Abstract
Distributions different from those predicted by equilibrium statistical mechanics are commonplace in a number of physical situations, such as plasmas and self-gravitating systems. The best strategy for probing these distributions and unavailing their origins consists in combining theoretical knowledge with experiments, involving both direct and indirect measurements, as those associated with dispersion relations. This paper addresses, in a quite general context, the signature of nonequilibrium distributions in dispersion relations. We consider the very general scenario of distributions corresponding to a superposition of equilibrium distributions, that are well-suited for systems exhibiting only local equilibrium, and discuss the general context of systems obeying the combination of the Schrödinger and Poisson equations, while allowing the Planck’s constant to smoothly go to zero, yielding the classical kinetic regime. Examples of media where this approach is applicable are plasmas, gravitational systems, and optical molasses. We analyse in more depth the case of classical dispersion relations for a pair plasma. We also discuss a possible experimental setup, based on spectroscopic methods, to directly observe these classes of distributions.
Highlights
Where β is some intensive parameter, which is distributed according to some unspecified distribution f (β), while f0 is the equilibrium distribution that can be identified with the Maxwell-Boltzmann (MB) distribution for classical systems or the relevant quantum generalization thereof for a quantum system
In the statistical mechanics literature, such an approach goes by the name of superstatistics[22], as it merely consists of a superposition of different statistics
Given the growing empirical evidence in favor of distributions different from those predicted by equilibrium statistical mechanics, it becomes increasingly important to better understand their origins in physical problems
Summary
Let us first state precisely the statistical conditions we are considering here, i.e., the circumstances under which Eq (1) holds. Provided that the temperature varies within a time-scale much larger than the local relaxation time (viz., the adiabatic Ansatz41), the long-term distribution arises as a superposition of local equilibrium statistics, averaged over the distribution of the inverse temperature β ≡ 1/kBT (hereafter, kB = 1 ) across the different cells, i.e., Eq (1). Which is equivalent to the Tsallis distribution (q-Gaussian), known in the paradigm of NSM This can be made more transparent by adopting a slightly different parametrization; by defining an entropic index q := 1 + 2/(n + d) and an effective inverse temperature β := β0(n + d)/n ≡ 1/T , Eq (3) can be re-expressed in the more familiar form[21] as. Β follows an inverse-χ 2 distribution, f (β) β0
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