Path Integral Representation of the Wigner Function in Canonical Ensemble

Abstract

Quantum effects can affect the shape of the particle kinetic energy distribution function, as the interaction of a particle with its surroundings restricts the volume of configuration space, which, due to the uncertainty relation, results in an increase in the volume of the momentum space, i.e., in a rise in the fraction of particles with higher momenta. Allowing for quantum effects at calculations of the equilibrium rate constants of inelastic processes is important in consideration of such phenomena as the transition of combustion into detonation, flame propagation, vibrational relaxation, and even thermonuclear fusion at high pressure and low temperatures. Quantum effects are also important in treatment of transport properties of the strongly interacting systems of many particles. In this work the new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in harmonic approximation. New quantum Monte-Carlo method for ab initio calculations of the average values of quantum operators, Wigner function, momentum and position distributions and wave functions of the ground state has been developed and tested. Obtained results are in a very good agreement with available analytical results and results of usual path-integral Monte-Carlo method. The developed approach allows simulation of thermodynamic and kinetic properties of quantum systems and calculation average values of quantum operators, when the usual path integral Monte Carlo methods in configurational space failed.

Peculiarities of the momentum distribution functions of strongly correlated charged fermions

“…Numerical valuee of this integral rapidly decreases as 1/24, 1/1920 and 1/322560 for n = 2, 4, 6 and are lesser for odd n. To check the accuracy of this approximation Monte Carlo simulations of the thermodynamic values of particles in different arbitrary potentials have been carried out in [17,18]. These calculations confirm our expectations.…”

“…When parameter of degeneracy is more than 10 (T /E F = 0.141) the thermal electron wave length is of order of Monte Carlo cell size and influence of finite number of particles and periodic boundary conditions becomes significant as has been tested in our calculations. Let us stress that detailed test calculation for interacting particle carried out within developed approach in [17,18] have demonstrated good agreement with results of independent calculations by standard path integral Monte Carlo method and available analytical results.…”

“…Since operators of kinetic and potential energy in Hamiltonian do not commutate, the exact explicit analytical expression for Wigner function does not in general exist. To overcome this difficulty let us represent Wigner function in form of path integral similarly to path integral representation of the partition function [1,2,[17][18][19][20]. As example of Coulomb system of particles, we consider a 3D two-component mass asymmetric electron -hole plasma consisting of N e electrons and N h heavier holes in equilibrium (N e = N h = N ) [21].…”

“…Here we are going to obtain new representation of Wigner functions in the path integral form [1,2,14,[17][18][19][20], which allows the numerical simulations of strongly coupled quantum systems of particles in canonical ensemble. Since operators of kinetic and potential energy in Hamiltonian do not commutate, the exact explicit analytical expression for Wigner function does not in general exist.…”

Phase Space Path Integral Representation for Wigner Function

“…In this paper we continue developing the ab initio path integral Monte Carlo approach in phase space by using the basic ideas suggested before in [22]. Here we have generalized the path integral representation for Wigner function to three-dimensional many-particle quantum system with Boltzmann and Fermi statistics.…”