<p><span style="font-size: 11.000000pt; font-family: 'CMR10';">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)). </span>
<span style="font-size: 11.000000pt; font-family: 'CMR10';">In the present paper we address the problem of generalizing the concept for infinite-dimensional systems, i.e., the random processes and fields. Apparently, rather well suited models are the Gibbs distributions (cf. e.g., Georgii (1088)). </span>
<span style="font-size: 11.000000pt; font-family: 'CMR10';">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. </span>
