Computational methods inspired by Tsallis statistics: Monte Carlo and molecular dynamics algorithms for the simulation of classical and quantum systems

Abstract

Tsallis's generalization of statistical mechanics is summarized. A modi cation of this formalism which employs a normalized expression of the q-expectation value for the computation of equilibrium averages is reviewed for the cases of pure Tsallis statistics and Maxwell-Tsallis statistics. Monte Carlo and Molecular Dynamics algorithms which sample the Tsallis statistical distributions are presented. These methods have been found to be e ective in the computation of equilibrium averages and isolation of low lying energy minima for low temperature atomic clusters, spin systems, and biomolecules. A phase space coordinate transformation is proposed which connects the standard Cartesian positions and momenta with a set of positions and momenta which depend on the potential energy. It is shown that pure Tsallis statistical averages in this transformed phase space result in the q-expectation averages of Maxwell-Tsallis statistics. Finally, an alternative n o vel derivation of the Tsallis statistical distribution is presented. The derivation begins with the classical density matrix, rather than the Gibbs entropy formula, but arrives at the standard distributions of Tsallis statistics. The result suggests a new formulation of imaginary time path integrals which m a y lead to an improvement in the simulation of equilibrium quantum statistical averages.

Generalized-ensemble algorithms for molecular simulations of biopolymers

“…For reviews of molecular simulations based on Tsallis statistics, see, e.g., Refs. 71–73.) In this generalized ensemble the weight factor is known, once the value of the global‐minimum energy is given 61.…”

A nonextensive method for spectroscopic data analysis with artificial neural networks

“…Nonextensive statistical mechanics [14] have successfully been applied in physics (astrophysics, astronomy, cosmology, nonlinear dynamics etc) [18,19], chemistry [3], biology [20], human sciences [21], economics [22], computer sciences [2,23,24], and others [25].…”

Reducing quasi-ergodicity in a double well potential by Tsallis Monte Carlo simulation

“…Andricioaei and Straub applied this Tsallis Monte Carlo algorithm to the two-dimensional Ising system [8], a classical one-dimensional potential and a 13-atom Lennard-Jones cluster [9], and showed that this new Monte Carlo algorithm is more effective than the standard Metropolis algorithm. They also proposed the generalized molecular dynamics based on Tsallis generalized statistical mechanics [9,10] 3 Application to a classical double well potential In order to check the performance of this new Monte Carlo algorithm [8,9] in a classical system such as molecules and liquids, we look at the problem of a classical particle in a one-dimensional double well potential defined by…”