The q-exponential family in statistical physics

The q-exponential family in statistical physics

The Boltzmann-Gibbs probability distribution, seen as a statistical model, belongs to the exponential family. Recently, the latter concept has been generalized. The q-exponential family has been shown to be relevant for the statistical description of small isolated systems. Two main applications are reviewed: 1. The distribution of the momentum of a single particle is a q-Gaussian, the distribution of its velocity is a deformed Maxwellian; 2. The configurational density distribution belongs to the q-exponential family. The definition of the temperature of small isolated systems is discussed. I…

Bistable stochastic processes in the q-exponential family

Stochastic bistable systems whose stationary distributions belong to the q-exponential family are investigated using two approaches: (i) the Langevin model subjected to additive and quadratic multiplicative noise, and (ii) the superstatistical model. Previously, the bistable Langevin model has been analyzed under linear multiplicative noise, whereas this paper reports on quadratic multiplicative noise, which is more physically meaningful. The stationary distribution of the Langevin model under quadratic multiplicative noise, which agrees with that derived by the maximum Tsallis entropy method,…

Exponential Families and MaxEnt Calculations for Entropy Measures of Statistical Physics

For a wide range of entropy measures, easy calculation of equilibria is possible using a principle of Game Theoretical Equilibrium related to Jaynes Maximum Entropy Principle. This follows previous work of the author and relates to works of Naudts and, partly, Abe and Bagci.

Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos

It is well known that, for chaotic systems, the production of relevant entropy (Boltzmann-Gibbs) is always linear and the system has strong (exponential) sensitivity to initial conditions. In recent years, various numerical results indicate that basically the same type of behavior emerges at the edge of chaos if a specific generalization of the entropy and the exponential are used. In this work, we contribute to this scenario by numerically analysing some generalized nonextensive entropies and their related exponential definitions using $z$-logistic map family. We also corroborate our findings…

κ分布作为指数族系综的渐近边缘分布

Kappa distributions as asymptotic marginals of exponential family ensembles

该研究以受驱经典系统超统计行为的前期成果为基础，证明总动能服从逆伽马分布的N粒子系统在N→∞时单粒子速度服从κ分布，并提出了生成κ速度的实用模拟方法。

横场Ising模型量子指数族的标量曲率与量子相变

Scalar Curvature of the Quantum Exponential Family for the Transverse-Field Ising Model and the Quantum Phase Transition

该研究解析了零维与一维横场Ising模型量子指数族吉布斯热态的标量曲率特性，揭示其高温收敛、零温指数发散的规律，验证了零温下指数为1的临界行为与量子-经典对应关系的一致性。