No abstract

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann-Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accomodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy SBG = −k i pi ln pi, and typically provides exponential laws for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form Sq = k(1 − i p q i )/(q − 1) (with S1 = SBG), and typically provides, for the same type of description, (asymptotic) power laws.Connections between dynamics and thermodynamics are far from being completely elucidated. Frequently, statistical mechanics is presented as a self-contained body, which could dispense dynamics from its formulation. This is an unfounded assumption (see, for instance, [1] and references therein). Questions still remain open even for one of its most well established equilibrium concepts, namely the Boltzmann-Gibbs (BG) factor e −Ei/kT , where E i is the energy associated with the ith microscopic state of a conservative Hamiltonian system, k is Boltzmann constant, and T the absolute temperature. For example, no theorem exists stating the necessary and sufficient conditions for the use of this celebrated and ubiquitous factor to be justified. In the mathematician F. Takens' words [2]:The values of p i are determined by the following dogma: if the energy of the system in the i th state is E i and if the temperature of the system is T then: One possible reason for this essential point having been poorly emphasized is that when dealing with short-range interacting systems, BG thermodynamical equilibrium may be formulated without much referring to the underlying dynamics of its constituents. One rarely finds in textbooks much more than a quick mention to ergodicity. A full analysis of the microscopic dynamical requirements for ergodicity to be ensured is still lacking, in spite of the pioneering studies of N. Krylov [3]. In his words:In Another possibly concomitant reason no doubt is the enormous success, since more than one century, of BG statistical mechanics for very many systems....

We analyze the equilibrium properties of a chain of ferromagnetically coupled rotators which interact through a force that decays as r −α where r is the interparticle distance and α ≥ 0. Our model contains as particular cases the mean field limit (α = 0) and the first-neighbor model (α → ∞). By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. Concerning the long-range order, we detect three different regimes at low energies, depending on whether α belongs to the intervals [0, 1), (1, 2) or (2, ∞). Moreover, for 0 ≤ α < 1, the microcanonical averages agree, after a simple scaling, with those obtained in the canonical ensemble for the mean-field XY model. This correspondence offers a mathematically tractable and computationally economic way of dealing with systems governed by slowly decaying long-range interactions.One of most important questions in statistical mechanics refers to the connection between dynamics and thermodynamics: To what extent a suitable ensemble average allows to predict the time average of a physical observable performed by our instruments in the laboratory? Or, in other words, which are the mechanical specifications of those systems to which the results of statistical physics can be applied? Within this context, while ergodicity and mixing have been analyzed intensively in the literature, there is another important point which has not deserved the same degree of attention, that is the possibility of defining a thermodynamically suitable energy function. In fact, for systems governed by sufficiently long-range interactions decaying as r −α with the interparticle distance r, there results a non-extensive Hamiltonian, i.e., the energy per particle diverges in the thermodynamics limit N → ∞ [1,2]. Gravitational (α = 1, d = 3) and monopole-dipole (α = 2, d = 3) interactions are only two well known instances among many others. Furthermore, such forces are particularly interesting since they can lead to equilibrium behaviors different from those observed in short-range systems and even give place to phase transitions otherwise absent, even in the d = 1 case.Our aim here is to investigate how to deal with systems governed by long-range interactions by analyzing a simple but rich prototype with adjustable α. The main goal of this letter is to show that the mean-field limit (α = 0) is able to describe the thermodynamics in the whole range 0 < α < 1. The model consists in a one

A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann-Gibbs statistical mechanics (based on S1 ≡ −k du p(u) ln p(u)). Similarly, other classes of models point toward nonextensive statistical mechanics (based on, where the value of the entropic index q ∈ ℜ depends on the specific model). We show here a family of models, with multiplicative noise, which belongs to the nonextensive class. More specifically, we consider Langevin equations of the typeu = f (u) + g(u)ξ(t) + η(t), where ξ(t) and η(t) are independent zero-mean Gaussian white noises with respective amplitudes M and A. This leads to the Fokker-Planck equation ∂tP (u, t) = −∂u[f (u)P (u, t)]+M ∂u{g(u)∂u[g(u)P (u, t)]}+ A∂uuP (u, t). Whenever the deterministic drift is proportional to the noise induced one, i.e., f (u) = −τ g(u)g ′ (u), the stationary solution is shown to be P (u, ∞) ∝ 1 − (1 − q)β[g(u)] 2 1 1−q

No abstract