f. \(E(t)\) and its spectral density \(|E_\omega|^2\), related to one another by the relations
\[ \overline{E^2}=\int_0^\infty |E_\omega|^2\,d\omega, \tag{2.1} \]
\[ |E_\omega|^2=\lim_{T\to\infty}\frac{1}{\pi T} \left|\int_0^T E(t)e^{-i\omega t}\,dt\right|^2 . \tag{2.2} \]
In exactly the same way, the fluctuating current \(I(t)\) is related to its spectral density \(|I_\omega|^2\), and the fluctuating charge \(q(t)\) to its spectral density \(|q_\omega|^2\).* It is then evident that
\[ |E_\omega|^2=|Z^2|\,|I_\omega|^2=\omega^2|Z^2|\,|q_\omega|^2, \tag{2.3} \]
where \(Z\) is the impedance of the circuit at frequency \(\omega\):
\[ Z=R+i\left(\omega L-\frac{1}{\omega C}\right). \tag{2.4} \]
It is easy to show (see^1) that the expression for the spectral density of the fluctuating e.m.f. has the form
\[ |E_\omega|^2=R f(\omega,T). \tag{2.5} \]
Here \(f(\omega,T)\) is a certain universal function, whose determination is precisely the task of the theory of thermal electrical fluctuations.
To solve it we shall use the results of the preceding section. From formulas (2.3), (2.5) and the definition of the spectral density it follows that the mean squares of the fluctuating current and fluctuating charge are equal to
\[ \overline{I^2}=\int_0^\infty |I_\omega|^2\,d\omega =\int_0^\infty \frac{R f(\omega,T)}{|Z|^2}\,d\omega, \qquad \overline{q^2}=\int_0^\infty |q_\omega|^2\,d\omega =\int_0^\infty \frac{R f(\omega,T)}{\omega^2|Z|^2}\,d\omega. \tag{2.6} \]
* Formulas (2.1) and (2.2) differ from formulas (1.3) of V. L. Ginzburg’s article^2 by the absence of the factor \(4\pi\), respectively, in the numerator and in the denominator. Calling \(|E_\omega|^2\) the spectral density, it is natural to normalize it according to formula (2.1). Sometimes another normalization is more convenient (see^3, § 117), in which formula (2.2), defining \(|E_\omega|^2\), contains no superfluous numerical factors, but in that case \(|E_\omega|^2\), strictly speaking, is no longer a density. The same may be said of the normalization proposed by V. L. Ginzburg.
According to the thermodynamic formulas (1.1), (1.2), and (1.7),
\[ \overline{I^{2}}=\frac{2T}{L}\,\varphi(LT,CT);\qquad \overline{q^{2}}=2CT\psi(LT,CT). \tag{2.7} \]
Substituting (2.7) into (2.6) and replacing \(|Z|^{2}\) by its explicit expression according to formula (2.4), we obtain:
\[ \left. \begin{aligned} 2T\varphi(LT,CT)&=\int_{0}^{\infty} \frac{\beta}{\beta^{2}+\left(\omega-\frac{\omega_0^{2}}{\omega}\right)^{2}}\, f(\omega,T)\,d\omega,\\ 2T\psi(LT,CT)&=\int_{0}^{\infty} \frac{\beta}{\beta^{2}+\left(\omega-\frac{\omega_0^{2}}{\omega}\right)^{2}}\, \frac{\omega_0^{2}}{\omega^{2}}\,f(\omega,T)\,d\omega, \end{aligned} \right\} \tag{2.8} \]
where
\[ \beta=\frac{R}{L};\qquad \omega_0^{2}=\frac{1}{LC}. \]
Equations (2.8) must be satisfied for any values of the circuit parameters \(L, C\), and \(R\). But as \(R\to 0\) the function
\[ G(\beta,\omega,\omega_0)=\frac{2}{\pi}\, \frac{\beta}{\beta^{2}+\left(\omega-\frac{\omega_0^{2}}{\omega}\right)^{2}} \tag{2.9} \]
passes into the improper Dirac function \(\delta(\omega-\omega_0)\), so that the right-hand sides of equations (2.8) become equal to \(\frac{\pi}{2}f(\omega_0,T)\). Consequently, the left-hand sides are also equal, and since they do not depend on \(R\), one always has \(\varphi=\psi\), and on the basis of (1.9)
\[ \varphi=\psi=\Phi\left(\frac{\omega_0}{T}\right), \tag{2.10} \]
so that
\[ f(\omega,T)=\frac{4}{\pi}\,T\Phi\left(\frac{\omega}{T}\right). \tag{2.11} \]
Formula (2.11) was obtained by V. L. Ginzburg\(^2\) under the additional assumption that the virial theorem is valid for an oscillatory circuit. The derivation given here shows that in this additional assumption there is no need: both the virial theorem (\(\varphi=\psi\)) and formula (2.11) are consequences of the second law of thermodynamics and of the general equations of the theory of alternating currents.
So long as \(\Phi\left(\frac{\omega}{T}\right)\) is an arbitrary function, expressions (2.10), (2.11) automatically satisfy relations (2.8) only for \(R\to 0\). In order that relations (2.8) be fulfilled for arbitrary values of \(R\), the function \(\Phi\)
must satisfy two integral equations
\[ \Phi\left(\frac{\omega_0}{T}\right) = \int_0^\infty G(\beta,\omega,\omega_0)\, \Phi\left(\frac{\omega}{T}\right)\,d\omega, \]
\[ \frac{1}{\omega_0^2}\Phi\left(\frac{\omega_0}{T}\right) = \int_0^\infty G(\beta,\omega,\omega_0)\, \frac{1}{\omega^2}\Phi\left(\frac{\omega}{T}\right)\,d\omega, \]
i.e., both the function \(\Phi\) and the function \(\dfrac{\Phi}{\omega^2}\) are solutions of the integral equation
\[ F(\omega_0)=\int_0^\infty G(\beta,\omega,\omega_0)F(\omega)\,d\omega. \tag{2.12} \]
The \(\beta\)-independent general solution of this equation has the form (see the Appendix)
\[ F(\omega)=C_1+\frac{C_2}{\omega^2}, \tag{2.13} \]
where \(C_1\) and \(C_2\) are arbitrary constants. Therefore, in order that both \(\Phi\) and \(\dfrac{\Phi}{\omega^2}\) satisfy equation (2.12), the function \(\Phi\) must be equal to a constant. But then from formulas (2.5) and (2.11) it follows at once that
\[ |E_\omega|^2=aRT, \tag{2.14} \]
where \(a\) is some constant. Up to the value of the constant \(a\), (2.14) is the classical Nyquist formula,\({}^{4}\) equivalent to the Rayleigh–Jeans formula for the spectral density of equilibrium radiation (see \({}^{5}\)).
Formula (2.14) was obtained by G. S. Gorelik\({}^{1}\) in a thermodynamic consideration of a degenerate \(CR\)-circuit. However, such a derivation of (2.14) is not entirely correct, since from formula (2.14) it immediately follows that in a \(CR\)-circuit \(\overline{I^2}=\infty\), and this contradicts the initial assumption of thermal equilibrium between the resistance and the thermostat.
3. ON G. S. GORELIK’S PARADOX
In proving the independence of the equilibrium energy of an oscillatory circuit from the resistance and in deriving formulas (1.7), we proceeded from the fact that \(W_m\) and \(W_e\) are certain functions of the inductance \(L\), capacitance \(C\), and resistance \(R\), and we treated \(L\), \(C\), and \(R\) as thermodynamic parameters of the system. We must, however, bear in mind that \(R\) (as well as \(L\) and \(C\)) does not depend on frequency, since otherwise \(W_m\) and \(W_e\) would not be functions but functionals, and the usual form of the thermodynamic equalities would have no meaning.
Moreover, the dependence of \(R\) on frequency may be due either to the skin effect or to dispersion of the conductivity. In the first case, the conductivity depends on frequency; in the second, the permittivity will be a complex quantity (the real part of the complex permittivity also changes with frequency). Therefore the purely electrodynamic equations (1.1) and (1.3) in fact cease to be valid when \(R(\omega)\ne\mathrm{const}\).
In finding the spectral density of the fluctuating e.m.f., we used the result of § 1 and, in addition, explicitly assumed that the parameters of the oscillatory circuit under consideration do not depend on frequency.
Thus, formula (2.14), whose usual derivation relies on the classical theorem on the uniform distribution of energy over degrees of freedom, is a consequence of thermodynamics, the theory of alternating currents, and the assumption of the existence of conductors with resistance independent of frequency. But in such a refined formulation the assertion of H. S. Callen1 contains, as we shall now show, nothing paradoxical.
A necessary condition for the independence of resistance from frequency is, evidently, the instantaneous character of the collisions undergone by the conduction electrons *). However, the assumption that collisions are instantaneous is incompatible, as is easily seen, with quantum ideas. Indeed, the duration of a collision has the same order of magnitude as the time of passage of a wave packet past a fixed point of space. But for motion in any potential force field (see 6, p. 306) this passage time \(\tau\) satisfies the inequality
\[ \tau \gtrsim \frac{\hbar}{2\Delta E}, \]
where \(\Delta E\) is the spread of energies in the packet. For conduction electrons \(2\Delta E \approx kT\) (the smearing band of the Fermi distribution), so that
\[ \tau \gtrsim \frac{\hbar}{kT} \]
and, consequently, in the quantum region of frequencies collisions cannot be regarded as instantaneous, and the resistance cannot be independent of frequency. For the classical region of frequencies, however, the instantaneous character of collisions entails the instantaneous character of the fluctuations of the fluctuating e.m.f., from which it follows at once that the universal function \(f(\omega,T)\) does not depend on frequency.
Thus, the assumption of the existence of conductors with resistance independent of frequency is equivalent to a very strong statistical assertion (the instantaneous character of collisions), which has a purely classical character. This explains the apparent paradoxical nature of the thermodynamic derivation of formula (2.14).
There is no paradox from a more formal point of view either. In fact, both the resistance of a conductor and the fluctuation currents in it have as their cause one and the same random process, whose spectral characteristics are \(|E_\omega|^2\) and \(R(\omega)\) (here, as before, we ignore the skin effect, considering the conductor sufficiently thin). Naturally, these two spectral characteristics of one process are related to each other, and therefore there is nothing surprising in the fact that, putting \(R(\omega)=\mathrm{const}\), we obtain with the help of thermodynamics a quite definite expression for \(|E_\omega|^2\).
If \(R=R(\omega)\), then, as has already been said, a thermodynamic consideration of electrical fluctuations is impossible: \(R(\omega)\) is not a parameter
*) In the opinion of V. L. Ginzburg2, the mechanism of non-instantaneous collisions is such that, as \(\omega\to\infty\), \(R(\omega)\to0\), since otherwise in the classical case the mean square of the fluctuating e.m.f. \(\overline{E^2}\sim\int R(\omega)d\omega\) would not be a finite quantity. However, on the same grounds one could assert that, as \(\omega\to\infty\), \(R\to\infty\), for only then would the expression for the mean square of the fluctuation current \(\overline{I^2}\sim\int \frac{d\omega}{R(\omega)}\) in a circuit containing only resistances not diverge (in work2 it is assumed that, in principle, such circuits can exist).
in the thermodynamic sense. An exception is the case considered by V. L. Ginzburg,^2 when \(R(\omega)\) vanishes slowly. Then one may neglect the dependence of \(W_m\) and \(W_e\) on \(R\), with the same degree of accuracy that they do not depend on the frequency \(L\) and \(C\), and consequently all the results of § 1 and the results of § 2 up to formula (2.11) inclusive remain valid. Therefore, for the quantum region of frequencies, where in principle \(R(\omega) \ne \mathrm{const}\), thermodynamics cannot yield anything more than the general result expressed by formula (2.11).
V. L. Ginzburg^2 arrives at the same conclusion in his paper; however, his argumentation contains, in our opinion, erroneous assertions on which we consider it necessary to dwell. First of all, V. L. Ginzburg believes that imaginary conductors whose resistance does not depend on frequency, even in the quantum region, but which are thermodynamically considered as real conductors (with nonvanishing resistance), are legitimate, since, generally speaking, \(R\) is a measure of the nonadditivity of the energy of the system circuit + thermostat. The latter assertion is based, as it seems to us, on a confusion of concepts: the resistance of a macroscopic circuit in thermal equilibrium is a measure of the nonadditivity of energy*), not a measure of the nonadiabaticity of this circuit. Therefore, on the same grounds one could assert that thermodynamics, generally speaking, is inapplicable to bodies with a nonvanishing coefficient of thermal conductivity. The general point of view developed in paper^2, according to which before applying thermodynamic laws to a macroscopic system in a state of thermal equilibrium it is in fact necessary first to substantiate such an application by means of a statistical consideration, leads ultimately to an undeserved discrediting of phenomenological thermodynamics. Of course, it is not thermodynamics but the initial assumption \(R(\omega)=\mathrm{const}\), incompatible with the quantum laws, that is responsible for the fact that formula (2.14) contradicts Nyquist’s quantum formula.
The thermodynamic consideration of thermal electric fluctuations in an oscillatory circuit with constant parameters carried out above in §§ 1, 2 has, as it seems to us, not only pedagogical interest. First, it shows that in the usual statistical derivation of Nyquist’s classical formula (see, for example,^4,8,9), when the assumption \(R(\omega)=\mathrm{const}\) is taken as the “noise” element, one can omit the rather lengthy proof of the independence of the universal function \(f(\omega,T)\) from frequency—this follows from thermodynamics. Secondly, it is clear from it that in the quantum region of frequencies one cannot, even without taking into account the skin effect, regard the resistance as independent of frequency. In particular, the calculation of various integrals that converge in the classical case, performed using Nyquist’s quantum formula but under the assumption \(R(\omega)=\mathrm{const}\) (see pp. 366–669 of paper^2), gives results which, in our opinion, have no physical meaning.
) In what follows, speaking of the nonadditivity of energy, V. L. Ginzburg asserts (^2, p. 376) that the energy of interaction of the circuit with the thermostat may be neglected only when \([E_\omega]^2\) does not depend on frequency, i.e., when the impulses of the integral fluctuational e.m.f. are instantaneous. But this is also incorrect. In order for the energy of interaction to be negligible, it is sufficient that the impulses of the differential* fluctuational fields be instantaneous (the instantaneous character of summation). In this case, however, the impulses of the integral e.m.f. due to the skin effect may well not be instantaneous: \([E_\omega]^2\) is proportional to the resistance, which varies, generally speaking, with frequency because of the skin effect (see^7). Thus, in this classical case the energy of interaction of the thermostat with the circuit may be neglected, but thermodynamics cannot be applied to the circuit: the resistance depends on frequency and, consequently, is not a thermodynamic parameter.
ADDENDUM
The solution of the integral equation
\[ \frac{2}{\pi}\int\limits_{0}^{\infty} \frac{\beta}{\beta^2+\left(x-\frac{y^2}{x}\right)^2}\,F(x)\,dx=F(y). \tag{A} \]
To find a solution of equation (A) independent of the parameter \(\beta\), let us expand its kernel in the following formal series in powers of \(\beta\):
\[ \frac{2}{\pi}\frac{\beta}{\beta^2+z^2} = \frac{2}{\pi}\int\limits_{0}^{\infty} e^{-\beta u}\cos zu\,du = 2\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{n!}K_n(z)\beta^n, \tag{B} \]
where
\[ K_n(z)=\frac{1}{\pi}\int\limits_{0}^{\infty}u^n\cos zu\,du, \tag{V} \]
and
\[ z=x-\frac{y^2}{x}. \]
Substituting the expansion (B) into equation (A) and equating the coefficients of like powers of \(\beta\), we obtain the system of equations
\[ 2\int\limits_{0}^{\infty} K_n(z)F(x)\,dx=\delta_{0n}F(y), \tag{G} \]
where \(\delta_{0n}\) is the Kronecker symbol.
If \(n\) is even, then the right-hand side of (V) is, up to sign, equal to the \(n\)-th derivative of Dirac’s \(\delta\)-function,
\[ K_{2m}(z)=(-1)^m\delta^{(2m)}(z). \]
Therefore, passing in the left-hand sides of equations (G) from the variable of integration \(x\) to the variable of integration \(z\) (with \(y=\mathrm{const}\)), we can, for even \(n\), rewrite these equations in the form
\[ \left\{\frac{d^n}{dz^n}[gF(x)]\right\}=\frac{1}{2}\delta_{0n}F(y), \tag{D} \]
where
\[ g=\left(\frac{\partial x}{\partial z}\right)_y=\frac{x^2}{x^2+y^2}, \]
and the brace means that \(z=0\), i.e. \(x=y\), has been put.
For \(n=0\) equation (D) is satisfied automatically, since \(\{g\}=\frac{1}{2}\).
For \(n=2\), passing again to the variable \(x\), we shall have:
\[ \left\{\frac{d}{dx}\left[g\frac{d}{dx}(gF)\right]\right\}=0, \]
or
\[ \{g^2\}F''+\{3gg'\}F'+\{g'^2+gg''\}F=0, \]
where the prime, as usual, denotes differentiation with respect to \(x\). But
\[ \{g\}=\frac{1}{2};\qquad \{g'\}=\frac{1}{x};\qquad \{g''\}=-\frac{1}{2x^2}, \]
so that finally \(F''+\dfrac{3}{x}F'=0\), and, consequently,
\[ F(x)=C_1+\frac{C_2}{x^2}, \tag{E} \]
where \(C_1, C_2\) are arbitrary constants. It is easy to verify that (E) indeed satisfies the original equation (A).
The formal device by means of which we have just found the solution of equation (A) cannot, of course, in any way claim mathematical rigor. It does, however, possess, as it seems to us, some heuristic value, since it makes it possible rather quickly to find solutions of integral equations whose kernels \(G(\beta, x, y)\) can be represented in the form of a series in eigenfunctions of some combination of the variables \(x\) and \(y\) corresponding to the region of irregularity of the kernel.
M. L. Levin
References
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