Generalized thermostatistics and mean-field theory

Abstract

The present paper studies a large class of temperature-dependent probability distributions and shows that entropy and energy can be defined in such a way that these probability distributions are the equilibrium states of a generalized thermostatistics, which is obtained from the standard formalism by deformation of exponential and logarithmic functions. Since this procedure is non-unique, specific choices are motivated by showing that the resulting theory is well-behaved. In particular, with the choices made in the present paper, the equilibrium state of any system with a finite number of degrees of freedom is, automatically, thermodynamically stable and satisfies the variational principle. The equilibrium probability distribution of open systems deviates generically from the Boltzmann–Gibbs distribution. If the interaction with the environment is not too strong then one can expect that a slight deformation of the exponential function, appearing in the Boltzmann–Gibbs distribution, can reproduce the observed distribution with its temperature dependence. An example of a system, where this statement holds, is a single spin of the Ising chain. However, because all systems of the present generalized thermostatistics are automatically stable, one must not expect that all open systems can be described in this way. Indeed, systems exhibiting a phase transition in their thermodynamic limit, can be unstable even when the interaction with the environment is weak. Therefore, their equilibrium probability distribution cannot be described by a simple deformation of the Boltzmann–Gibbs distribution. In order to be able to handle such systems as well the second part of the paper discusses a further extension of the class of probability distributions using mean-field techniques. Connections are discussed that exist between the present formalism, Tsallis’ thermostatistics, and the superstatistics formalism of Beck and Cohen. In particular, the present generalization sheds some light onto the historical development of the Tsallis formalism. It is pointed out that temperature dependence of the Tsallis distribution has hardly been verified experimentally.