Non-equilibrium statistical mechanics of complex systems: An overview

Summary

The ensemble formalism has been extremely successful in the handling of the grandiose theoretical scheme of statistical mechanics and thermodynamics initiated by Maxwell, Boltzmann and Gibbs, which has been given concrete and consistent foundations to the study of the many situations present in condensedmatter physics. However, its use is hampered when dealing with certain complex phenomena for which the researcher may not have an access to the information on all the constraints relevant to the problem in hands (so-called hidden constraints), which leads to poor predictions. In an attempt to improve predictions there have been introduced, beginning in the past 1950s, and pioneered by P. Levy in the 1930s, auxiliary approaches which attempt to assuage the difficulty, but at the price of not being fully consistent and depending on free parameters. This is done in the framework of the variational (extremum principle) approach in statistical mechanics founded on information theory. In it, the general and well-established Boltzmann-Gibbs canonical scheme follows from maximization with given constraints of Gibbs-Boltzmann-Shannon information-theoretic entropy (better called measure of uncertainty of information): it is considered to be the only consistent probability measure of information. The other (say non-canonical or heterotypical) auxiliary approaches are based on replacing GBS information-theoretic entropy by others, which are used to derive non-conventional probability distributions for non-equilibrium systems. We present a detailed description of their construction and a clarification of their scope, interpretation and utility. Also, resorting to the particular case of Renyi’s approach the construction of a non-equilibrium ensemble formalism is described. The non-conventional distribution functions of fermions and bosons are discussed. The use of the formalism is illustrated via the analysis of experimental results in the case of fractal-like structured systems. Also a purely theoretical analysis is done in the cases of an ideal gas and of radiation comparing the conventional and non-conventional approaches. In all of these situations it is discussed which are the difficulties involved (hidden constraints in an insufficient description) which require to resort to the non-conventional approach, and what determines the value of the parameter(s) that the formalism introduces in each case, namely, its (their) dependence on the system characteristics and the experimental protocol.

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Luzzi, R., Vasconoellos, Á.R. & Ramos, J.G. Non-equilibrium statistical mechanics of complex systems: An overview. Riv. Nuovo Cim. 30, 95–157 (2007). https://doi.org/10.1393/ncr/i2007-10018-6

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Key words

  • PACS 05.70.Ln–Nonequilibrium and irreversible thermodynamics
  • PACS 82.20.-w–Chemical kinetics and dynamics
  • PACS 82.20.Db–Transition state theory and statistical theories of rate constants