Non-Anonymous Referee Reply to Tsallis's Comment

Benjamin P. Vollmayr-Lee and Erik Luijten

Main Page | Nonintegrable Interactions

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

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

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

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.

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