On the Tsallis Entropy for Gibbs Random Fields∗

The Tsallis entropy, as a generalization of the standard Shannon-type entropy, was introduced by Constantino Tsallis (1988). Since that the concept has been extensively studied (see, e.g., Tsallis (2009)). In the present paper we address the problem of generalizing the concept for infinitedimensional systems, i.e., the random processes and fields. Apparently, rather well suited models are the Gibbs distributions (cf. e.g., Georgii (1988)). We construct the appropriate Tsallis entropy rate either asymptotically by limit over a sequence of expanding volumes or by analogy with the exponential finite-dimensional distributions. Basic properties, taking into account the possible phase transitions, are also introduced.