Topical Collection "Foundations of Statistical Mechanics"
Dr. Antonio M. Scarfone
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
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Interests: nonextensive statistical mechanics; nonlinear Fokker–Planck equations; geometry information; nonlinear Schroedinger equation; quantum groups and quantum algebras; complex systems
Statistical mechanics, covered in the section of Statistical Physics, aims to relate the microscopic to the macroscopic properties of matter by using the concepts developed in the field of probability theory and thermodynamics. It is a successful combination of the statistics and mechanics arising from the union of the basic laws of classical or quantum mechanics, with the laws of large numbers.
The foundations of statistical mechanics lie in the thermodynamics theory developed at the end of the nineteenth century. The first person to analyse transport phenomena with statistical methods was Clausius, who introduced the concept of a mean free path. He also introduced the famous “Stosszahlansatz” hypothesis, which played a prominent role in the succeeding works of Boltzmann. In the pioneering paper, “Zusammeuhang zwischen den Satzen iiber das Verhalten mehratomiger Gasmolekiile mit Jacobi's Princip des letzten Multiplicators”, Boltzmann considers explicitly a great number of systems, their distribution in phase space, and the permanence of this distribution in time. Another impressive contribution to the theory is represented by Maxwell’s work on the kinetic theory of gases derived from what is now called the Maxwell velocity distribution. Finally, Gibbs, in his book “Elementary principles in statistical mechanics”, published in 1902, definitively established the equivalence between the statistical physics and thermodynamics.
From then, statistical mechanics has been developed in several aspects, becoming so general that its methods still hold in a much wider context than that on which the original theory was developed. In fact, thanks to its impressive success, considerable efforts have been made in recent years to extend the formalism of statistical mechanics beyond its application limits. Traditional statistical mechanics focuses on systems with many degrees of freedom, and has become exact in the thermodynamic limit, although, nowadays, an increasing amount of physical systems seem to not comply with this limit imposed by the large numbers. Definitively, such systems reach a meta-equilibrium configuration, which appears to be better described by generalized entropic forms different from the traditional Boltzmann–Gibbs one.
This collection intends to present mainly theoretical oriented material (even purely mathematical) on the foundation of statistical mechanics. It focuses on the challenges of modern theory incorporating a high degree of mathematical rigor, in order to provide relevance not only to statistical physicists, but also to mathematicians and theoretical physicists. The papers submitted should have real and concrete applications in statistical mechanics, or provide clear evidence of possible applications.
Dr. Antonio M. Scarfone
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- foundations of classical and quantum statistical mechanics
- Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics
- exotic statistics—Haldane, Gentile, and Quons
- generalizations of statistical mechanics
- non-Gibbsian distributions and power–law distributions
- statistical mechanics of non-equilibrium and meta-equilibrium—critical phenomena and phase transitions
- geometric foundations of statistical mechanics