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. This generalized thermostatistics 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, 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 temperature dependence. An example of a system, where this statement holds, is a single spin of the Ising chain. The connection between the present formalism and Tsallis' thermostatistics is discussed. In particular, the present generalization sheds some light onto the historical development of the latter formalism.