What should a statistical mechanics satisfy to reflect nature

There is no compelling reason imposing that the methods of statistical mechanics should be restricted to the dynamical systems which follow the usual Boltzmann–Gibbs prescriptions. More specifically, ubiquitous natural and artificial systems exhibit generic stationary states which are not ergodic nor close to it, in any geometrically simple subset of the a priori allowed phase space, in any (even extended) trivial sense. A vast class of such systems appears, nevertheless, to be tractable within thermostatistical methods completely analogous to the usual ones. The question posed in the title arises then naturally. Some answer to this complex question is advanced in the present review of nonextensive statistical mechanics and its recent connections.

[1]  C. Tsallis,et al.  Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity , 1997, cond-mat/9701096.

[2]  Asymmetric unimodal maps at the edge of chaos. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  C. E. Budde,et al.  Superdiffusion in decoupled continuous time random walks , 2000 .

[4]  S. Ruffo,et al.  Clustering and relaxation in Hamiltonian long-range dynamics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Peter V. Coveney,et al.  Galilean-invariant multi-speed entropic lattice Boltzmann models , 2004 .

[6]  Celia Anteneodo,et al.  Nonextensive statistical mechanics and economics , 2003, ArXiv.

[7]  F. Baldovin,et al.  Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions , 2002 .

[8]  S. Gheorghiu,et al.  Power-law distribution of pressure fluctuations in multiphase flow. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  E. Lutz.  Anomalous diffusion and Tsallis statistics in an optical lattice , 2003 .

[10]  Funabashi,et al.  Nonuniqueness of Canonical Ensemble Theory arising from Microcanonical Basis , 1999 .

[11]  C. Tsallis,et al.  Nonextensive Entropy: Interdisciplinary Applications , 2004 .

[12]  M. Fisher.  The free energy of a macroscopic system , 1964 .

[13]  Chaotic dynamics and superdiffusion in a Hamiltonian system with many degrees of freedom , 1999, cond-mat/9911317.

[14]  Turbulence and Tsallis statistics , 2003, nlin/0305040.

[15]  G. Nicolis,et al.  Probabilistic and thermodynamic aspects of dynamical systems. , 1998, Chaos.

[16]  Rates of convergence of non-extensive statistical distributions to Lévy distributions in full and half-spaces , 2000, cond-mat/0009399.

[17]  Anomalous diffusion as a signature of a collapsing phase in two-dimensional self-gravitating systems , 1998, cond-mat/9801008.

[18]  Constantino Tsallis,et al.  Classical spin systems with long-range interactions: universal reduction of mixing , 2001 .

[19]  D. Gross,et al.  Microcanonical Thermodynamics: Phase Transitions in 'Small' Systems , 2001 .

[20]  Tsallis,et al.  Anomalous diffusion associated with nonlinear fractional derivative fokker-planck-like equation: exact time-dependent solutions , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  M. Cáceres,et al.  Computing a non-Maxwellian velocity distribution from first principles. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Evaldo M F Curado,et al.  Derivation of nonlinear Fokker-Planck equations by means of approximations to the master equation. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  N. KRYLOV,et al.  Relaxation Processes in Statistical Systems , 1944, Nature.

[24]  H. Kangro.  Planck''s Original Papers in Quantum Physics , 1972 .

[25]  Tsallis,et al.  Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  V. Latora,et al.  The rate of entropy increase at the edge of chaos , 1999, cond-mat/9907412.

[27]  Albert,et al.  Topology of evolving networks: local events and universality , 2000, Physical review letters.

[28]  Chaos and statistical mechanics in the Hamiltonian mean field model , 1998, chao-dyn/9803019.

[29]  C. Tsallis,et al.  Nonextensive foundation of Lévy distributions. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[30]  Sumiyoshi Abe,et al.  Justification of power law canonical distributions based on the generalized central-limit theorem , 2000 .

[31]  C. Tsallis,et al.  Border between regular and chaotic quantum dynamics. , 2002, Physical review letters.

[32]  Bernhard Lesche,et al.  Instabilities of Rényi entropies , 1982 .

[33]  C. Tsallis,et al.  Crossover in diffusion equation: anomalous and normal behaviors. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Funabashi,et al.  Microcanonical foundation for systems with power-law distributions , 2000, cond-mat/0002159.

[35]  Sumiyoshi Abe,et al.  Stability of Tsallis entropy and instabilities of Rényi and normalized Tsallis entropies: a basis for q-exponential distributions. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Constantino Tsallis,et al.  Nonequilibrium probabilistic dynamics of the logistic map at the edge of chaos. , 2002, Physical review letters.

[37]  C. Beck,et al.  Thermodynamics of chaotic systems , 1993 .

[38]  Constantino Tsallis,et al.  Fluxes of cosmic rays: a delicately balanced stationary state , 2002 .

[40]  Sumiyoshi Abe.  Tsallis entropy: how unique? , 2004 .

[41]  Refractor.  Vision , 2000, The Lancet.

[42]  Nikolai SergeevichHG Krylov,et al.  Works on the foundations of statistical physics , 1979 .

[43]  Vito Latora,et al.  Lyapunov Instability and Finite Size Effects in a System with Long-Range Forces , 1998 .

[44]  Włodarczyk,et al.  Interpretation of the nonextensivity parameter q in some applications of tsallis statistics and Levy distributions , 2000, Physical review letters.

[45]  M. Baranger,et al.  Why Tsallis statistics , 2002 .

[46]  Constantino Tsallis,et al.  Nonextensive Statistical Mechanics: Some Links with Astronomical Phenomena , 2004 .

[47]  Stability of the entropy for superstatistics , 2003, cond-mat/0301304.

[48]  A. Baldassarri,et al.  Influence of correlations on the velocity statistics of scalar granular gases , 2001, cond-mat/0111066.

[49]  Michael F. Shlesinger,et al.  Strange kinetics , 1993, Nature.

[50]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[51]  Thermal measurements of stationary nonequilibrium systems: a test for generalized thermostatistics , 2002, cond-mat/0212327.

[52]  Classical infinite-range-interaction Heisenberg ferromagnetic model: metastability and sensitivity to initial conditions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  V Latora,et al.  Non-Gaussian equilibrium in a long-range Hamiltonian system. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  J. Morán-López,et al.  New trends in magnetism, magnetic materials, and their applications , 1994 .

[55]  A. Piperno,et al.  2003 , 2003, Intensive Care Medicine.

[56]  F Baldovin,et al.  Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  A. Compagner.  On Metastable States , 1969 .

[58]  Michael E. Fisher,et al.  THE STABILITY OF MANY-PARTICLE SYSTEMS , 1966 .

[59]  Renio S. Mendes,et al.  Is re-association in folded proteins a case of nonextensivity? , 1999 .

[60]  A. R. Plastino,et al.  Non-extensive statistical mechanics and generalized Fokker-Planck equation , 1995 .

[61]  Celia Anteneodo,et al.  Multiplicative noise: A mechanism leading to nonextensive statistical mechanics , 2002 .

[62]  Ugur Tirnakli.  Two-dimensional maps at the edge of chaos: numerical results for the Henon map. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[63]  SUPERDIFFUSION AND OUT-OF-EQUILIBRIUM CHAOTIC DYNAMICS WITH MANY DEGREES OF FREEDOMS , 1999, cond-mat/9904389.

[64]  C. Tsallis,et al.  Anomalous sensitivity to initial conditions and entropy production in standard maps: Nonextensive approach , 2004 .

[65]  M. Fisher.  Correlation Functions and the Coexistence of Phases , 1965 .

[66]  Trying to Make Sense of Disorder , 2002, Science.

[67]  Michael Nauenberg.  Critique of q-entropy for thermal statistics. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[68]  C. Tsallis,et al.  Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems , 1997, cond-mat/9709226.

[69]  D. Prato.  Generalized statistical mechanics: Extension of the Hilhorst formula and application to the classical ideal gas , 1995 .

[70]  Satosi Watanabe,et al.  Knowing and guessing , 1969 .

[71]  Kac-potential treatment of nonintegrable interactions. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[72]  M. Fisher.  Bounds for the Derivatives of the Free Energy and the Pressure of a Hard‐Core System near Close Packing , 1965 .

[73]  Funabashi,et al.  Scale-free network of earthquakes , 2002 .

[74]  J. Doye.  Network topology of a potential energy landscape: a static scale-free network. , 2002, Physical review letters.

[75]  C. Tsallis,et al.  The role of constraints within generalized nonextensive statistics , 1998 .

[76]  Revisiting Disorder and Tsallis Statistics , 2003, Science.

[77]  M. Fisher,et al.  Asymptotic free energy of a system with periodic boundary conditions , 1970 .

[78]  H. Posch,et al.  STATISTICAL MECHANICS AND COMPUTER SIMULATION OF SYSTEMS WITH ATTRACTIVE POSITIVE POWER-LAW POTENTIALS , 1998 .

[79]  Christian Beck.  Dynamical Foundations of Nonextensive Statistical Mechanics , 2001 .

[80]  C. Tsallis,et al.  Power-law sensitivity to initial conditions—New entropic representation , 1997 .

[81]  Zanette,et al.  Fractal random walks from a variational formalism for Tsallis entropies. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[82]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[83]  Metastable states, anomalous distributions and correlations in the HMF model , 2003, cond-mat/0303081.

[84]  C. Tsallis,et al.  Breakdown of Exponential Sensitivity to Initial Conditions: Role of the Range of Interactions , 1998 .

[85]  Andrea Rapisarda,et al.  Non Extensive Thermodynamics and physical applications , 2002 .

[86]  Constantino Tsallis,et al.  Constructing a statistical mechanics for Beck-Cohen superstatistics. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[87]  Constantino Tsallis,et al.  Metastability and weak mixing in classical long-range many-rotator systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[88]  N. Lemke,et al.  Tsallis entropy production for diffusion on the diluted hypercube , 2003 .

[89]  de Moura FA,et al.  Convergence to the critical attractor of dissipative maps: log-periodic oscillations, fractality, and nonextensivity , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[90]  J. Spencer,et al.  EVOLUTION OF THE n-CUBE , 1979 .

[91]  H. Barlow.  Conditions for versatile learning, Helmholtz's unconscious inference, and the task of perception , 1990, Vision Research.

[92]  C. Tsallis,et al.  On the Emergence of Nonextensivity at the Edge of Quantum Chaos , 2003, quant-ph/0305086.