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An explicit analytical expression of the Wigner function has been proposed to account for Fermi statistical effects using an effective pair pseudopotential in phase space. The derived pseudopotential depends on coordinates, momenta, and the degeneracy parameter of fermions and takes into account Pauli blocking of fermions in phase space. A new quantum Monte Carlo method (WPIMC) is put forward to calculate average values of arbitrary quantum operators in phase space. When calculated using the WPIMC method, the momentum distributions and pair distribution functions for degenerate ideal fermions are in good agreement with the analytical distribution over a wide range of values of the degeneracy parameter. Generalization of this approach for treating strongly correlated fermions is in progress.

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.

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. (© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Over the last decades the fixed-node method has been used for a numerical treatment of thermo-dynamic properties of strongly correlated Fermi systems. In this work correctness of the fixed-node method for ideal Fermi systems is analytically analyzed. It is shown that the fixed-node prescription of calculation of the density matrix leads to contradictions even for two ideal fermions. The main conclusion of this work is that the fixed-node method can not reproduce the fermion density matrices and should be considered as uncontrolled empirical approach in treatment of thermodynamics of Fermi systems.

We are going to study the kinetic properties of dense hydrogen plasma by a new quantum dynamics method in the Wigner representation of quantum mechanics. This method combines the Feynman and Wigner formulation of quantum mechanics and uses for calculation the direct path integral Monte Carlo (PIMC) and molecular dynamics methods. We are going to solve the Wigner–Liouville equation for dense degenerate hydrogen with the initial condition sampled by the PIMC method from equilibrium plasma state. We hope to do calculations of the hydrogen plasma properties under extreme conditions.

The uniform electron gas (UEG) at finite temperature has recently attracted substantial interest due to the epxerimental progress in the field of warm dense matter. To explain the experimental data accurate theoretical models for high density plasmas are needed which crucially depend on the quality of the thermodynamic properties of the quantum degenerate correlated electrons. Recent fixed node path integral Monte Carlo (RPIMC) data are the most accurate for the UEG at finite temperature, but they become questionable at high degeneracy when the Brueckner parameter $r_s$ becomes smaller than $1$. Here we present new improved direct fermionic PIMC simulations that are exptected to be more accurate than RPIMC at high densities., arXiv admin note: text overlap with arXiv:cond-mat/0702049

A new numerical approach for the treatment of quantum dynamics and for the computation of average values of quantum operators and of time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. For electrons in a disordered system of scatterers, new numerical results have been obtained for various average values of quantum operators, including the position and momentum dispersions, the average energy, the energy distribution function, and frequency-dependent tensors, such as the electron conductivity and the permittivity, which follow from the quantum Kubo formula.

The new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For electrons in disordered systems of scatterers the numerical results have been obtained for series of the average values of the quantum operators including position and momentum dispersions, average energy, energy distribution function as well as for the frequency dependencies of tensor of electron conductivity and permittivity according to quantum Kubo formula. Zero or very small value of static conductivity have been considered as the manifestation of Anderson localization of electrons in 1D case.

Based on the quasiparticle model of the quark-gluon plasma (QGP), a color quantum path-integral Monte-Carlo (PIMC) method for calculation of thermodynamic properties and -- closely related to the latter -- a Wigner dynamics method for calculation of transport properties of the QGP are formulated. The QGP partition function is presented in the form of a color path integral with a new relativistic measure instead of the Gaussian one traditionally used in the Feynman-Wiener path integral. It is shown that the PIMC method is able to reproduce the lattice QCD equation of state at zero baryon chemical potential at realistic model parameters (i.e. quasiparticle masses and coupling constant) and also yields valuable insight into the internal structure of the QGP. Our results indicate that the QGP reveals quantum liquid-like (rather than gas-like) properties up to the highest considered temperature of 525 MeV. The pair distribution functions clearly reflect the existence of gluon-gluon bound states, i.e. glueballs, at temperatures just above the phase transition, while meson-like $q\bar{q}$ bound states are not found. The calculated self-diffusion coefficient agrees well with some estimates of the heavy-quark diffusion constant available from recent lattice data and also with an analysis of heavy-quark quenching in experiments on ultrarelativistic heavy ion collisions, however, appreciably exceeds other estimates. The lattice and heavy-quark-quenching results on the heavy-quark diffusion are still rather diverse. The obtained results for the shear viscosity are in the range of those deduced from an analysis of the experimental elliptic flow in ultrarelativistic heavy ions collisions, i.e. in terms the viscosity-to-entropy ratio, $1/4��< ��/S < 2.5/4��$, in the temperature range from 170 to 440 MeV., 23 pages, 17 figures, version accepted by Phys. Rev. C, technical problems with file fixed

The integral Wigner - Liouwille equation describing time evolution of the semi-relativistic quantum 1D harmonic oscillator have been exactly solved by combination of the Monte-Carlo procedure and molecular dynamics methods. The strong influence of the relativistic effects on the time evolution of the momentum, velocity and coordinate Wigner distribution functions and the average values of quantum operators have been studied. Unexpected 'protuberances' in time evolution of the distribution functions were observed. Relativistic proper time dilation for oscillator have been calculated., 10 pages, 26 figures. arXiv admin note: text overlap with arXiv:hep-th/0110114, arXiv:hep-ph/0501268, arXiv:math-ph/0702082 by other authors

The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. 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 linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics ("tails') of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.

A new method for solving the Wigner-Liouville equation allowing a treatment of quantum dynamics has been developed. The method combines two widely known approaches: molecular dynamics and Monte Carlo. Numerical results have been obtained for one-, two- and three-dimensional potential wells of realistic type. The quantum dynamics method was tested by a comparison of the numerical results with analytical estimates for such values as the average position x¯(t), average momentum p¯(t), energy Ē(t), position and momentum dispersions, and Heisenberg inequality.

A strongly coupled plasma of quark and gluon quasiparticles at temperatures from 1.1Tc to 3Tc is studied by path integral Monte Carlo simulations. This method extends previous classical nonrelativistic simulations based on a color Coulomb interaction to the quantum regime. We present the equation of state and find good agreement with lattice results. Further, pair distribution functions and color correlation functions are computed indicating strong correlations and liquid-like behavior (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper the numerical procedure combining both molecular dynamics and Monte Carlo methods for solving the integral Wigner- Liouville equation has been developed and applications of this approach to the treatment of kinetic properties of quantum dense hydrogen plasma has been considered. To study the influence of the Coulomb interaction on kinetic properties of dense plasma the quantum dynamics in a canonical ensemble at finite temperature for both weakly and strongly coupled plasmas was simulated. The main quantities calculated are the temporal momentum-momentum correlation functions and their frequency-domain Fourier transforms.

In the standard formulation of quantum mechanics states of a system are described by wave functions or density operators . However quantum description of the system can be given in many other ways , for example, in Wigner-Moyal or Feynman formulations . All these representation are equivalent in the the sense of physical results, but the form of presentation of quantum mechanics is different as wave functions, Wigner-Liouville functions or other representations of density operators differ in essential way from each other. In this work we are dealing with Wigner and probability representation of quantum mechanics. The probability representations or tomography representation was recently proposed in terms of marginal distribution functions (MDF). MDF describing the quantum states are positive distribution functions connected with wave functions or density matrices by known integral transformation. Sign conservation of the of the MDF can be valuable in computer simulations to overcome the "sign problem". Important advantage of the probability representation is that the quantum transitions between quantum states are descried by non-negative probabilities. This work is devoted to the development of a new stochastic approaches to numerical solution of the evolution equations for Wigner-Liouviile (WL) and marginal distribution functions. To solve the WL equation we combine both molecular dynamics and Monte Carlo methods and compute traces of the dynamical operators. The results obtained for a system of electrons and random scatterers clearly demonstrate that the many-particle interaction between the electrons leads to an enhancement of the conductivity compared to the case of noninteracting electrons. To obtain evolution of MDF we developed new stochastic approach to solution of the generalized Langevin equation (GLE). GLE is derived from Kolmogorov equations for Green function of evolution equation for MDF. We discuss the basic relations and main ideas of this approach, compare obtained numerical results with results of independent finite-difference calculations for quantum oscillator and quantum particles crossing the finite well and tunneling through the Gaussian barrier.

The uniform electron gas (UEG) at finite temperature has recently attracted substantial interest due to the experimental progress in the field of warm dense matter. To explain the experimental data accurate theoretical models for high density plasmas are needed which crucially depend on treatment of quantum effects in electron-electron interaction as well as in the interaction of electrons with uniform positive background. To comply with these requirements we have developed the new quantum path integral model of the UEG and present the results of related direct path integral Monte-Carlo (DPIMC) simulations. Contrary to the known in literature approaches treating the electron-background interaction classically our simulations take into account the quantum effects in this interaction. We have observed very good agreement with known in literature results only up to moderate densities when the ratio of the average interparticle distance to the Bohr radius is of order four (rs ≥ 4) and observe deviations for higher densities. At very high electron density (rs ≈ 1) presented in literature approaches as well as our simulations are problematic due to the strong degeneracy of electrons and increasing fermion sign problem.

The correlation length exponent ? has been calculated for the hard-sphere Bose-Einstein condensation by path integral Monte Carlo simulations in the Feynman formulation of quantum mechanics. This new calculation method allows to determine the density dependence of the correlation length exponent. It is shown that ?, being equal to unity for an ideal gas, is close to 2/3 for boson hard spheres in a wide range of gas densities with an accuracy of 10%.

Path integrals and complex Monte Carlo method are formulated for the system of noninteracting electrons in a random potential of scatterers and external electrical field. The ensemble-averaged probability for electron with energy E to travel a distance \( \left| {r - r'} \right| \) have been calculated. The numerical results for 1D, 2D and 3D disordered systems have been obtained and compared with analytical results. The unwaited sharp peaks, which interrupt the exponential decay of the mean squared value of the Green function modulus and delocalization of electrons in external electrical field have been also obtained.

This paper presents a first step towards combining two well-established methods used in semiconductor physics—semiconductor Bloch equations and the Wigner approach to quantum transport. This combination provides the possibility of including spontaneous emission, i.e., the spontaneous recombination of excited electron–hole pairs in semiconductors, into the Wigner approach, which so far has been used only for systems with fixed particle number. The theory is presented and first numerical results for a three-dimensional system are shown.