Stationary solutions of Liouville equations for non-Hamiltonian systems

Abstract

We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by Hamiltonian. The constant temperature systems, canonical-dissipative systems, and Fermi-Bose classical systems are the special cases of this class of non-Hamiltonian systems.

Measure-preserving integrators for molecular dynamics in the isothermal–isobaric ensemble derived from the Liouville operator

“…In addition, it is not necessary to derive closed-form expressions for integrators based on factorizations of the classical propagator, such as Eq. (24). In fact, the same algorithm results if each operator in Eq.…”

“…(7) and (10) constitute a non-Hamiltonian dynamical systems. The phase space distributions generated by classical non-Hamiltonian systems can be determined using a framework introduced by some of us [13,14] and subsequently elaborated by others [20][21][22][23][24]. Briefly, if a dynamical system _ x ¼ nðxÞ, where x is the phase space vector and nðxÞ is a vector field, has a nonvanishing compressibility jðxÞ ¼ r x Á nðxÞ, then a generalization of Liouville's theorem is needed [13,14].…”

Nonholonomic constraints with fractional derivatives

“…These systems are non-Hamiltonian ones and they are described by the non-potential forces that are proportional to the velocity, and the Gaussian nonholonomic constraint. Note that this constraint can be represented as an addition term to the non-potential force [60].…”

“…In molecular dynamics calculations, nonholonomic systems can be exploited to generate statistical ensembles as the canonical, isothermalisobaric and isokinetic ensembles [50,51,52,53,54,55,56,57,58,59,60]. Using fractional nonholonomic constraints, we can consider a fractional extension of the statistical mechanics of conservative Hamiltonian systems to a much broader class of systems.…”

“…(2) In the papers [59,60], the canonical distribution is considered as a stationary solution of the Liouville equation for a wide class of non-Hamiltonian system. This class is defined by a very simple condition: the power of the non-potential forces must be proportional to the velocity of the Gibbs phase (elementary phase volume) change.…”