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SOME APPLICATIONS OF THE SECOND LAW OF THERMODYNAMICS TO ELECTRICAL FLUCTUATIONS
G. S. Gorelik
1. INTRODUCTION
What can be said about electrical fluctuations, which play such an important role in modern technology, on the basis of phenomenological thermodynamics alone, without using statistical considerations? In particular, what consequences pertaining to electrical fluctuations can be derived from the equation*)
\[ T\,dS=dU+\sum X_i\,dx_i, \tag{1} \]
which expresses the second law of thermodynamics for quasistatic processes?
These are the questions to which we shall try to give an answer here. These questions may seem strange: it is customary to say that fluctuation phenomena lie outside the scope of phenomenological thermodynamics. However, such an opinion is incorrect, as will be explained in § 2**). After that, the questions posed at the beginning will appear, we hope, quite natural and not devoid of a certain—at least pedagogical—interest.
*) Notation: \(T\) is the temperature on the thermodynamic scale, \(S\) is the entropy, \(U\) is the internal energy, \(x_i, X_i\) are generalized coordinates and the corresponding generalized forces.
**) Already in Nyquist’s paper\({}^{1}\) there is a discussion concerning electrical fluctuations based on phenomenological thermodynamics. This discussion is reproduced (with insignificant changes) in § 4. We note, in order to avoid misunderstanding, that what is constructed here is not, as has sometimes been done, a new formulation of the second law of thermodynamics that takes into account fluctuations of physical quantities about their mean values. Here the second law of thermodynamics is applied to fluctuation phenomena in its classical formulation.
2. THERMAL RADIATION AND FLUCTUATION NOISE
The usual terminology relating to fluctuations suffers from inconsistency. This is connected with the fact that it is impossible to carry out an unambiguous demarcation of physical phenomena into two classes: fluctuations and “non-fluctuations”; the assignment of a given physical phenomenon to fluctuations or to “non-fluctuations” may depend on the choice of variables*).
We shall be able to justify these assertions most clearly if we draw a parallel between thermal radiation in a cavity and thermal noise in an electrical oscillatory circuit**). The discussion in this paragraph will be conducted from the standpoint of statistical thermodynamics.
As is known (see, for example,\(^{2}\)), if a closed system in a state of thermodynamic equilibrium consists of two parts having energies \(E\) and \(E'\) (the total energy \(E+E'\) being constant), and if \(\overline{E}\ll \overline{E'}\), then the mean square deviation
\[ \varepsilon = E-\overline{E} \]
of the energy \(E\) from its mean value \(\overline{E}\) is equal to:
\[ \overline{\varepsilon^{2}} = kT^{2}\frac{d\overline{E}}{dT} \tag{2} \]
(\(k\) is Boltzmann’s constant). On the basis of formula (2) and Planck’s formula for the mean value of the energy \(\overline{E}_{\mathrm{и}}\) of equilibrium radiation of temperature \(T\) in a volume \(V\) and in the frequency interval \(\Delta\nu\), we have:
\[ \overline{E}_{\mathrm{и}} = N\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}, \tag{3} \]
where \(h\) is Planck’s constant and \(N\) is the number of degrees of freedom of this radiation, namely:
\[ N=\frac{8\pi\nu^{2}}{c^{3}}\,V\Delta\nu \tag{4} \]
(\(c\) is the speed of light), Einstein calculated the mean square deviation of the energy \(E_{\mathrm{и}}\) from its mean value: on the basis of (2), (3) we obtain, as a result of a simple calculation,
\[ \overline{\varepsilon_{\mathrm{и}}^{2}} = h\nu\,\overline{E}_{\mathrm{и}} + \frac{1}{N}\overline{E}_{\mathrm{и}}^{2}. \tag{5} \]
*) My attention was drawn to this circumstance, and to the related impossibility of demarcating the domains of applicability of phenomenological and statistical thermodynamics, by M. A. Leontovich.
**) Shot electrical fluctuations are not considered in this article.
Let us apply Einstein’s reasoning to an electric oscillatory circuit in thermodynamic equilibrium with the surrounding matter and/or radiation. The mean value of the energy of the circuit
\[ E_k=\frac{q^2}{2C}+\frac{L\dot q^2}{2} \tag{6} \]
(\(q\) is charge, \(C\) capacitance, \(L\) inductance), if \(h\nu\), where
\[ \nu=\frac{1}{2\pi\sqrt{LC}}, \tag{7} \]
is not neglected in comparison with \(kT\), is equal to:
\[ \overline{E}_k=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}. \tag{8} \]
On the basis of (2), (8) we have:
\[ \overline{\varepsilon_k^{\,2}}=h\nu\overline{E}_k+\overline{E}_k^{\,2}. \tag{9} \]
From the thermodynamic and statistical points of view, the case of an oscillatory circuit differs from the case of equilibrium radiation \((V,\Delta\nu)\) only in that in the former the distinguished system has the number of degrees of freedom \(N=1^*)\).
The terminology usually employed is as follows. The quantity \(\overline{E}_i\) is called the mean energy of thermal radiation; the quantity \(\overline{E}_k\) is spoken of as the mean energy of electrical fluctuations; in the case of radiation the quantity of fluctuation is called \(\varepsilon_i\). With such terminology one should speak of \(\varepsilon_k\) as a quantity characterizing the fluctuation of the energy of fluctuations.
It is clear that this terminology is at variance with the true relation between the physical meaning of the quantities \(E_i, E_k, \varepsilon_i, \varepsilon_k\): the quantity \(E_k\) is analogous to \(E_i\), and the quantity \(\varepsilon_k\) to the quantity \(\varepsilon_i\). This gives us the right to assert that the usual terminology is inconsistent.
A consistent terminology is, for example, the following: \(E_i\) is the energy of thermal radiation \((V,\Delta\nu)\), \(E_k\) is the energy of thermal oscillation in the circuit; \(\varepsilon_i\) is the fluctuation of the energy of thermal radiation \((V,\Delta\nu)\), \(\varepsilon_k\) is the fluctuation of the energy of thermal oscillation in the circuit. But one can also construct another consistent terminology, in which the name “fluctuation” would refer to what is usually called thermal radiation.
*) Formula (9) can be obtained directly from (5) by putting \(N=1\) in it.
Indeed, we shall consider not the energy, but the intensities of the electric and magnetic fields \(\mathcal{E}\), \(\mathcal{H}\). As in the case of an oscillatory circuit, so also in the case of radiation, we have:
\[ E=\frac{1}{8\pi}\int\left(\mathcal{E}^{2}+\mathcal{H}^{2}\right)dV . \tag{10} \]
Moreover, in thermodynamic equilibrium everywhere
\[ \overline{\mathcal{E}}=0,\qquad \overline{\mathcal{H}}=0, \tag{11} \]
and, consequently, \(\mathcal{E}=\mathcal{E}-\overline{\mathcal{E}}\), \(\mathcal{H}=\mathcal{H}-\overline{\mathcal{H}}\).
Thus one may say that both the quantity \(\overline{E}_{\mathrm{k}}\) and the quantity \(\overline{E}_{\mathrm{i}}\) characterize the mean square fluctuation of the quantities \(\mathcal{E}\), \(\mathcal{H}\), while the quantities \(\overline{\varepsilon_{\mathrm{k}}^{2}}\), \(\overline{\varepsilon_{\mathrm{i}}^{2}}\) are the mean square fluctuation of the energy fluctuation.
Thus, one and the same phenomenon (the presence, under thermodynamic equilibrium, of radiation in a cavity or of oscillations in a circuit) belongs to the class of fluctuations if the consideration is carried out in “linear” variables \(\mathcal{E}\), \(\mathcal{H}\) (or \(q,\dot q\)), and to the class of “non-fluctuations” if the consideration is carried out in “quadratic” variables \(E_{\mathrm{i}}\) or \(E_{\mathrm{k}}\). The term “fluctuations” cannot serve as an unambiguous characteristic of one phenomenon or another; only the term has an unambiguous meaning: “fluctuations of such-and-such a physical quantity”: the charge of a capacitor, the energy of a field, etc.
Let us now formulate the main idea of the present article. We know that the laws of phenomenological thermodynamics are applicable to the mean values of physical quantities. Thus, for example, with the aid of phenomenological thermodynamics (without statistics) it is derived that for the energy density of radiation \(u\), equal by definition to
\[ u=\frac{1}{V}\int_{0}^{\infty}\frac{\overline{E}}{\Delta\nu}\,d\nu, \tag{12} \]
the relation holds
\[ u=\sigma T^{4}, \tag{13} \]
where \(\sigma\) is a universal constant (the Stefan–Boltzmann law), that the spectral density of the quantity \(u\) and of the proportional energy flux of equilibrium radiation is a universal function \(F(\nu,T)\) of frequency and temperature (Kirchhoff’s law), moreover
\[ F(\nu,T)=\nu^{3}\Psi\!\left(\frac{\nu}{T}\right). \tag{14} \]
(Wien’s law). Here it will be shown how, from phenomenological thermodynamics, one can obtain statements analogous to a certain extent for the mean quantities characterizing—if the usual terminology is used—equilibrium electrical fluctuations.
Phenomenological thermodynamics will be applied by us in combination with classical electrodynamics. Thus, quantum phenomena (thermal fluctuations for \(h\nu \gg kT\)) lie outside the scope of our investigation.
3. THERMODYNAMIC THEOREM ON THE MEAN SQUARE OF CHARGE FLUCTUATION
Let it be known to us that the radiation pressure is
\[ p=\frac{u}{3}. \tag{15} \]
Then, applying (1), we obtain (“thermodynamics doubles our knowledge”) the Stefan–Boltzmann law. Let us carry out reasoning of the same type for the mean square of charge fluctuations on a capacitor.
Let a capacitor of capacitance \(C\) be in equilibrium with a thermostat at temperature \(T\) (in the circuit of Fig. 1 the equilibrium is ensured by means of a resistance \(R\) immersed in the thermostat). Let us take \(T\), \(C\), and \(R\) as independent variables. To the independent variable \(C\) there corresponds the generalized force
\[ X=-\frac{\partial}{\partial C}\left(\frac{\overline{q^2}}{2C}\right) =\frac{\overline{q^2}}{2C^2}. \tag{16} \]
Fig. 1.
(the generalized force in the thermodynamic sense is the time average over a long time of the generalized force in the mechanical sense, equal, in turn, to minus the derivative with respect to the generalized coordinate \(C\) of the electrostatic energy \(q^2/2C\)). The quantity \(X\) characterizes, on average over a long time, the attraction that exists as a result of the charge fluctuations \(q\) between the plates of the capacitor. (The pressure of a gas or radiation \(p\) considered in thermodynamics is likewise the time average of the “true” pressure over a long time.)
The generalized force corresponding to the variable \(R\) is equal to zero. Unlike the displacement of the capacitor plates, displacement of the slider of the rheostat—in the absence of friction—takes place without expenditure of work.)
The quantity \(\overline{q^2}\) must be regarded as an unknown function of the independent variables \(T, C, R\)
\[ \overline{q^2}=\varphi(T,C,R). \tag{17} \]
The equation
\[ X=\frac{\varphi(T,C,R)}{2C^2} \tag{18} \]
is the equation of state of our system, just as (15), if \(u\) is expressed as a function of \(T, V\), is the equation of state of the radiation.
The internal energy of the system (in the thermodynamic sense)
\[ U=\frac{\overline{q^2}}{2C}+F(T,R)=\frac{\varphi(T,C,R)}{2C}+F(T,R) \tag{19} \]
(the mean value over a long time of the electrostatic energy plus a function of temperature and resistance \(R\)).
Consequently, for the system under consideration equation (1) has the form
\[ T\,dS=d\left[\frac{\varphi(T,C,R)}{2C}+F(T,R)\right] +\frac{\varphi(T,C,R)}{2C^2}\,dC \tag{20} \]
or
\[ dS=\frac{1}{T}\left(\frac{1}{2C}\frac{\partial\varphi}{\partial T} +\frac{\partial F}{\partial T}\right)dT +\frac{1}{2TC}\frac{\partial\varphi}{\partial C}\,dC + \frac{1}{T}\left(\frac{1}{2C}\frac{\partial\varphi}{\partial R} +\frac{\partial F}{\partial R}\right)dR. \tag{21} \]
Since \(dS\) is a complete differential, we have:
\[ \frac{\partial}{\partial C}\left[\frac{1}{T} \left(\frac{1}{2C}\frac{\partial\varphi}{\partial T} +\frac{\partial F}{\partial T}\right)\right] = \frac{\partial}{\partial T}\left[\frac{1}{2TC} \frac{\partial\varphi}{\partial C}\right], \tag{22a} \]
\[ \frac{\partial}{\partial T}\left[\frac{1}{T} \left(\frac{1}{2C}\frac{\partial\varphi}{\partial R} +\frac{\partial F}{\partial R}\right)\right] = \frac{\partial}{\partial R}\left[\frac{1}{T} \left(\frac{1}{2C}\frac{\partial\varphi}{\partial T} +\frac{\partial F}{\partial T}\right)\right], \tag{22b} \]
\[ \frac{\partial}{\partial R}\left[\frac{1}{2TC} \frac{\partial\varphi}{\partial C}\right] = \frac{\partial}{\partial C}\left[\frac{1}{T} \left(\frac{1}{2C}\frac{\partial\varphi}{\partial R} +\frac{\partial F}{\partial R}\right)\right], \tag{22c} \]
whence it is easy to obtain
\[ T\frac{\partial\varphi}{\partial T} = C\frac{\partial\varphi}{\partial C}, \tag{23a} \]
\[ \frac{1}{2C}\frac{\partial\varphi}{\partial R} +\frac{\partial F}{\partial R} =0, \tag{23b} \]
\[ \frac{\partial\varphi}{\partial R}=0. \tag{23c} \]
Equation (23в) indicates that \(\varphi\) does not depend on \(R\), (23б) indicates that \(F\) also does not depend on \(R\), and equation (23а) that the function \(\varphi\) has the form
\[ \varphi=\Phi(CT), \tag{24} \]
i.e., on the basis of (17),
\[ \overline{q^{2}}=\Phi(CT). \tag{25} \]
Thus, from relation (18) (from what was known beforehand about the structure of the equation of state) and from relation (19) (from what was known beforehand about the structure of the expression for the internal energy), the second law of thermodynamics makes it possible to conclude that the mean square of the charge of a capacitor (the mean square fluctuation of the quantity \(q\)) depends only on the product of the capacitance of the capacitor and the temperature.
4. THERMODYNAMIC THEOREM ON THE SPECTRAL DENSITY OF FLUCTUATIONAL E.M.F.
One of the assertions constituting the second law of thermodynamics may be formulated as follows. If a system enclosed in an adiabatic envelope is in a state of thermodynamic equilibrium (with its parts having the same temperature), then no temperature difference can be created between its parts without the expenditure of work from outside. Applying this assertion to the exchange of energy between two furnaces of equal temperature by means of radiation (Fig. 2) and mentally placing in the path of the radiation nonabsorbing plates whose reflection and transmission coefficients depend on frequency, we arrive at Kirchhoff’s law.
Fig. 2.
An analogous argument can be carried out for electrical fluctuations.
Let us imagine two linear two-terminal networks located in adiabatically insulated thermostats at temperatures \(T_1, T_2\) and having, respectively, impedances
\[ Z_1=R_1+iX_1,\qquad Z_2=R_2+iX_2 \tag{26} \]
(generally speaking, not only the reactances \(X_1, X_2\), but also the active resistances \(R_1, R_2\) depend on frequency). Let us connect these two-terminal networks to one another (Fig. 3). Owing to the presence of a fluctuating (thermal) voltage between the connecting wires \(A, B\) and of a fluctuating (thermal) current in them, there occurs,
G. S. GORELIK
generally speaking, the transfer of energy from one two-terminal network to another, which leads to a change in their temperatures.
Let us introduce, for the description of fluctuation electrical phenomena in the circuit of Fig. 3, equivalent fluctuation e.m.f.’s \(\mathcal{E}_1(t)\), \(\mathcal{E}_2(t)\) (\(t\) is time), imagining their sources connected in series with the two-terminal networks and located inside the corresponding thermostats (Fig. 4). What properties must these equivalent e.m.f.’s possess so that there is no contradiction with the thermodynamic statement given at the beginning of this section?
Fig. 3.
Fig. 4.
If the system is in thermodynamic equilibrium, then, according to the second law of thermodynamics, it must be
\[ P_{12}=P_{21}, \tag{27} \]
where \(P_{12}\), \(P_{21}\) are, respectively, the powers delivered by the source of e.m.f. \(\mathcal{E}_1\) to the two-terminal network \(Z_2\) and by the source of e.m.f. \(\mathcal{E}_2\) to the two-terminal network \(Z_1\).* In this case
\[ P_{12}=\int_{0}^{\infty} R_2 \frac{w_1(\nu)\,d\nu}{|Z|^2}, \quad P_{21}=\int_{0}^{\infty} R_1 \frac{w_2(\nu)\,d\nu}{|Z|^2}, \tag{28} \]
* Statistical interpretation:
\[ P_{12}=\overline{V_1(t)I_1(t)}, \qquad P_{21}=\overline{V_2(t)I_2(t)}, \]
where \(I_1(t)\), \(I_2(t)\) are the current components in wire \(A\) (Fig. 4), and \(V_1(t)\), \(V_2(t)\) are the corresponding potential differences between wires \(A\), \(B\), caused by the respective sources \(\mathcal{E}_1\), \(\mathcal{E}_2\). The positive directions for \(V_1\) and for \(V_2\) are from \(A\) to \(B\). The positive direction for \(I_1\) is from \(Z_1\) to \(Z_2\), and for \(I_2\) from \(Z_2\) to \(Z_1\). Averaging is performed over a very long time.
where \(w_1(\nu), w_2(\nu)\) are the spectral densities*) of the e.m.f. \(\mathcal E_1(t)\), \(\mathcal E_2(t)\), and
\[ Z=Z_1+Z_2 . \tag{29} \]
Substituting (28) into (27), we obtain the equation
\[ \int_0^\infty R_2 \frac{w_1(\nu)\,d\nu}{|Z|^2} = \int_0^\infty R_1 \frac{w_2(\nu)\,d\nu}{|Z|^2}. \tag{30} \]
From this it follows, in particular, as a lemma: if at all frequencies \(R_2=0\), \(R_1\ne0\), then at all frequencies \(w_2(\nu)=0\); in other words, if one of the two-terminal networks does not absorb, then it contains no fluctuating e.m.f.
Using this lemma, it is easy to strengthen assertion (30).
Let us insert between the two-terminal networks a nonabsorbing (purely reactive) filter (Fig. 5). Now
\[ P_{12}=\int_0^\infty R_1 \frac{w_2(\nu)\,d\nu}{|Z'|^2}, \]
\[ P_{21}=\int_0^\infty R_2 \frac{w_1(\nu)\,d\nu}{|Z'|^2}, \tag{31} \]
where
\[ Z'=Z+iX_0 \tag{32} \]
(\(X_0\) is the reactance of the filter). Since, according to the lemma, the filter “does not make noise,”**) in thermodynamic equilibrium it must still
Fig. 5.
*) Statistical interpretation:
\[ w(\nu)=\lim_{\tau\to\infty}\frac{2A(\nu)A^*(\nu)}{\tau}, \]
where
\[ A(\nu)=\int_{t_0}^{t_0+\tau}\mathcal E(t)e^{-i2\pi\nu t}\,dt,\qquad \nu=\frac{n}{\tau}\qquad (n=0,1,2,\ldots). \]
The averaging is carried out over a large number of time intervals \(\tau\). The quantity \(|A(\nu)|\) is proportional to the amplitude of the harmonic component of frequency \(\nu\) of the Fourier series representing the function \(\mathcal E(t)\) in the interval \(t_0, t_0+\tau\).
**) That is, it contains no fluctuating e.m.f. In order to apply the lemma, one must regard the circuit of Fig. 5 as the connection of the filter with the two-terminal network \(Z_1+Z_2\).
be satisfied (27), whence
\[ \int\limits_{0}^{\infty} R_1 \frac{w_2(\nu)\,d\nu}{|Z'|^2} = \int\limits_{0}^{\infty} R_2 \frac{w_1(\nu)\,d\nu}{|Z'|^2}. \tag{33} \]
In order that this equality be satisfied for any form of the filter characteristic \(X_0(\nu)\), the integrand functions must be identically equal to one another. Hence it follows that
\[ R_1(\nu)\,w_2(\nu)=R_2(\nu)\,w_1(\nu) \tag{34} \]
or
\[ \frac{w_1(\nu)}{R_1(\nu)}=\frac{w_2(\nu)}{R_2(\nu)}. \tag{35} \]
Let us now consider all possible linear two-terminal networks \(Z_1, Z_2,\ldots, Z_i,\ldots, Z_k,\ldots\) at temperature \(T\). Extending the arguments just presented to any pair of two-terminal networks \(Z_i, Z_k\), we find that the ratio \(\dfrac{w_i(\nu)}{R_i(\nu)}\) is the same (at the given temperature) for all two-terminal networks. But it may depend on the temperature.
Thus we arrive at the theorem:
\[ \frac{w(\nu)}{R(\nu)}=f(\nu,T), \tag{36} \]
where \(f(\nu,T)\) is a universal function of frequency and temperature.
The similarity of this theorem to Kirchhoff’s law is obvious. Rewriting (36) in the form
\[ w(\nu)=R(\nu)\,f(\nu,T), \tag{37} \]
we may say: a two-terminal network is noisier at a given frequency the greater its active resistance at that frequency. Although there is no complete analogy between \(R\) and absorptive capacity, nor between \(w\) and emissive capacity, one should note the similarity of this latter formulation to another (“second”) formulation of Kirchhoff’s theorem: the emissive powers of bodies (for given \(\nu,T\)) are related to one another in the same way as their absorptive powers.
5. SOME CONSEQUENCES OF RELATIONS (25) AND (37) AND OF THE THEORY OF ALTERNATING CURRENTS *)
A. Let us return to the circuit of Fig. 1. Applying to it the representation of an equivalent fluctuational e.m.f., we can write for the spectral density of the current strength in the \(RC\)-circuit, using
* The author is grateful to M. L. Levin and M. A. Leontovich for discussions that had a substantial influence on the content of §§ 5, 6.
quasistationary classical electrodynamics (the theory of alternating currents), the expression
\[ \frac{w(\nu)}{R^2+(1/2\pi\nu C)^2}. \]
Dividing it by \((2\pi\nu)^2\), we find the spectral density of the charge of the capacitor. Integrating then over all frequencies, we obtain for the mean square of this charge the formula
\[ \overline{q^2} = \int_0^\infty \frac{1}{(2\pi\nu)^2}\, \frac{w(\nu)\,d\nu}{R^2+(1/2\pi\nu C)^2}, \tag{38} \]
or
\[ \overline{q^2} = \int_0^\infty \frac{C^2 w(\nu)\,d\nu}{4\pi^2\nu^2 R^2 C^2+1}. \tag{39} \]
Let us take into account relation (37). Substituting (37) into (39), we have:
\[ \overline{q^2} = C\int_0^\infty \frac{RC f(\nu,T)\,d\nu}{4\pi^2\nu^2 R^2 C^2+1}. \tag{40} \]
Let \(R\) not depend on frequency. Then the integrand depends on two parameters: the temperature \(T\) and the product \(RC\). Consequently, the integral is some function \(\Psi\) of two arguments: \(T\) and \(RC\), and
\[ \overline{q^2}=C\Psi(T,RC). \tag{41} \]
Let us now also take into account (25). We have:
\[ C\Psi(T,RC)=\Phi(CT). \tag{42} \]
Since the right-hand side does not depend on \(R\), the left-hand side also must not depend on \(R\); consequently, \(\Psi\) does not depend on the argument \(RC\):
\[ \Psi(T,RC)=\psi(T). \tag{43} \]
Substituting (43) into (42), we obtain:
\[ C\psi(T)=\Phi(CT). \tag{44} \]
Since the left-hand side of (44) depends linearly on \(C\), the right-hand side must also depend linearly on \(C\), and consequently also on its argument \(CT\). Thus,
\[ \Phi(CT)=\alpha CT, \tag{45} \]
where \(\alpha\) is a constant (the same for all capacitors), or
\[ \overline{q^2}=\alpha CT, \tag{46} \]
or also
\[ \boxed{\frac{\overline{q^2}}{2C}=\frac{\alpha T}{2}.} \tag{47} \]
Thus, the theory of alternating currents + relation (25) + relation (37) lead to the conclusion that in \(RC\)-systems with frequency-independent \(R\) and \(C\), the mean energy due to charge fluctuations is proportional to the temperature, and the coefficient of proportionality does not depend on \(R\) or \(C\).
B. Substituting (46) into (40), we obtain, after division by \(CT\):
\[ \int_0^\infty \frac{\tau F(\nu,T)\,d\nu}{4\pi^2\nu^2\tau^2+1}=\alpha, \tag{48} \]
where the notations
\[ \tau=RC,\qquad F(\nu,T)=\frac{f(\nu,T)}{T} \tag{49} \]
have been introduced.
The quantity \(\alpha\) does not depend on \(\tau\) (see Section A). Equation (48) is an integral equation which the function \(F(\nu,T)\) must satisfy. The unique solution of this equation is
\[ F(\nu,T)=4\alpha\;{}^{*}). \tag{50} \]
\({}^{*})\) Proof: Introducing new variables
\[ x=\frac{1}{2\pi\tau},\qquad u=\frac{\nu}{x}, \]
we rewrite (48) in the form
\[ \int_0^\infty \frac{F(ux,T)}{u^2+1}\,du=2\pi\alpha. \tag{a} \]
Differentiating (a) with respect to \(x\) and denoting
\[ v=(ux)^2,\qquad \Phi(v,T)=F'_{\nu}(ux,T), \tag{б} \]
we obtain:
\[ \int_0^\infty \frac{\Phi(v,T)\,dv}{v+x^2}=0. \tag{в} \]
Equation (в) is a degenerate case of the Stieltjes integral equation (the right-hand side is equal to 0). It has the unique solution (see, for example, \(^{4}\), p. 404)
\[ \Phi(v,T)=0, \]
whence, on the basis of (б),
\[ F(\nu,T)=\chi(T). \tag{г} \]
Substituting (г) into (a), we obtain:
\[ \chi(T)\int_0^\infty \frac{du}{u^2+1}=2\pi\alpha, \]
i.e.
\[ F(\nu,T)=\chi(T)=4\alpha. \]
From this, on the basis of (49), (37), there follows the relation
\[ w = 4aRT. \tag{51} \]
This is (up to the interpretation of the constant \(a\)) Nyquist’s formula
\[ w = 4kRT. \tag{52} \]
6. CONCLUDING REMARKS
Let us consider a system consisting of radiation and conductors. Let the system be in thermodynamic equilibrium. Then, as is known, from Nyquist’s formula (52) (taking into account the radiation resistance of the conductors) + classical electrodynamics + the second principle of thermodynamics, one can obtain, for the distribution of energy in the radiation spectrum, the Rayleigh–Jeans formula (see, for example, \(^{5}\)).
In the usual derivations of Nyquist’s formula or of the Rayleigh–Jeans formula, as well as of the assertion expressed by formula (47), a statistical assertion on the uniform distribution of energy over the degrees of freedom is used explicitly. In the usual derivation of Nyquist’s formula, the transition from thermodynamic positions to statistical ones occurs immediately after obtaining formula (37).
Here it has been proved that formulas (47) and (51) are consequences of the initial relations (25), (37), (39). As it seems to us, this circumstance is of some interest independently of the arguments that led to the initial relations. As for these arguments themselves, the following may be said: either they implicitly contain assumptions equivalent to the assertion of the uniform distribution of energy over the degrees of freedom, or else, without resorting to this assertion, one can advance in constructing the theory of thermal fluctuations and thermal radiation considerably further than is commonly believed. We hope to return to this question.
CITED LITERATURE
- H. Nyquist, Phys. Rev., 32, 110 (1928).
- G. A. Lorentz, Statistical Theories in Thermodynamics, ONTI (1935).
- A. Einstein, Phys. Zeits., 10, 185, 817 (1909).
- E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Gostekhizdat (1948).
- R. E. Burgess, Proc. Phys. Soc., 53, I, 293 (1941).