LETTERS TO THE EDITOR

ON THE APPLICATION OF THERMODYNAMICS TO ELECTRICAL FLUCTUATIONS*)

In the March 1954 issue of Uspekhi Fizicheskikh Nauk my article on the thermodynamics of electrical fluctuations and the reply by V. L. Ginzburg were published. The entire physical argumentation of this reply rests on the “real” example, refuting M. L. Levin’s “scholastic” reasoning: a conductor that is thin [in comparison with the thickness of the skin layer] at a temperature of 0.1° K, for which the classical frequency region lies below \(10^{10}\). Using the known estimates of the number of collisions in metals, V. L. Ginzburg concludes that the dispersion of the conductivity \(\sigma\), and with it also of the resistance \(R\), begins only at frequencies of order \(10^{12}\). However, such a conclusion, as is well known to V. L. Ginzburg [see his review article in the November 1950 issue of Uspekhi Fizicheskikh Nauk], is unfounded. At limiting temperatures, even at meter wavelengths, the skin-layer thickness \(\delta\) is smaller than the mean free path \(l\). Consequently, the thickness of the conductor \(a\) is also small in comparison with \(l\). Therefore one cannot in general speak of the conductivity \(\sigma\), which is a coefficient in Ohm’s law, valid only for \(l \ll \delta\), \(l \ll a\).

Since the thickness of the conductor is small in comparison with the mean free path, to find the resistance \(R\) one must use the methods of electron theory. Without anticipating the results of such microscopic calculations, let us note, however, that—since at low radio frequencies \(\delta \gg l\), while at high ones \(\delta \ll l\)—there is no basis for asserting in advance that the resistance \(R\) will be constant over the entire radio-frequency interval.

M. L. Levin

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In note 1, in my opinion, the erroneousness of the basic proposition contained in M. L. Levin’s article 2 was clarified. However, in his letter 3 M. L. Levin completely bypasses the questions discussed earlier 1, 2, makes new incorrect assertions, and at the same time, without any grounds for doing so, declares that the “untenability” of my own conclusions is “well known” to me. All this compels me to make several remarks.

1. The basic assertion contained in M. L. Levin’s article reduced to the fact that “in the quantum frequency region it is impossible, even without taking account of the skin effect, to regard the resistance as independent of frequency” 2. In 1 it was,

*) From the editors. In publishing the letters of M. L. Levin and V. L. Ginzburg, the editors consider the discussion on the application of thermodynamics to electrical fluctuations concluded.

first, it is indicated that M. L. Levin’s approach to resolving the question of the dependence of the resistance \(R\) on the frequency \(\omega\) is not physical, since M. L. Levin gives no quantitative estimates of the magnitude of the change of \(R\) with \(\omega\); at the same time, a sufficiently weak dependence of \(R\) on \(\omega\) cannot violate the applicability of calculations in which resistance is assumed constant. Second, and this is the main point, in [1] it is shown that the basic assertion cited by M. L. Levin is erroneous, as is, in my view, quite obvious from the following considerations. We enter the quantum region only when \(\hbar\omega \gg kT\), where \(\omega\) is the frequency under consideration and \(T\) is the absolute temperature. Therefore, by choosing a sufficiently low temperature, one can make the quantum region encompass arbitrarily low frequencies, at which the change of \(R\) with \(\omega\) is sufficiently small, as is well known both from theory and from experiment. Thus there is no basis for denying, as M. L. Levin does [2], the fundamental possibility of realizing a circuit lying in the quantum region and possessing a constant resistance.

All this is contained in [1], but is simply ignored by M. L. Levin, who wants to regard as physical only the specific example given in [1], which has a purely illustrative character*). This example, however, is also entirely correct, as will be said below. As for whether the argumentation set out above and in [1] is physical or not, I leave this to the readers to judge. Here I should merely like to point out that the mention in [1] of “heat death” is not felicitous, since it may, if desired, be incorrectly interpreted. I must therefore note that I had no intention of attributing to M. L. Levin a defense of the ideas of the “heat death” of the Universe, and this question was mentioned in [1] only as an example having no direct bearing on the problem under discussion.

1. As is already clear from the statement by M. L. Levin quoted above, in his article [2] he did not take the skin effect into account. Therefore, naturally, in [1] the question of the skin effect was not discussed either, although the corresponding qualification was made. In his letter [3] M. L. Levin completely abandons references to quantum phenomena (and thereby, as far as I can judge, in fact acknowledges the erroneousness of his article [2]), but brings to the fore precisely the question of the skin effect. It is easy to see, however, that allowing for the skin effect cannot in any way change the conclusion of [1, 4] concerning the possibility in the quantum region of regarding the resistance as completely independent of frequency. This would seem to be perfectly clear from what was said earlier, since at a sufficiently low temperature quantum phenomena set in at a sufficiently low frequency, when the role of the skin effect can be made arbitrarily small. The question of the anomalous character of the skin effect for thin specimens, on which only M. L. Levin dwells in his letter [3], has no bearing on the essence of the problem under discussion, since, by varying the temperature, the material (conductivity) of the wire, and its radius, one can always choose conditions under which the skin effect in the circuit considered in [1, 2, 4] is normal.

The preceding remarks, in my opinion, exhaust the question. But one cannot fail also to touch here on the example from [1] discussed in M. L. Levin’s letter [3], since this example alone is regarded as “physical” and at the same time as incorrect, allegedly following from my own article [5]. In reality, the question of the resistance of thin wires is not even raised in [5]. If, however, one considers the question of whether Ohm’s law is valid for the direct current flowing through a wire, and the question of interest—

*) In this example (see [1], p. 496, line 10 from the bottom), \(\omega \ll \omega_0\) is incorrectly printed instead of \(\omega \gg \omega_0\).

the resistance of the wire \(R\) has a quite definite meaning; there is no need to make use of the region of the anomalous skin effect and of Ohm’s law in its ordinary understanding of conductivity (see, for example, 6). Further, it can be shown that the frequency dependence of \(R\) for thin wires under conditions of the anomalous skin effect is weak, analogously to what takes place in the case of the normal skin effect. Therefore the example cited in 1 meets with no objections in this respect either. To give the corresponding estimates here is difficult because of lack of space and, chiefly, quite unnecessary in view of the possibility mentioned above of indicating other examples in which the skin effect is normal and the determination of the dependence \(R(\omega)\) is carried out by well-known formulas.

From all that has been said it is clear that the letter of M. L. Levin 3 contains only new erroneous assertions, but in no way leads to a change in the conclusions drawn earlier 1 with regard to his article 2.

V. L. Ginzburg

References Cited

1. V. L. Ginzburg, UFN 52, 494 (1954).
2. M. L. Levin, UFN 52, 486 (1954).
3. M. L. Levin, UFN (see this issue, p. 146).
4. V. L. Ginzburg, UFN 46, 348 (1952).
5. V. L. Ginzburg, UFN 42, 333 (1950).
6. E. H. Sondheimer, Adv. in Phys. 1, 1 (1952).