Nonextensive Statistical Mechanics

14 1 Historical Background and Physical Motivationsture) of a many-body system composed by say N interacting elements or fields. Itcould also be conservative or dissipative coupled maps, or even cellular automata.Consistently, time t could be continuous or discrete. The same is valid for space.The quantity which is defined in space-time could itself be continuous or discrete.For example, in quantum mechanics, the quantity is a complex continuous variable(the wave function) defined in a continuous space-time. On the other extreme, wehave cellular automata, for which all three relevant variables – time, space, andthe quantity therein defined – are discrete. In the case of a Newtonian mechanicalsystem of particles, we may think of N Dirac delta functions localized in continuousspatial positions which depend on a continuous time.Langevin-like equations (and associated Fokker-Planck-like equations) are typicallyconsidered not microscopic, but mesoscopic instead. The reason of courseis the fact that they include at their very formulation, i.e., in an essential manner,some sort of (stochastic) noise. Consequently, they should not be used as astarting point if we desire the foundation of statistical mechanics to be from firstprinciples.(ii) Then assume some set of initial conditions and let the system evolve intime. These initial conditions are defined in the so-called phase-space of the microscopicconfigurations of the system, for example Gibbs’ space for a NewtonianN-particle system (the space for point masses has 2dN dimensions if theparticles live in a d-dimensional space). These initial conditions typically (but notnecessarily) involve one or more constants of motion. For example, if the system isa conservative Newtonian one of point masses, the initial total energy and the initialtotal linear momentum (d dimensional vector) are such constants of motion. Thetotal angular momentum might also be a constant of motion. It is quite frequent touse coordinates such that both total linear momentum and total angular momentumvanish.If the system consists of conservative coupled maps, the initial hypervolume of anensemble of initial conditions near a given one is preserved through time evolution.By the way, in physics, such coupled maps are frequently obtained through Poincarésections of Newtonian dynamical systems.(iii) After some sufficiently long evolution time (which typically depends on bothN and the spatial range of the interactions), the system might approach some stationaryor quasi-stationary macroscopic state. 2 In such a state, the various regions ofphase-space are being visited with some probabilities. This set of probabilities eitherdoes not depend anymore on time or depends on it very slowly. More precisely, if itdepends on time, it does so on a scale much longer than the microscopic time scale.The visited regions of phase-space that we are referring to typically correspond to apartition of phase-space with a degree of (coarse or fine) graining that we adopt forspecific purposes. These probabilities can be either insensitive or, on the contrary,very sensitive to the ordering in which t →∞(asymptotic) and N →∞(thermodynamic)limits are taken. This can depend on various aspects such as the range of2 When the system exhibits some sort of aging, the expression quasi-stationary is preferable tostationary.