The new numerical version of Wigner approach to quantum thermodynamics of strongly coupled systems of particles has been developed for extreme conditions, when analytical approximations obtained in different kind of perturbation theories can not be applied. Explicit analytical expression of Wigner function has been obtained in linear and harmonic approximations. Fermi statistical effects are accounted by effective pair pseudopotential depending on coordinates, momenta and degeneracy parameter of particles and taking into account Pauli blocking of fermions. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Calculations of the momentum distribution functions and pair correlation functions of the degenerate ideal Fermi gas have been carried out for testing the developed approach. Comparison of obtained momentum distribution functions of strongly correlated Coulomb systems with MaxwellBoltzmann and Fermi distributions shows the significant influence of interparticle interaction both at small momenta and in high energy quantum 'tails'.

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.

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Quantum interference and exchange statistical effects can affect the momentum distribution functions making them non-Maxwellian. Such effects may be important in studies of kinetic properties of matter at low temperatures and under extreme conditions. In this work we have generalized the path integral representation for Wigner function to strongly coupled three-dimensional quantum system of particles with Boltzmann and Fermi statistics. In suggested approach the explicit expression for Wigner function was obtained in harmonic approximation and Monte Carlo method allowing numerical calculation of Wigner function, distribution functions and average quantum values has been developed. As alternative more accurate single-momentum approach and related Monte Carlo method have been developed to calculation of the distribution functions of degenerate system of interacting fermions. It allows partially overcoming the well-known sign problem for degenerate Fermi systems.

In this paper, the single-momentum path integral Monte Carlo method, previously developed for simple quantum systems and hydrogen plasma, is adapted to simulations of the uniform electron gas. The developed method is based on the combination of Wigner formalism and the path integral approach and is able to calculate various thermodynamic values and distribution functions without differentiation of the partition function. Since the exchange interaction between electrons is taken into account by the Gram determinants of the exchange matrix, the fermionic sign problem is reduced significantly, and in the case of coordinate-depending variables, is completely eliminated. The method was applied to study thermodynamic properties of the uniform electron gas in warm dense matter regime. Average kinetic, potential, and exchange-correlation energy were calculated in a wide range of states.