How, Why and When Tsallis Statistical Mechanics Provides Precise Descriptions of Natural Phenomena

Abstract

The limit of validity of ordinary statistical mechanics and the pertinence of Tsallis statistics beyond it is explained considering the most probable evolution of complex systems processes. To this purpose we employ a dissipative Landau–Ginzburg kinetic equation that becomes a generic one-dimensional nonlinear iteration map for discrete time. We focus on the Renormalization Group (RG) fixed-point maps for the three routes to chaos. We show that all fixed-point maps and their trajectories have analytic closed-form expressions, not only (as known) for the intermittency route to chaos but also for the period-doubling and the quasiperiodic routes. These expressions have the form of q-exponentials, while the kinetic equation’s Lyapunov function becomes the Tsallis entropy. That is, all processes described by the evolution of the fixed-point trajectories are accompanied by the monotonic progress of the Tsallis entropy. In all cases the action of the fixed-point map attractor imposes a severe impediment to access the system’s built-in configurations, leaving only a subset of vanishing measure available. Only those attractors that remain chaotic have ineffective configuration set reduction and display ordinary statistical mechanics. Finally, we provide a brief description of complex system research subjects that illustrates the applicability of our approach.

A Nonlinear Dynamical View of Kleiber’s Law on the Metabolism of Plants and Animals

“…Recently [ 25 ], we have demonstrated that the trajectories of all RG fixed-point maps for the three known routes to chaos (intermittency, period doubling, and quasi-periodicity [ 24 ]) can be couched in the statistical–mechanical language of the (discrete time) Landau–Ginzburg (LG) equation. Additionally, the associated Lyapunov function [ 40 ] is precisely the expression for the Tsallis entropy [ 25 ]. Equation ( 10 ) is a particular case of the LG equation used to describe the most probable evolution of processes in statistical–mechanical systems.…”

“…This driving force is the (functional) derivative of the Lyapunov function. This function represents a generalized thermodynamic potential and evolves monotonically as t , or N , increases along the solution of the LG equation [ 25 ]. In the case of Equations ( 10 ) and ( 17 ), it is given by The Tsallis entropy above corresponds to a uniformly distributed set of events.…”

“…Other issues addressed that involve Tsallis generalized entropy and related quantities are [ 34 ] within condensed matter physics: The formation of glasses, the transformation of a conductor into an insulator, and critical point fluctuations; concerning complex systems problems, the phenomenon of self-organization and the development of diversity (biological or social, like languages); and, as described here, the comprehension of empirical laws, like those relating to the universality of ranked data or the power-law scalings present in allometry. A common feature in all these cases is that access to their configurational space is severely hindered to a point where the allowed configurational space has a vanishing measure with respect to the initial setup [ 25 ]. This restriction is naturally provided by the attractors at the transitions to chaos present in the nonlinear dissipative maps employed to model these subjects [ 25 ].…”

“…A common feature in all these cases is that access to their configurational space is severely hindered to a point where the allowed configurational space has a vanishing measure with respect to the initial setup [ 25 ]. This restriction is naturally provided by the attractors at the transitions to chaos present in the nonlinear dissipative maps employed to model these subjects [ 25 ].…”

“…It is worth mentioning that the fixed-point maps for the other (and only) two routes to (or out of) chaos, period doubling and quasi-periodicity [ 24 ], were originally obtained numerically via approximations of their power series representation [ 21 ]. Their analytical closed-form expressions, also in terms of the q -exponential function , have become known only very recently [ 25 ]. The inverse function of the q -exponential, the q -logarithm, is given by .…”