Main Page | Nonintegrable Interactions
To: pre@aps.org Date: Mon, 26 Nov 2001 09:11:41 -0600 (CST) We think this Comment is unsuitable for publication in Physical Review E. According to the guidelines, the Comments section is "solely for publications that criticize or correct papers," which we understand to mean that a valid criticism or correction is required to justify publication. This Comment contains what might be considered a valid supplemental observation, as we will clarify below, but the main points raised in criticism of our paper we find to be unjustified and unreasonable. There are also some relatively minor points raised which criticize our choice of words. For the most part, we find these to be unjustified as well. There is one case, at least, where our wording could have been improved, but we believe this case is insignificant enough that it fails to "be directed to the physics in the paper being criticized" (quoting from the guidelines). To support our opinion, we first provide a statement on the main points of our objection to this Comment, and then follow this with a more detailed description. Finally, we provide at the end a brief introduction to the topic for the benefit of any non-specialist reviewers, which also serves to clarify our terminology. Our Main Objections ------------------- The author of the Comment (henceforth referred to by name for brevity) claims that we made a "misleading oversimplification" that can "severely misguide the reader" by neglecting the "crucial point" of equilibrium relaxation times. We did indeed neglect equilibrium relaxation times and simply studied the equilibrium thermodynamics. Tsallis also claims that our choice of subject was "rather surprising since the key role of t has been strongly emphasized in several occasions." That is, Tsallis contends that we should have known not to study the equilibrium thermodynamics because of some established literature indicating that only metastable states are accessible. However, Tsallis fails to support his claims that the equilibrium relaxation times are crucial, or that our omission is one of negligence (as opposed to simply following the established interests of a field of study, including many papers of Tsallis himself). Regarding the equilibrium relaxation times, Tsallis presents evidence for some models with nonintegrable interactions (the rotor model of his ref. [8,11] and a related study in his ref. [7]) that relaxation times can diverge with the system size. He concludes from this that our equilibrium solution for nonextensive thermodynamics requires waiting times that "could be longer than the age of the universe" when N is of the order of Avogadro's number. His estimation is severely flawed on two accounts and a more accurate estimate (details provided below) demonstrates equilibration times on the order of seconds. This is curiously long for a molecular fluid, but it is a negligible experimental challenge compared to the 10^23 K temperatures necessary for nonextensive thermodynamics. Tsallis accepts these unphysical temperatures and yet contends that our equilibrium solution is not "physically meaningful" based on a waiting time of seconds. We find this unreasonable. In our view, the nonintensive temperatures (scaled with some positive power of the system size) of nonextensive thermodynamics renders it trivial (as we showed, the free energy may be obtained exactly) and unphysical. Any interest in this system should derive its motivation from essentially mathematical purposes. The second main criticism is that we were negligent not to mention these curiously long relaxation times because this issue has been "profusely detected and stressed in the related literature." To support this claim Tsallis cites our reference [31] and his refs. [3,6-8]. The latter group of references is truly misleading. Reference [3] contains a group of papers all but one of which either contain no reference to N,t limits or time at all, or else post-date our paper. The single exception is the 1999 Brazilian Journal of Physics paper (also our ref. [31]), which we discuss at the end of the paragraph. Tsallis's ref. [6] is completely inappropriate, as it regards a system we did not study: we considered pairwise interactions that decayed as 1/r^tau for large r, with 0 <= tau <= d, while this paper considers only negative values of tau. Tsallis's ref. [7,8] both find relaxation times that grow with the system size, but none of the cited papers in these references have made the final step of correspondence to nonextensive thermodynamics, which appears to be made for the first time by us (in the next section). Finally, our ref. [31] is a 35 page paper in which one paragraph and one figure contain speculative comments on the non-commutivity of the N->infinity and t-> infinity limits. There is no evidence presented to support the speculation in this paper. Perhaps there is more literature to support Tsallis's claims, but we find the literature that he cites to be strikingly insufficient for support his case that we were negligent. To make one further observation, many papers by Tsallis, including our ref. [8] which was the primary stimulus for our work, study the equilibrium thermodynamics of exactly the same system we considered. To conclude this section, we add that the relaxation times that grow with the system size as found in Tsallis's refs. [7,8] are of some interest for this academic field of study. While a connection between the statistics of the metastable state and Tsallis's q-statistics has not been fully demonstrated, we can understand the author's interest in this study and raise no objections to it. Our objections lie with the misinterpretation of these results and the misrepresentation of the literature presented in Tsallis's comment, as we describe above and below. More Detailed Description ------------------------- * Relaxation Times. Tsallis's references [7,8] find relaxation times that grow linearly with the number of particles N for the case tau=0. These works study a system where the temperature is independent of system size but the coupling constant in the potential energy is scaled by a factor of 1/N. However, nonextensive thermodynamics is instead concerned with N-independent coupling constants and a "temperature" that scales as a positive power of N (for tau=0 the nonintensive temperature grows linearly with N). Of course, there is a direct mathematical equivalence of the two viewpoints, but this equivalence is direct only when ALL the energy terms are scaled by the power of N. In ref. [7,8] the kinetic energy terms are NOT scaled by 1/N, so they do not yet correspond to nonextensive thermodynamics. We can put the factor of N back into the denominator of the kinetic energy to restore the equivalence simply by rescaling time by a factor of sqrt(N). Thus, we end up with the result that for nonextensive thermodynamics the relaxation times for tau=0 grow as sqrt(N), rather than linearly with N. This correspondence was not in the literature when we published our paper, and in fact does not appear to be in the literature today. This is curious, given that Tsallis has repeatedly stated his preference for nonintensive temperatures over rescaled couplings. We add that neither case, N-dependent couplings or N-dependent temperature, has any particular value over the other. However, it is a fact that the two cases differ in their time scale. It is also a fact that we made clear that we considered the nonintensive temperature formulation of nonequilibrium thermodynamics (see our second paragraph, or the first two paragraphs of our section IV), thus it would seem necessary to mention the nonintensive temperature results (relaxation going as sqrt(N) for tau=0) when criticizing our work. Even with a sqrt(N) dependence on the relaxation time, Tsallis is formally correct that the N->infinity limit leads to an unreachable equilibration time. But it also corresponds to an unreachable temperature, so let us keep N finite but large and see what results. Tsallis cites N=10^23, for which he must be thinking of molecular particles. His refs. [8,11] find relaxation times of roughly 10*sqrt(N) (translated to nonextensive thermodynamics), but uses a unit of time derived from setting the molecular rotational inertia and molecular coupling to unity. If we put in the typical molecular values of 10^(-24) eVs^2 for the rotational inertia and 1 eV for the couplings, we find the unit of time in Tsallis's refs. [8,11] corresponds to 1 picosecond. Thus for Avogadro's number of particles, we have relaxation times on the scale of seconds. To reach his conclusion that relaxation time is longer than the age of the universe, Tsallis evidently overlooked both the proper N-scaling for nonextensive thermodynamics as well as the characteristic time scales for molecules. There is another point to be made: some systems with nonintegrable interactions exhibit finite relaxation times as N->infinity, including the (equilibrium) molecular dynamics study of Tsallis and Curilef (our ref. [8]). Another simple example is the Glauber dynamic Ising model: the ratio of flip rates for a spin is given by the ratio of Boltzmann factors. Because of the non-intensive temperature, the Boltzmann factor remains of order unity in the thermodynamic limit even for nonintegrable interactions. For Glauber dynamics there is still a phenomenological parameter characterizing the flip time for a free spin, but there is no reason to assume that it depends on the system size. Thus there are at least two cases where the equilibrium relaxation time is of order unity, i.e. independent of N, which weakens considerably the case made by Tsallis that our results are not "physically meaningful." It would be interesting to understand the difference between these cases and those of Tsallis's ref. [7,8]. We believe the difference is probably due to how the degrees of freedom are coupled to the temperature bath: in both Glauber dynamics and Curilef and Tsallis's MD study (our ref. [8]) the degrees of freedom are coupled additively to a distinct temperature bath. In Tsallis's ref. [7,8] there is no distinct temperature bath, rather the other N-1 particles serve as the bath for a given particle. Since these degrees of freedom are coupled non-additively, this latter case is fundamentally different. And again, we have not found a discussion of this distinction in this Comment or its cited literature. * Criticisms of our terminology Tsallis takes exception to our statement that Boltzmann-Gibbs (BG) statistics are "sufficient" for the class of nonintegrable interactions we studied. This perhaps a misunderstanding due to the imprecision of language. Our statement was intended as a response to statements such as those in our ref. [30], by Nobre and Tsallis, that "we no doubt need nonextensive statistics" for interactions 1/r^tau with 0 < tau < d (this paper contains no mention of time or nonequilibrium states). We had a direct calculation showing that, in fact, there was no need to consider Tsallis's nonextensive statistics; rather, Boltzmann-Gibbs statistics aided by the non-intensive temperatures encountered no difficulty with nonintegrable interactions and reproduced directly all the scaling results of nonextensive thermodynamics. Thus we said that BG statistics were "sufficient for a standard description." We stand by the term. Tsallis also provides in his endnote [4] a defense of his choice of the term "long-range interactions" in place of "nonintegrable interactions," a choice we were critical of in our paper. We stand by our reasons, which requires us to comment on this endnote. Tsallis contends that our statement that integrable interactions are "considerably more important" than nonintegrable interactions (stated in the context of purely attractive large distance interactions, i.e. excluding Coulombic interactions where charge neutrality is sufficient for a well-defined thermodynamic limit) is merely a communication of our preferences. We believe our statement is simply fact, and to believe anything to the contrary would require a large burden of proof that Tsallis has not come close to producing. However, he also cites our use of "true" long-range interactions (a subclass of integrable interactions) as expressing our preferences, and here we agree. A better choice of words on our part would be "the long-range interactions used in study of critical phenomena" or something similar. * General remark on layout of comment To our view, the comment is set up in a way that does not follow the guidelines of Physical Review E. Large parts reiterate (rather than refer to) results that have been published before by the author, specifically the example of planar rotors with nonintegrable interactions (see ref. [8,11]). Also, large parts of the comment are NOT directed to the physics in the paper being criticized (namely our Kac-potential treatment of nonintegrable interactions) but are statements on other matters (e.g., side remarks on turbulence, electron-positron annihilation, motion of living bodies such as Hydra viridissima, and many other topics mentioned on p. 7 of this comment, and the associated references [20-27], which alone take a full page). Also, the page of quotations from the works of Krylov, Balescu, and Dorfman (contained in Refs. [28-30]) may or may not be interesting in its own right, but surely is absolutely unrelated to our work. To a large part, this comment does not seriously comment on our paper at all, but is an - admittedly very eloquent! - advertisement for the concept and applications of nonextensive statistical mechanics in general. Background Material and Definitions ----------------------------------- For the benefit of potential reviewers not familiar with the nonextensive thermodynamics, we offer a brief summary, which also serves to define our terminology here: "Tsallis statistics" or "q-statistics" or "nonextensive statistics" refers to a generalized entropy introduced by Tsallis (ref. [6] of our paper, with some later revisions) and its consequences for statistics. "Nonintegrable interactions" refers to a pair potential coupling microscopic degrees of freedom that does not give a finite value when integrated over all space. A system of particles or spins interacting via a nonintegrable potential (with purely attractive large distance tails, i.e., excluding the Coulombic case) will have an energy per particle that cannot be bounded from below, and so the standard thermodynamic limit does not exist. "Nonextensive thermodynamics" refers to recent studies of these nonintegrable systems in which a type of thermodynamic limit has been restored. While the energy per particle cannot be bounded from below, it can be bounded by some negative constant times a power of the system size or number of particles. Thus a non-intensive temperature is introduced that contains a factor of this power of the system size, with the result that the ratio E/kT per particle remains finite in the thermodynamic limit. Many of these papers presume a connection between Tsallis statistics and nonextensive thermodynamics. To illustrate with an example, gravitational systems in three dimensions have a non-integrable potential that goes as 1/r. In this case, a collection of N particles (with a short distance regulator) will have a lower bound on the energy per particle that scales as -const*N^(2/3) for large N. The nonextensive thermodynamics approach to this system is to consider temperatures that grow with the system size as T=const*N^(2/3). The diverging temperature greatly simplifies the study of this system, in fact it allows for an exact solution of the free energy, as we showed. This case should not be confused with more realistic studies of gravitational systems which are limited to finite temperatures.
Main Page | Nonintegrable Interactions
This page maintained by Ben
Vollmayr-Lee. Last updated January 11, 2002.