The H-theorem in κ-statistics : influence on the molecular chaos hypothesis

We rediscuss recent derivations of kinetic equations based on the Kaniadakis' entropy concept. Our primary objective here is to derive a kinetical version of the second law of thermodynamics in such a κ-framework. To this end, we assume a slight modification of the molecular chaos hypothesis. For the Hκ-theorem, it is shown that the collisional equilibrium states (null entropy source term) are described by a κ-power law extension of the exponential distribution and, as should be expected, all these results reduce to the standard one in the limit κ→0.

[1]  Colin J. Thompson,et al.  Mathematical Statistical Mechanics , 1972 .

[2]  A. M. Scarfone,et al.  A new one-parameter deformation of the exponential function , 2002 .

[3]  A Maxwellian path to the q-nonextensive velocity distribution function , 1998, cond-mat/0201503.

[4]  G. Kaniadakis.  H-theorem and generalized entropies within the framework of nonlinear kinetics , 2001 .

[5]  S. Burbury.  Boltzmann's Minimum Function , 1894, Nature.

[6]  A. Sommerfeld,et al.  Lectures on theoretical physics , 1950 .

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

[8]  Canonical quantization of nonlinear many-body systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  A. M. Scarfone,et al.  Lesche stability of κ-entropy , 2004 .

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

[11]  A. Sommerfeld.  Thermodynamics and Statistical Mechanics , 1956 .

[12]  Giorgio Kaniadakis,et al.  A κ-entropic approach to the analysis of the fracture problem , 2004 .

[13]  A. Plastino,et al.  Nonextensive thermostatistics and the H theorem. , 2001, Physical review letters.

[14]  A. M. Scarfone,et al.  Kinetical foundations of non-conventional statistics , 2001 .

[15]  S. Brush,et al.  Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager , 1986 .

[16]  Sumiyoshi Abe,et al.  Stabilities of generalized entropies , 2004 .

[17]  H. G. Miller,et al.  κ-DEFORMED STATISTICS AND THE FORMATION OF A QUARK-GLUON PLASMA , 2003 .

[18]  Sumiyoshi Abe,et al.  A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics , 1997 .

[19]  A M Scarfone,et al.  Two-parameter deformations of logarithm, exponential, and entropy: a consistent framework for generalized statistical mechanics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  A. M. Scarfone,et al.  Deformed logarithms and entropies , 2004, cond-mat/0402418.

[21]  P. Laurençot,et al.  Chapman–Enskog derivation of the generalized Smoluchowski equation , 2004, cond-mat/0403610.

[22]  Ludwig Boltzmann,et al.  Lectures on Gas Theory , 1964 .

[23]  H. Dieter Zeh,et al.  The Problem of Time. (Book Reviews: The Physical Basis of the Direction of Time.) , 1989 .

[24]  G. Kaniadakis,et al.  Non-linear kinetics underlying generalized statistics , 2001 .

[25]  Denis Bolduc,et al.  The K-deformed multinomial logit model , 2005 .

[26]  A. M. Scarfone,et al.  Generalized kinetic equations for a system of interacting atoms and photons: theory and simulations , 2004 .