ONCE AGAIN ON THE APPLICATION OF THERMODYNAMICS TO ELECTRICAL FLUCTUATIONS

Acquaintance with the article by M. L. Levin1 has led me to the conclusion that in it the question of the thermodynamic treatment of electrical fluctuations is not only not “set forth with due clarity,” but, on the contrary, is treated in an entirely erroneous manner. It therefore seems necessary once again, if only briefly, to dwell on this range of questions.

G. S. Gorelik2 proposed a derivation of the classical Nyquist formula based, as it seemed at first sight, only on the use of thermodynamics and the theory of alternating currents. The result obtained in 2, however, is paradoxical, since it contradicts, in particular, quantum theory and therefore must be connected with certain additional assumptions. In my article3 (cited below as I), among some other questions, G. S. Gorelik’s paradox is also considered, and it is shown that the derivation of the Nyquist formula proposed in 2 is indeed not purely thermodynamic, but also contains a very strong assumption that the interaction energy between an arbitrarily strongly damped “subsystem” (circuit) and its surroundings is equal to zero (for more detail and precision see I)*). Only such an assumption,

*) I take this opportunity to point out misprints noticed in I. In formula (1.3), in the first integral the upper limit is equal to \(T\), and not to \(\infty\). In formula (1.4), the factor \(4\pi\) is in the denominator, and not in the numerator. In formula (2.6), in the exponent before \(i\) a minus sign must be placed. At the bottom of p. 363, in the expres-

which can correspond to reality only in the classical region (the case of instantaneous impulses), makes it possible to apply thermodynamics to the circuit in the way this is done in [^2] (and also in [^1]). In the quantum region, neglect of the interaction under strong damping is impossible, for which reason the conclusion [^2, ^1] is applicable only in the classical case and is not, in this form, thermodynamic in the usual sense of the word. The conclusion made in [^1] concerning the inapplicability of the thermodynamic consideration contained in [^2, ^1] to a strongly damped quantum subsystem rests first of all on the rigorously proved quantum Nyquist formula and on its consequences (see [^1] and [^4]). It follows from the quantum Nyquist formula that in the quantum region, if the resistance of the circuit \(R \ne 0\), then the mean values of the magnetic and electric energies \(\overline{K}\) and \(\overline{U}\) are not equal to one another and depend on \(R\). Meanwhile, in the thermodynamic approach the expressions \(\overline{K}\) and \(\overline{U}\) never depend on \(R\) (see [^1, ^2] and [^1], pp. 373–374) and, according to [^1], one always has \(\overline{K}=\overline{U}\). Thus, the inapplicability of thermodynamics is undoubted and is interpreted in [^1] as the result of a violation of the assumption of additivity of the energy of the “subsystem” and its surroundings, used both in the phenomenological construction of thermodynamics (see [^5], §§ 7, 16) and in its statistical justification.

What, then, does M. L. Levin assert? He reduces the whole matter to a single proposition, namely that “in the quantum region of frequencies the resistance cannot be regarded as independent of frequency.” It is precisely for this reason, according to [^1], that in the quantum case thermodynamics cannot be applied to the circuit and that the conclusions obtained in [^2, ^1] under the assumption that \(R=\mathrm{const}\) are invalid.

However, this main thesis of M. L. Levin—and, moreover, the only point in his article which, in my opinion, could be of interest—withstands no criticism whatever. Indeed, physical statements of the type of the assertion “the independence of \(R\) from \(\omega\),” as is well known, are not absolute in character, but mean only that for the given problem the dependence of \(R\) on \(\omega\) may be neglected. There are no grounds for believing that \(R=\mathrm{const}\) with absolute precision either in the classical or in the quantum regions, and the question is only what dependence of \(R\) on \(\omega\) may still be considered inessential. Meanwhile, M. L. Levin makes no estimates of the magnitude of the variation of \(R\) with \(\omega\) and confines himself to the general assertion that in the quantum region \(R=R(\omega)\); at the same time it is quite obvious that a sufficiently weak dependence of \(R\) on \(\omega\) must be inessential in any theory. From this alone the vulnerability of M. L. Levin’s position is clear, which will become quite beyond doubt from what follows. First of all, let us recall that the “paradoxical conclusions” cited in [^1, ^2] are based entirely on considering expressions of the form

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

in the expression for \(H_0\) there is \(F(t)\), not \(F(x)\). On p. 364, in line 8 from the top, in the expression for \(Z(\omega)\) there is \(\omega m\) instead of \(m\). On p. 365, in line 12 from the bottom, \(\Delta\omega \gg 1/\Delta t\) instead of \(\Delta\omega \gg \Delta\tau\). In the formulas at the bottom of p. 372 and at the top of p. 377, in one place the differentials \(dC\) and \(dL\), respectively, have been omitted.

Let us also point to the new works [^7, ^8], which appeared after article [^1] had gone to press and which concern the quantum theory of electrical fluctuations.

assuming that \(R=\mathrm{const}\) (it is also assumed, here and below, that the parameters \(L\) and \(C\) are likewise constant). The expression given for \(U(\omega)\), and analogous expressions at very high frequencies, are obviously unjustified, since in that case the introduction of the notions of resistance, capacitance, etc. is inadmissible. It is therefore in fact assumed that, for real systems, at least in the classical case, the region of very high frequencies does not affect the value of the integral, and that the resistance \(R\) is, with sufficient accuracy, constant not at all frequencies in the interval \((0,\infty)\), but only in some finite frequency region, where the difference of the subintegral expression from zero is appreciable. Such an assumption, typical for physical problems containing integrals with an infinite limit, has known grounds, since in the classical case

\[ f(\omega,T)=\frac{2kT}{\pi}=\mathrm{const} \]

and the subintegral function decreases rapidly with increasing \(\omega\). But, accepting this assumption (and otherwise there are no paradoxes at all and nothing to discuss), it is easy to see that the transition to the quantum region not only does not strengthen, but, on the contrary, weakens the requirements on the constancy of \(R\) with respect to \(\omega\). Indeed, on passing to the quantum case, which for a given system is attained as a result of a sufficiently strong lowering of the temperature, the condition for applicability of the classical theory \(\hbar\omega \ll kT\) is no longer fulfilled for all frequencies that are still important in computing \(\overline{U}\), and

\[ f(\omega,T)=\frac{2}{\pi}\cdot \frac{\hbar\omega}{\exp(\hbar\omega/kT)-1}<\frac{2kT}{\pi} \]

(see I; the term in \(f(\omega,T)\) independent of \(T\) is neglected, which is quite justified). Hence it is clear that in the quantum case, owing to the decrease of the factor \(f\) with increasing \(\omega\), the region of high frequencies contributes still less to the integral than in the classical case. It is true that with decreasing temperature the resistivity \(R\) and its frequency dependence may also change, but, relying on the properties of real metals well known from experiment and quantum theory, it is easy to show that this circumstance over wide limits does not change our conclusion (see also below).

In this point the scholastic character of M. L. Levin’s whole approach to the problem stands out especially clearly. After considering a certain model of a conductor (incidentally, a model having no relation to reality and discussed incorrectly) and concluding that \(R=R(\omega)\), M. L. Levin, without any estimates of the magnitude of the change of \(R\) with \(\omega\), not only draws far-reaching conclusions of the kind mentioned above, but at the end of his article states, moreover, that calculations using the Nyquist formula with \(R=\mathrm{const}\) (see I, pp. 366—369) give “results having no physical meaning.” Meanwhile it is enough to turn to real examples to be convinced of the contrary. Thus, for example, at

\[ T=0.1^\circ\ \mathrm{K} \]

the frequency range

\[ \omega>\omega_0=\frac{kT}{\hbar}\simeq 1.3\cdot 10^{10} \left(\lambda_0=\frac{2\pi c}{\omega_0}\simeq 15\ \mathrm{cm}\right) \]

makes practically no contribution to the values of \(K\) and \(\overline{U}\) calculated in I. Further, for metals the dispersion of conductivity is usually appreciable only at \(\omega>\nu\), where \(\nu\) is the collision frequency (see, for example, \({}^{6}\), Chap. IV), and in many cases, even at low temperatures, \(\nu>10^{12}\div 10^{13}\). Therefore, under the conditions considered, the calculations mentioned are quite legitimate, and in carrying them out one may in fact put \(R=\mathrm{const}\) with enormous accuracy. For example, for \(\nu\sim 3\cdot 10^{12}\), allowance for the dispersion of \(R\) is necessary only if an accuracy is required determined by a factor of order

\[ \exp\left(-\frac{\nu}{\omega_0}\right)\sim 10^{-100}\ (!). \]

One may doubt that

there will be found a physicist whom M. L. Levin will succeed in convincing of the fundamental significance of taking into account such a dispersion \(R\) for the theory of electric fluctuations.

For completeness it remains only to note that the “erroneous assertions” contained, according to M. L. Levin, in I consist “above all” precisely in the fact that, in the opinion of V. L. Ginzburg, “conceivable conductors whose resistance does not depend on the frequency are possible in the quantum region.” As is clear from what was said earlier, such conductors are not only conceivable, but actually exist—they are the most ordinary nonsuperconducting metallic conductors at a sufficiently low temperature (in order that there be no substantial skin effect, the conductors must, in addition, be sufficiently thin).

Finally, one cannot pass over in silence M. L. Levin’s attempt, in essence, to deny the limited applicability of thermodynamics on the grounds that clarification of the regularity of its application in one case or another “leads ultimately to an undeserved discrediting of phenomenological thermodynamics.” Yes, I do indeed believe that thermodynamics cannot be used “without thinking,” and in the case of applying thermodynamics to electrical fluctuations we have a vivid example of this. The history of physics knows other such examples as well—it is enough to recall the “heat death” of the universe and the limitations of the nonstatistical interpretation of the second law. Therefore M. L. Levin’s defense of the unlimited application of phenomenological thermodynamics can only cause surprise. This surprise increases still more if one takes into account that M. L. Levin himself, in his article, asserts—although from erroneous positions—that in the quantum region “a thermodynamic treatment of electric fluctuations is impossible” (since, in his opinion, in this case always \(R=R(\omega)\)).

The limitation of space granted to me by the editors of UFN makes impossible an analysis of a number of other critical remarks and assertions by M. L. Levin, whose article abounds in logical and factual inaccuracies. There is, however, no special need for such an analysis, since I am sure that an attentive reader, having familiarized himself with articles I and the present note, will easily convince himself both of the groundlessness of M. L. Levin’s criticism of article I (see *), and, most importantly, of the untenability—shown above—of his own argumentation.

At the same time I would like to emphasize that I by no means consider the exposition in I of the question of the application of thermodynamics to electrical fluctuations sufficiently complete and rigorous from all points of view. On the contrary, in this area and, in general, in the question of the limits of applicability of the thermodynamic approach from the standpoint of quantum statistics, there are points requiring further analysis. However, I did not consider it necessary in the past, and do not intend in the future, to engage in such an analysis, because this whole problem seems to me to have only very slight, and moreover only methodological or perhaps pedagogical, interest.

V. L. Ginzburg

*) As an example I shall point out that in I the skin effect is assumed to be absent, as is evident from what is said on p. 359 and in the note to p. 365; meanwhile in his criticism M. L. Levin assumes that the skin effect is taken into account. Of the remarks made in I, I agree only that, in normalizing the spectral densities, it is more convenient not to write the coefficient \(4\pi\), and that there are no grounds for the conclusion made in I on p. 365 about the convergence of the quantity \(\overline{\xi^2}\) when the instantaneousness of collisions is taken into account. Both these points, however, are quite secondary.

References

1. M. L. Levin, see UFN, this issue, p. 486.
2. G. S. Gorelik, UFN 44, 33 (1951).
3. V. L. Ginzburg, UFN 46, 348 (1952).
4. H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).
5. M. A. Leontovich, Introduction to Thermodynamics, Gostekhizdat (1951).
6. A. Wilson, Quantum Theory of Metals, GTTI (1941).
7. J. L. Jackson, Phys. Rev. 87, 470 (1952).
8. J. Weber, Phys. Rev. 90, 977 (1953).