Nonextensive Statistical Mechanics

Chapter 4Stochastic Dynamical Foundationsof Nonextensive Statistical MechanicsSi l’action n’a quelque splendeur de liberté, elle n’a point degrâce ni d’honneurMontaigne, Essais4.1 IntroductionIn this chapter we focus on mesoscopic-like nonlinear dynamical systems, in thesense that the time evolution explicitly includes, in addition to deterministic ingredients,stochastic noise.A paradigmatic path in statistical physical systems is as follows. We assume theknowledge of the Hamiltonian of a classical or quantum many-body system. Thisis referred to as the microscopic level or microscopic description. If the system isclassical, the time evolution is given by Newton’s law F = m a, and is thereforecompletely deterministic. The equations of motion of the system are completelydetermined by the Hamiltonian and the initial conditions. However, it is in generaltremendously difficult to solve the corresponding equations. So, as a simpler alternative,Langevin introduced the following phenomenological approach. We focuson one molecule or element of the system, and its motion is described in terms ofthe combination of two ingredients. The first ingredient is deterministic, comingtypically from the existence of a possible external potential acting on the entire system,as well as from the average action of all the other molecules or elements. Thesecond ingredient is stochastic, introduced in an ad hoc manner into the equationsas a noise. This noise represents the rapidly fluctuating effects of the rest of thesystem onto the single molecule we are observing. This level and the associated descriptionare referred to as the mesoscopic ones, and the basic equation is of coursethe Langevin equation (as well as the Kramers equation, of similar nature). Thetime evolution is determined by the initial conditions and the particular stochasticsequence. When we conveniently average over many initial conditions and manystochastic sequences [29], we obtain a probabilistic description of the system. Moreprecisely, we obtain the time evolution of its probability distribution in the phasespaceof the system. The basic equation is the so-called Fokker–Planck equation,or, for quantum and discrete systems, the master equation (whose continuous limitC. Tsallis, Introduction to Nonextensive Statistical Mechanics,DOI 10.1007/978-0-387-85359-8 4, C○ Springer Science+Business Media, LLC 2009109