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

Tsallis's generalization of statistical mechanics is summarized. A modifica tion of this formalism which employs a normalized expression of the q-expec tation value for the computation of equilibrium averages is reviewed for th e cases of pure Tsallis statistics and Maxwell-Tsallis statistic, Monte Car lo and Molecular Dynamics algorithms which sample the Tsallis statistical d istributions are presented, These methods have been found to be effective i n 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 st andard Cartesian positions and momenta with a set of positions and moments which depend on the potential energy. It is shown that pure Tsallis statist ical averages in this transformed phase space result in the q-expectation a verages of Maxwell-Tsallis statistics. Finally, an alternative novel deriva tion of the Tsallis statistical distribution is presented. The derivation b egins with the classical density matrix, rather than the Gibbs entropy form ula, but arrives at. the standard distributions of Tsallis statistics, The result suggests a new formulation of imaginary time path integrals which ma y lead to an improvement in the simulation of equilibrium quantum statistic al averages.