Geometric Foundation of Thermo-Statistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit

A geometric foundation thermo-statistics is presented with the only axiomatic assumption of Boltzmann's principle S(E,N,V)=k\ln W. This relates the entropy to the geometric area e^{S(E,N,V)/k} of the manifold of constant energy in the finite-N-body phase space. From the principle, all thermodynamics and especially all phenomena of phase transitions and critical phenomena can unambiguously be identified for even small systems. The topology of the curvature matrix C(E,N) of S(E,N) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Within Boltzmann's principle, Statistical Mechanics becomes a geometric theory addressing the whole ensemble or the manifold of all points in phase space which are consistent with the few macroscopic conserved control parameters. This interpretation leads to a straight derivation of irreversibility and the Second Law of Thermodynamics out of the time-reversible, microscopic, mechanical dynamics. This is all possible without invoking the thermodynamic limit, extensivity, or concavity of S(E,N,V). The main obstacle against the Second Law, the conservation of the phase-space volume due to Liouville is overcome by realizing that a macroscopic theory like Thermodynamics cannot distinguish a fractal distribution in phase space from its closure.Comment: 26 pages, 6 figure

Non-extensive Trends in the Size Distribution of Coding and Non-coding DNA Sequences in the Human Genome

We study the primary DNA structure of four of the most completely sequenced human chromosomes (including chromosome 19 which is the most dense in coding), using Non-extensive Statistics. We show that the exponents governing the decay of the coding size distributions vary between $5.2 \le r \le 5.7$ for the short scales and $1.45 \le q \le 1.50$ for the large scales. On the contrary, the exponents governing the decay of the non-coding size distributions in these four chromosomes, take the values $2.4 \le r \le 3.2$ for the short scales and $1.50 \le q \le 1.72$ for the large scales. This quantitative difference, in particular in the tail exponent $q$, indicates that the non-coding (coding) size distributions have long (short) range correlations. This non-trivial difference in the DNA statistics is attributed to the non-conservative (conservative) evolution dynamics acting on the non-coding (coding) DNA sequences.Comment: 13 pages, 10 figures, 2 table

What Is a Macrostate? Subjective Observations and Objective Dynamics

We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both; more specifically, that the set of macrostates forms the unique maximal partition of phase space which 1) is consistent with our observations (a subjective fact about our ability to observe the system) and 2) obeys a Markov process (an objective fact about the system's dynamics). We review the ideas of computational mechanics, an information-theoretic method for finding optimal causal models of stochastic processes, and argue that macrostates coincide with the ``causal states'' of computational mechanics. Defining a set of macrostates thus consists of an inductive process where we start with a given set of observables, and then refine our partition of phase space until we reach a set of states which predict their own future, i.e. which are Markovian. Macrostates arrived at in this way are provably optimal statistical predictors of the future values of our observables.Comment: 15 pages, no figure

The Statistical Physics of Athermal Materials

At the core of equilibrium statistical mechanics lies the notion of statistical ensembles: a collection of microstates, each occurring with a given a priori probability that depends only on a few macroscopic parameters such as temperature, pressure, volume, and energy. In this review article, we discuss recent advances in establishing statistical ensembles for athermal materials. The broad class of granular and particulate materials is immune from the effects of thermal fluctuations because the constituents are macroscopic. In addition, interactions between grains are frictional and dissipative, which invalidates the fundamental postulates of equilibrium statistical mechanics. However, granular materials exhibit distributions of microscopic quantities that are reproducible and often depend on only a few macroscopic parameters. We explore the history of statistical ensemble ideas in the context of granular materials, clarify the nature of such ensembles and their foundational principles, highlight advances in testing key ideas, and discuss applications of ensembles to analyze the collective behavior of granular materials

Extensive nonadditive entropy in quantum spin chains

We present details on a physical realization, in a many-body Hamiltonian system, of the abstract probabilistic structure recently exhibited by Gell-Mann, Sato and one of us (C.T.), that the nonadditive entropy $S_q=k [1- Tr \hat{\rho}^q]/[q-1]$ ($\hat{\rho}\equiv$ density matrix; $S_1=-k Tr \hat{\rho} \ln \hat{\rho}$) can conform, for an anomalous value of q (i.e., q not equal to 1), to the classical thermodynamical requirement for the entropy to be extensive. Moreover, we find that the entropic index q provides a tool to characterize both universal and nonuniversal aspects in quantum phase transitions (e.g., for a L-sized block of the Ising ferromagnetic chain at its T=0 critical transverse field, we obtain $\lim_{L\to\infty}S_{\sqrt{37}-6}(L)/L=3.56 \pm 0.03$). The present results suggest a new and powerful approach to measure entanglement in quantum many-body systems. At the light of these results, and similar ones for a d=2 Bosonic system discussed by us elsewhere, we conjecture that, for blocks of linear size L of a large class of Fermionic and Bosonic d-dimensional many-body Hamiltonians with short-range interaction at T=0, we have that the additive entropy $S_1(L) \propto [L^{d-1}-1]/(d-1)$ (i.e., $\ln L$ for $d=1$, and $L^{d-1}$ for d>1), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy $S_q(L)\propto L^d$ ($\forall d$), i.e., it is extensive. The present discussion neatly illustrates that entropic additivity and entropic extensivity are quite different properties, even if they essentially coincide in the presence of short-range correlations.Comment: 9 pages, 4 figures, Invited Paper presented at the international conference CTNEXT07, satellite of STATPHYS23, 1-5 July 2007, Catania, Ital