On the definition of temperature using time--averages

This paper is a natural continuation of a previous one by the author, which was concerned with the foundations of statistical thermodynamics far from equilibrium. One of the problems left open in that paper was the correct definition of temperature. In the literature, temperature is in general defined through the mean kinetic energy of the particles of a given system. In this paper, instead, temperature is defined "a la Caratheodory", the system being coupled to a heat bath, and temperature being singled out as the ``right'' integrating factor of the exchanged heat. As a byproduct, the ``right'' expression for the entropy is also obtained. In particular, in the case of a q-distributions the entropy turns out to be that of Tsallis, which we however show to be additive, at variance with what is usually maintained

Comment on "Typicality for Generalized Microcanonical Ensemble"

The validity of the so-called "typicality" argument for a generalised microcanonical ensemble proposed recently is examined.Comment: Version to appear in PR

Thermodynamics and time-average

For a dynamical system far from equilibrium, one has to deal with empirical probabilities defined through time-averages, and the main problem is then how to formulate an appropriate statistical thermodynamics. The common answer is that the standard functional expression of Boltzmann-Gibbs for the entropy should be used, the empirical probabilities being substituted for the Gibbs measure. Other functional expressions have been suggested, but apparently with no clear mechanical foundation. Here it is shown how a natural extension of the original procedure employed by Gibbs and Khinchin in defining entropy, with the only proviso of using the empirical probabilities, leads for the entropy to a functional expression which is in general different from that of Boltzmann--Gibbs. In particular, the Gibbs entropy is recovered for empirical probabilities of Poisson type, while the Tsallis entropies are recovered for a deformation of the Poisson distribution.Comment: 8 pages, LaTex source. Corrected some misprint

Axioms and uniqueness theorem for Tsallis entropy

The Shannon-Khinchin axioms for the ordinary information entropy are generalized in a natural way to the nonextensive systems based on the concept of nonextensive conditional entropy, and a complete proof of the uniqueness theorem for the Tsallis entropy is presented.Comment: 14 pages. To appear in Physics Letters

On and Off-diagonal Sturmian operator: dynamic and spectral dimension

We study two versions of quasicrystal model, both subcases of Jacobi matrices. For Off-diagonal model, we show an upper bound of dynamical exponent and the norm of the transfer matrix. We apply this result to the Off-diagonal Fibonacci Hamiltonian and obtain a sub-ballistic bound for coupling large enough. In diagonal case, we improve previous lower bounds on the fractal box-counting dimension of the spectrum.Comment: arXiv admin note: text overlap with arXiv:math-ph/0502044 and arXiv:0807.3024 by other author

Simple observations concerning black holes and probability

It is argued that black holes and the limit distributions of probability theory share several properties when their entropy and information content are compared. In particular the no-hair theorem, the entropy maximization and holographic bound, and the quantization of entropy of black holes have their respective analogues for stable limit distributions. This observation suggests that the central limit theorem can play a fundamental role in black hole statistical mechanics and in a possibly emergent nature of gravity.Comment: 6 pages Latex, final version. Essay awarded "Honorable Mention" in the Gravity Research Foundation 2009 Essay Competitio

On the Limiting Cases of Nonextensive Thermostatistics

We investigate the limiting cases of Tsallis statistics. The viewpoint adopted is not the standard information-theoretic one, where one derives the distribution from a given measure of information. Instead the mechanical approach recently proposed in [M. Campisi, G.B. Bagci, Phys. Lett. A (2006), doi:10.1016/j.physleta.2006.09.081], is adopted, where the distribution is given and one looks for the associated physical entropy. We show that, not only the canonical ensemble is recovered in the limit of $q$ tending to one, as one expects, but also the microcanonical ensemble is recovered in the limit of $q$ tending to minus infinity. The physical entropy associated with Tsallis ensemble recovers the microcanonical entropy as well and we note that the microcanonical equipartition theorem is recovered too. We are so led to interpret the extensivity parameter q as a measure of the thermal bath heat capacity: $q=1$ (i.e. canonical) corresponds to an infinite bath (thermalised case, temperature is fixed), $q=-\infty$ (microcanonical) corresponds to a bath with null heat capacity (isolated case, energy is fixed), intermediate $q's$ (i.e. Tsallis) correspond to the realistic cases of finite heat capacity (both temperature and energy fluctuate).Comment: 5 pages, 2 figure

Central limit behavior of deterministic dynamical systems

We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A Central Limit Theorem (CLT) is only valid if the dynamical system under consideration is sufficiently mixing. For the fully developed logistic map and a cubic map we analytically calculate the leading-order corrections to the CLT if only a finite number of iterates is added and rescaled, and find excellent agreement with numerical experiments. At the critical point of period doubling accumulation, a CLT is not valid anymore due to strong temporal correlations between the iterates. Nevertheless, we provide numerical evidence that in this case the probability density converges to a $q$-Gaussian, thus leading to a power-law generalization of the CLT. The above behavior is universal and independent of the order of the maximum of the map considered, i.e. relevant for large classes of critical dynamical systems.Comment: 6 pages, 5 figure

Fairness Is an Emergent Self-Organized Property of the Free Market for Labor

The excessive compensation packages of CEOs of U.S. corporations in recent years have brought to the foreground the issue of fairness in economics. The conventional wisdom is that the free market for labor, which determines the pay packages, cares only about efficiency and not fairness. We present an alternative theory that shows that an ideal free market environment also promotes fairness, as an emergent property resulting from the self-organizing market dynamics. Even though an individual employee may care only about his or her salary and no one else's, the collective actions of all the employees, combined with the profit maximizing actions of all the companies, in a free market environment under budgetary constraints, lead towards a more fair allocation of wages, guided by Adam Smith's invisible hand of self-organization. By exploring deep connections with statistical thermodynamics, we show that entropy is the appropriate measure of fairness in a free market environment which is maximized at equilibrium to yield the lognormal distribution of salaries as the fairest inequality of pay in an organization under ideal conditions

Thermodynamics with generalized ensembles: The class of dual orthodes

We address the problem of the foundation of generalized ensembles in statistical physics. The approach is based on Boltzmann's concept of orthodes. These are the statistical ensembles that satisfy the heat theorem, according to which the heat exchanged divided by the temperature is an exact differential. This approach can be seen as a mechanical approach alternative to the well established information-theoretic one based on the maximization of generalized information entropy. Our starting point are the Tsallis ensembles which have been previously proved to be orthodes, and have been proved to interpolate between canonical and microcanonical ensembles. Here we shall see that the Tsallis ensembles belong to a wider class of orthodes that include the most diverse types of ensembles. All such ensembles admit both a microcanonical-like parametrization (via the energy), and a canonical-like one (via the parameter $\beta$). For this reason we name them ``dual''. One central result used to build the theory is a generalized equipartition theorem. The theory is illustrated with a few examples and the equivalence of all the dual orthodes is discussed.Comment: 20 pages, 4 figures. Minor improvement