Notebooks
http://bactra.org/notebooks
Cosma's NotebooksenTsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions
http://bactra.org/notebooks/2021/05/13#tsallis
<P>A standard assumption of <a href="stat-mech.html">statistical mechanics</a>
is that quantities like energy are "extensive" variables, meaning that the
total energy of the system is proportional to the system size; similarly the
entropy is also supposed to be extensive. Generally, at least for the energy,
this is justified by appealing to the short-range nature of the interactions
which hold matter together, form chemical bonds, etc. But suppose one deals
with long-range interactions, most
prominently <a href="astrophysics.html">gravity</a>; one can then find that
energy is <em>not</em> extensive. This makes the life of the statistical
mechanic much harder.
<P>Constantino Tsallis is a physicist who came up with a supposed solution,
based on the idea of <a href="max-ent.html">maximum entropy</a>. One popular
way to derive the (canonical) equilibrium probability distribution is the
following. One purports to know the average values of some quantities, such as
the energy of the system, the number of molecules, the volume it occupies, etc.
One then searches for the probability distribution which maximizes the entropy,
subject to the constraint that it give the right average values for your
supposed givens. Through the magic of Lagrange multipliers, the
entropy-maximizing distribution can be shown to have
the <a href="exponential-families.html">right, exponential, form</a>, and the
Lagrange multipliers which go along with your average-value constraints turn
out to be the "intensive" variables paired with (or "conjugate to") the
extensive ones whose means are constrained (energy <=> temperature, volume <=>
pressure, molecular number <=> chemical potential, etc.). But, as I said, the
entropy is an extensive quantity. What Tsallis proposed is to replace the
usual (Gibbs) entropy with a new, non-extensive quantity, now commonly called
the Tsallis entropy, and maximize <em>that</em>, subject to constraints. There
is actually a whole infinite family of Tsallis entropies, indexed by a
real-valued parameter <em>q</em>, which supposedly quantifies the degree of
departure from extensivity (you get the usual entropy back again
when <em>q</em> = 1). One can then grind through and show that many of the
classical results of statistical mechanics can be translated into the new
setting. What has really caused this framework to take off, however, is that
while normal entropy-maximization gives you exponential, Boltzmann
distributions, Tsallis statistics give you power-law, Pareto distributions, and
<a href="power-laws.html">everyone loves a power-law</a>. (Strictly speaking,
Tsallis distributions are type II generalized Pareto distributions, with
power-law tails.) Today you'll find physicists applying Tsallis statistics to
nearly anything with a heavy right tail.
<P>I have to say I don't buy this at all. Leaving to one
side <a href="max-ent.html">my skepticism about the <em>normal</em> maximum
entropy story</a>, at least as it's usually told (e.g. by E. T. Jaynes), there
are a number of features which make me deeply suspicious of Tsallis statistics.
<ol>
<li>It's simply not true that one maximizes the Tsallis entropy subject
to constraints on the mean energy \( \langle E \rangle =\sum_{i}{p_i
E_i} \). Rather, to get things to work out, you have to fix the value of a
"generalized mean" energy, \( { \langle E \rangle }_{q} = \sum_{i}{p_i^q E_i}
/ \sum_{i}{p^q_i} \). (This can be interpreted as replacing the usual
average, an expectation take with respect to the actual probability
distribution, by an expectation taken with respect to a new, "escort"
probability distribution.) I have yet to encounter anyone who can explain why
such generalized averages should be either physically or probabilistically
natural; the usual answer I get is "OK, yes, it's weird, but it works, doesn't
it?"
<li>There is
no <a href="information-theory.html">information-theoretic</a> justification
for the Tsallis entropy, unlike the usual Gibbs entropy. The Tsallis
form <em>is</em>, however, a kind of low-order truncation of the Rényi
entropy, which <em>does</em> have information-theoretic interest. (The Tsallis
form has been independently rediscovered many times in the literature, going
back to the 1960s, usually starting from the Rényi entropy. A brief
review of the "labyrinthic history of the entropies" can be found in one of
Tsallis's
papers, <a href="http://arxiv.org/abs/cond-mat/0010150">cond-mat/0010150</a>.)
Maximizing the Rényi entropy under mean-value constraints leads to
different distributions than maximizing the Tsallis entropy.
<li>I have pretty severe doubts about the backing story here, about
long-range interactions leading to a non-extensive form for
the <em>entropy</em>, particularly when, in derivations which begin with such a
story, I often see people blithely factoring the probability that a system is
in some global state into the product of the probabilities that its components
are in various states, i.e., assuming independent sub-systems.
<li>There are alternative, non-max-ent derivations of the usual
statistical-mechanical distributions; such derivations do not seem forthcoming
for Tsallis statistic. In particular, <a href="large-deviations.html">large
deviations</a> arguments, which essentially show how to get such distributions
as emergent, probabilistic consequences of individual-level interactions, do
not seem to ever lead to Tsallis statistics, <em>even</em> when one has the
kind of long-range interactions which, supposedly, Tsallis statistics ought to
handle.
<li>There is no empirical evidence that Tsallis statistics correctly
gives the microscopic energy distribution for any known system.
<li>Zanette and Montemurro have shown that you can get <em>any</em>
distribution you like out of the Tsallis recipe, simply by changing the
function whose generalized average you take as your given. The usual power-law
prescription only holds if you constrain either <i>x</i>
or <i>x</i><sup>2</sup>, but one of the more "successful" applications requires
constraining the generalized mean of \( x^{2\alpha}/2 -
c\mathrm{sgn}{x}({|x|}^{\alpha} - {|x|}^{3\alpha}/3) \), with <i>c</i> and \(
\alpha \) as adjustable parameters! (In fairness, I should point out that if
you're willing to impose sufficiently weird constraints, you can generate
arbitrary distributions from the usual max. ent. procedure, too; this is one of
the reasons why I don't put much faith in that procedure.)
</ol>
<P>I think the extraordinary success of what is, in the end, a slightly dodgy
recipe for generating power-laws illustrates some important aspects, indeed
unfortunate weaknesses, in the social and intellectual organization of
"<a href="complexity.html">the sciences of complexity</a>". But <em>that</em>
rant will have to wait for my book on <cite>The Genealogy of Complexity</cite>,
which, prudently, means waiting until I'm safely tenured. (Update, 2021: I am
indeed now safely tenured, but I have better, or at least more pressing, things
to do, so <a href="genealogy-of-complexity.html">enjoy</a>.)
<P>I should also discuss the "superstatistics" approach here, which tries to
generate non-Boltzmann statistics as mixtures of Boltzmann distributions,
physically justified by appealing to fluctuating intensive variables, such as
temperature. I will only remark that the superstatistics approach severes all
connections between the use of these distributions and non-extensivity and
long-range interactions; and that results in the statistical literature on
getting generalized Pareto distributions from mixtures of exponentials go back
to <a href="http://www.jstor.org/pss/2332475">1952</a> at least.
<P>Finally, it has come to my attention that some people are citing this
notebook as though it had some claim to authority. Fond though I am of my own
opinions, this seems to me to be deeply wrong. The validity of Tsallis
statistics, as a scientific theory, ought to be settled in the usual way, by
means of the peer-reviewed scientific literature, subject to all its usual
conventions and controls. It's obvious from the foregoing that I have pretty
strong beliefs in how that debate ought to go, and (this may not be so clear)
enough faith in the scientific community that I think, in the long run, it will
go that way, but no one should confuse my opinion with a scientific finding.
For myself, this page is a way to organize my own thoughts; for everyone else,
it's either entertainment, or at best an opinionated collection of pointers to
the real discussion.
<ul>Recommended, big picture:
<li>Tsallis & co. maintain a pretty comprehensive and
ever-growing <a href="http://tsallis.cat.cbpf.br/biblio.htm">bibliography</a>
on Tsallis statistics. This includes replies to many of the papers I list
here.
<li>Julien Barré, Freddy Bouchet, <a
href="http://perso.ens-lyon.fr/thierry.dauxois/">Thierry Dauxois</a> and
Stefano Ruffo, "Large deviation techniques applied to systems with long-range
interactions", <a
href="http://arxiv.org/abs/cond-mat/0406358">cond-mat/0406358</a> = <a
href="http://dx.doi.org/10.1007/s10955-005-3768-8"><cite>Journal of Statistics
Physics</cite> <strong>119</strong> (2005): 677--713</a> [What large deviation
results for long-range interactions look like]
<li>Christian Beck, "Superstatistics: Recent developments and
applications", <a
href="http://arxiv.org/abs/cond-mat/0502306">cond-mat/0502306</a>
<li>Freddy Bouchet and Thierry Dauxois, "Prediction of anomalous
diffusion and algebraic relaxations for long-range interacting systems, using
classical statistical mechanics", <a
href="http://dx.doi.org/10.1103/PhysRevE.72.045103"><cite>Physical Review
E</cite> <strong>72</strong> (2005): 045103</a>
= <a href="http://arxiv.org/abs/cond-mat/0407703">cond-mat/0407703</a>
<li>Thierry Dauxois, "Non-Gaussian distributions under scrutiny",
<a href="http://dx.doi.org/10.1088/1742-5468/2007/08/N08001"><cite>Journal of
Statistical Mechanics</cite> (2007) N08001</a>
<li>Brian R. La Cour and William C. Schieve, "A Comment on the Tsallis
Maximum Entropy Principle", <cite>Physical Review E</cite> <strong>62</strong>
(2000): 7494--7496, <a
href="http://arxiv.org/abs/cond-mat/0009216">cond-mat/0009216</a>
<li>Michael Nauenberg, "Critique of <em>q</em>-entropy for thermal
statistics", <cite>Physical Review E</citE> <strong>67</strong> (2003): 036114
[From the abstract: "[I]t is shown here that the joint entropy for systems
having <em>different</em> values of <em>q</em> is not defined in this
formalism, and consequently fundamental thermodynamic concepts such as
temperature and heat exchange cannot be considered for such systems. Moreover,
for <em>q</em> &neq; 1 the probability distribution for weakly interacting
systems does not factor into the product of the probability distribution for
the separate systems, leading to spurious correlations and other unphysical
consequences, e.g., nonextensive energy, that have been ignored in various
applications given in the literature." That the probabilities for sub-systems
do not factor is, I think, especially devastating, because almost all of the
work on the subject assumes that it <em>does</em>. See also comment by
Tsallis, <a href="http://arxiv.org/abs/cond-mat/305091">cond-mat/305091</a>,
and reply by Nauenberg, <a
href="http://arxiv.org/abs/cond-mat/0305365">cond-mat/0305365</a>, which I
believe to be correct.]
<li>Hugo Touchette, "Comment on 'Towards a large deviation theory for strongly correlated systems' ", <a href="http://arxiv.org/abs/1209.2611">arxiv:1209.2611</a>
<li>Damién H. Zanette and Marcelo A. Montemurro
<ul>
<li>"A note on non-therrmodynamical applications of
non-extensive statistics",
<a href="http://arxiv.org/abs/cond-mat/0305070">cond-mat/0305070</a>
= <cite>Physics Letters A</cite> <strong>324</strong> (2004): 383--387 [An
amusing and quite conclusive assault, culminating in a demonstration that you
can use the non-extensive formalism to "derive" any probability distribution
whatsoever.]
<li>"Thermal measurement of stationary nonequilibrium systems:
A test for generalized thermostatistics", <Cite>Physics Letters
A</cite> <strong>316</strong> (2003): 184--189 = <a
href="http://arxiv.org/abs/cond-mat/0212327">cond-mat/0212327</a> [And it
doesn't even work for for thermodynamic systems.]
</ul>
</ul>
<ul>Recommended, close-ups:
<li>Freddy Bouchet, Thierry Dauxois, Stefano Ruffo, "Controversy about
the applicability of Tsallis statistics to the HMF
model", <a href="http://arxiv.org/abs/cond-mat/0605445">cond-mat/0605445</a> =
<cite>Europhysics News</cite> <strong>37</strong> (2006): 9--10
<li>G. Baris Bagci, Thomas Oikonomou, "Do Tsallis distributions really originate from the finite baths?", <a href="http://arxiv.org/abs/1305.2493">arxiv:1305.2493</a>
<li>A. G. Bashkirov, "Comment on <a
href="http://dx.doi.org/10.1103/PhysRevE.66.046134">'Stability of Tsallis
entropy and instabilities of Rényi and normalized Tsallis entropies: A
basis for q-exponential distributions'</a>," <a
href="http://dx.doi.org/10.1103/PhysRevE.72.028101"><cite>Physical Review
E</cite> <strong>72</strong> (2005): 028101</a> [There is also a reply by
S. Abe, the author of the original article, which, predictably, I find
unconvincing: <a
href="http://dx.doi.org/10.1103/PhysRevE.72.028102"><cite>Physical Review
E</cite> <strong>72</strong> (2005): 028102</a>.]
<li>Alice M. Crawford, Nicolas Mordant, Andy M. Reynolds, Eberhard
Bodenschatz, "Comment on 'Dynamical Foundations of Nonextensive Statistical
Mechanics'", <a href="http://arxiv.org/abs/physics/0212080">physics/0212080</a>
<li>Peter Grassberger
<ul>
<li>"Temporal scaling at Feigenbaum points and
non-extensive thermodynamics", <a
href="http://arxiv.org/abs/cond-mat/0508110">cond-mat/0508110</a> =
<a href="http://dx.doi.org/10.1103/PhysRevLett.95.140601"><cite>Physical Review
Letters</cite> <strong>95</strong> (2005): 140601</a> [I can't resist quoting
the abstract in full, if only because I enjoy Prof. Grassberger's
no-quarter-asked-or-given tone: "We show that recent claims for the
non-stationary behaviour of the logistic map at the Feigenbaum point based on
non-extensive thermodynamics are either wrong or can be easily deduced from
well-known properties of the Feigenbaum attractor. In particular, there is no
generalized Pesin identity for this system, the existing 'proofs' being based
on misconceptions about basic notions of ergodic theory. In deriving several
new scaling laws of the Feigenbaum attractor, thorough use is made of its
detailed structure, but there is no obvious connection to non-extensive
thermodynamics." One point made here (but passed over in the abstract) is that
there are nearly as many estimates of the "right" value of the non-extensivity
parameter <em>q</em> at the period-doubling accumulation point as there are
papers on the system. This tends to reduce one's confidence that any of them
is a physically meaningful parameter.]
<li>"Proposed central limit behavior in deterministic dynamical systems", <a href="https://doi.org/10.1103/PhysRevE.79.057201"><cite>Physical Review E</cite> <strong>79</strong> (2009): 057201</a><
</ul>
<li>H. J. Hilhorst
<ul>
<li>"Central limit theorems for correlated variables: some critical remarks", <cite>Brazilian Journal of Physics</cite> <strong>39</strong>
(2000): 371--379, <a href="http://arxiv.org/abs/0901.1249">arxiv:0901.1249</a>
<li>"Note on a <i>q</i>-modified central
limit theorem", <cite>Journal of Statistical Mechanics</cite> (2010): P10023,
<a href="http://arxiv.org/abs/1008.4259">arxiv:1008.4259</a>
</ul>
<li>H. J. Hilhorst and G. Schehr, "A note on q-Gaussians and
non-Gaussians in statistical
mechanics", <a
href="http://dx.doi.org/10.1088/1742-5468/2007/06/P06003"><cite>Journal of
Statistical Mechanics</cite> (2007) P06003</a> [Analytical results on the
limiting distributions of certain sums of correlated random variables, supposed
to follow "q-Gaussians", but not actually doing so. It strikes me as
extraordinary that no one in this literature, on either side, pays any
attention to actual results in probability theory about generalizations of the
central limit theorem; one searches these bibliographies in vain for names like
Lévy and Rosenblatt.]
<li>B. H. Lavenda and J. Dunning-Davies, "Additive Entropies of
degree-q and the Tsallis
Entropy", <a href="http://arxiv.org/abs/physics/0310117">physics/0310117</a>
</ul>
<ul>Modesty forbids me to recommend:
<li>CRS, "Maximum Likelihood Estimation for q-Exponential (Tsallis)
Distributions", <a
href="http://arxiv.org/abs/math.ST/0701854">math.ST/0701854</a> [If you have to
use these things, you really should estimate their parameters this way, and not
try to fit curves to the sample distribution function.]
<li>CRS and Alessandro Rinaldo, "Consistency under Sampling of Exponential Random Graph Models", <a href="https://doi.org//10.1214/12-AOS1044"><cite>Annals of Statistics</cite> <strong>41</strong> (2013): 508--535</a>, <a href="http://arxiv.org/abs/1111.3054">arxiv:1111.3054</a> [While we didn't emphasize
it in the paper, considering our audience and primary application, one way to
understand this is as a study of the weirdness that can result within a
canonical ensemble where interactions are <em>not</em> short range, so that the total energy of a system divided into two large sub-systems doesn't break down into the internal, volume-proportional energy of each sub-system, plus a comparatively small term strictly from the boundary.]
</ul>
<ul>To read:
<li>Andrea Antoniazzi, Francesco Califano, Duccio Fanelli, and Stefano
Ruffo, "Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to
the Vlasov Equation", <a
href="http://dx.doi.org/10.1103/PhysRevLett.98.150602"><cite>Physical Review
Letters</cite> <strong>98</strong> (2007): 150602</a>
<li>R. Bachelard, C. Chandre, D. Fanelli, X. Leoncini and S. Ruffo,
"Abundance of Regular Orbits and Nonequilibrium Phase Transitions in the
Thermodynamic Limit for Long-Range Systems", <a
href="http://dx.doi.org/10.1103/PhysRevLett.101.260603"><cite>Physical Review
Letters</cite> <strong>101</strong> (2008): 260603</a>
<li>Fulvio Baldovin, Pierre-Henri Chavanis and Enzo Orlandini,
"Microcanonical quasistationarity of long-range interacting systems in contact with a heat bath", <a href="http://dx.doi.org/10.1103/PhysRevE.79.011102"><cite>Physical Review E</cite>
<strong>79</strong> (2009): 011102</a>
<li>Fulvio Baldovin and Enzo Orlandini
<ul>
<li>"Hamiltonian Dynamics Reveals
the Existence of Quasistationary States for Long-Range Systems in Contact with
a
Reservoir", <a
href="http://dx.doi.org/10.1103/PhysRevLett.96.240602"><cite>Physical Review
Letters</cite> <strong>96</strong> (2006): 240602</a>
= <a href="http://arxiv.org/abs/cond-mat/0603383">cond-mat/0603383</a> ["We
introduce a Hamiltonian dynamics for the description of long-range interacting
systems in contact with a thermal bath (i.e., in the canonical ensemble). The
dynamics confirms statistical mechanics equilibrium predictions for the
Hamiltonian mean field model and the equilibrium ensemble equivalence. We find
that long-lasting quasistationary states persist in the presence of the
interaction with the environment. Our results indicate that quasistationary
states are indeed reproducible in real physical experiments."]
<li>"Quasi-stationary states in long-range interacting systems
are incomplete equilibrium
states", <a href="http://arxiv.org/abs/cond-mat/0603659">cond-mat/0603659</a>
= <a href="http://dx.doi.org/10.1103/PhysRevLett.97.100601"><cite>Physical
Review Letters</cite> <strong>97</strong> (2006): 100601</a> ["Despite the
presence of an anomalous single-particle velocity distribution, we find that
ordinary Central Limit Theorem leads to the Boltzmann factor in Gibbs'
$\Gamma$-space. We identify the non-equilibrium sub-manifold of $\Gamma$-space
responsible for the anomalous behavior and show that by restricting the
Boltzmann-Gibbs approach to such sub-manifold we obtain the statistical
mechanics of the quasi-stationary states."]
</ul>
<li>Christian Beck, "Generalized information and entropy measures in physics", <a href="http://arxiv.org/abs/0902.1235">arxiv:0902.1235</a>
<li>Christian Beck, Ezechiel G. D. Cohen and Harry L. Swinney, "From
time series to superstatistics", <a
href="http://dx.doi.org/10.1103/PhysRevE.72.056133"><cite>Physical Review
E</cite> <strong>72</strong> (2005): 056133</a>
<li>Freddy Bouchet, "Stochastic process of equilibrium fluctuations of
a system with long-range interactions", <a
href="http://dx.doi.org/10.1103/PhysRevE.70.036113"><cite>Physical Review
E</cite> <strong>70</strong> (2004): 036113</a>
<li>F. Bouchet and J. Barré, "Classification of Phase
Transitions and Ensemble Inequivalence, in Systems with Long Range
Interactions",
<a href="http://dx.doi.org/10.1007/s10955-004-2059-0"><cite>Journal of
Statistical Physics</cite> <strong>118</strong> (2005): 1073--1105</a>
<li>Pierre-Henri Chavanis
<ul>
<li>"Dynamics and thermodynamics of systems with long-range interactions: interpretation of the different functionals", <a href="http://arxiv.org/abs/0904.2729">arxiv:0904.2729</a>
<li>"Statistical mechanics of geophysical
turbulence: application to jovian flows and Jupiter's great red spot", <a
href="http://dx.doi.org/10.1016/j.physd.2004.11.004"><cite>Physica
D</cite> <strong>200</strong> (2005): 257--272</a> [Listed here because this is
(judging by the abstract) an instance of Chavanis's more general non-Tsallisite
(non-Tsallisian?) approach to statistical mechanics with long-range
interactions]
<li>"Generalized Fokker-Planck equations and effective
thermodynamics", <a
href="http://arxiv.org/abs/cond-mat/0504716">cond-mat/0504716</a>
= <cite>Physica A</cite> <strong>340</strong> (2004): 57
<li>"Quasi-stationary states and incomplete violent relaxation
in systems with long-range interactions", <a
href="http://arxiv.org/abs/cond-mat/0509726">cond-mat/0509726</a>
<li>"Lynden-Bell and Tsallis distributions for the HMF model",
<a href="http://arxiv.org/abs/cond-mat/0604234">cond-mat/0604234</a>
</ul>
<li>Pierre-Henri Chavanis, C. Rosier and C. Sire, "Thermodynamics of
self-gravitating systems," <a
href="http://arxiv.org/abs/cond-mat/0107345">cond-mat/0107345</a>
<li>Thierry Dauxois, Stefano Ruffo, Ennio Arimondo and Martin Wilkens
(eds.), <cite><a href="https://doi.org/10.1007/3-540-45835-2">Dynamics and Thermodynamics of Systems With Long Range
Interactions</a></cite>
<li>Davide Ferrari and Yuhong Yang, "Maximum Lq-likelihood
estimation", <a href="http://projecteuclid.org/euclid.aos/1266586613"><cite>Annals of Statistics</cite> <strong>38</strong>
(2010): 753--783</a>
<li>V. Garcia-Morales, J. Pellicer, "Statistical mechanics and
thermodynamics of complex
systems", <a href="http://arxiv.org/abs/math-ph/0304013">math-ph/0304013</a>
<li>Toshiyuki Gotoh, Robert H. Kraichnan, "Turbulence and Tsallis
Statistics", <a href="http://arxiv.org/abs/nlin.CD/0305040">nlin.CD/0305040</a>
<li>D. H. E. Gross, "Non-extensive Hamiltonian systems follow
Boltzmann's principle not Tsallis statistics," <a
href="http://arxiv.org/abs/cond-mat/0106496">cond-mat/0106496</a>
<li>Shamik Gupta, David Mukamel, "Slow relaxation in long-range interacting systems with stochastic dynamics", <a href="http://arxiv.org/abs/1006.0233">arxiv:1006.0233</a>
<li>Rudolf Hanel and Stefan Thurner, "On the Derivation of power-law
distributions within standard statistical mechanics", <a
href="http://arxiv.org/abs/cond-mat/0412016">cond-mat/0412016</a> = <a
href="http://dx.doi.org/10.1016/j.physa.2004.11.055"><cite>Physica
A</cite> <strong>351</strong> (2005): 260--268</a>
<li>Petr Jizba and Toshihico Arimitsu, "The world according to Rényi:
Thermodynamics of multifractal systems," <a
href="http://arxiv.org/abs/cond-mat/0207707">cond-mat/0207707</a>
<li>Ramandeep S. Johal, Antoni Planes, and Eduard Vives, "Equivalence
of nonadditive entropies and nonadditive energies in long range interacting
systems under macroscopic equilibrium", <a
href="http://arxiv.org/abs/cond-mat/0503329">cond-mat/0503329</a>
<li>T. Kodama, H.-T. Elze, C. E. Aguiar, T. Koide, "Dynamical
Correlations as Origin of Nonextensive Entropy", <a
href="http://arxiv.org/abs/cond-mat/0406732">cond-mat/0406732</a>
<li>Hiroko Koyama, Tetsuro Konishi, and Stefano Ruffo, "Clusters die
hard: Time-correlated excitation in the Hamiltonian Mean Field
model", <a href="http://arxiv.org/abs/nlin.CD/0606041">nlin.CD/0606041</a>
<li>Bernard H. Lavenda
<ul>
<li>"Fundamental inconsistencies of 'superstatistics'", <a
href="http://arxiv.org/abs/cond-mat/0408485">cond-mat/0408485</a>
<li>"Information and coding discrimination of
pseudo-additive entropies (PAE)", <a
href="http://arxiv.org/abs/cond-mat/0403591">cond-mat/0403591</a>
</ul>
<li>Massimo Marino, "Power-law distributions and equilibrium
thermodynamics", <a
href="http://arxiv.org/abs/cond-mat/0605644">cond-mat/0605644</a> [Makes the
interesting claims that if you want a consistent thermodynamics with power-law
distributions, then the entropy is uniquely determined to be the Rényi
entropy, not the Tsallis entropy]
<li>David Mukamel
<ul>
<li>"Statistical Mechanics of systems with long range
interactions", <a href="http://arxiv.org/abs/0811.3120">arxiv:0811.3120</a>
<li>"Notes on the Statistical Mechanics of Systems with Long-Range Interactions", <a href="http://arxiv.org/abs/0905.1457">arxiv:0905.1457</a>
</ul>
<li>D. Mukamel, S. Ruffo and N. Schreiber, "Breaking of ergodicity and
long relaxation times in systems with long-range interactions", <a
href="http://arxiv.org/abs/cond-mat/0508604">cond-mat/0508604</a>
<li>Jan Naudts, "Parameter estimation in nonextensive thermostatistics",
<a href="http://arxiv.org/abs/cond-mat/0509796">cond-mat/0509796</a>
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</ul>
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<em>Previous versions</em>: 27 Feb 2017 16:30; 2007-01-29 23:22; first version several years older (2003? earlier?)