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Contributions to Plasma Physics

Calculation of Canonical Properties of a Quantum System by Path Integral Numerical Methods

Mikhail Voznesenskiy, P. N. Vorontsov‐Velyaminov

Published May 1, 2013• 0 citations
Publisher

Abstract

Abstract We propose a new approach in obtaining the partition function of a system of several interacting particles (fermions) in external field within path integral Monte Carlo method based on direct averaging of the exchange contributions over the positive weight determined by non‐closed trajectories. This procedure is performed at fixed temperatures, ranging to rather low values. The complete partition function for each temperature is then obtained within an expanded ensemble procedure. We found a good agreement of data obtained by the new method with results of previously proposed path‐integral‐expanded‐ensemble Monte Carlo calculations [1, 2]. The new approach allows to reach observably lower temperatures compared to method [1, 2], thus providing a significant reduce of the sign problem. It yields an independent way to treat thermal properties of quantum systems, so the good agreement with previous data [1, 2] allows us to test and state the validity of both approaches. Simulations for systems of 2 up to 5 non‐interacting particles and particles interacting with Coulomb repulsion in 3D harmonic field were performed. The presented approach allows one to carry out calculations for low temperatures that makes it possible to extract data for the ground‐state energy and low‐temperature thermodynamics. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/ctpp.201200092

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It is shown from first principles that, in spite of the large interatomic forces, liquid 4 He should exhibit a transition analogous to the transition in an ideal Bose–Einstein gas. The exact partition function is written as an integral over trajectories, using the space-time approach to quantum mechanics. R.P. Feynman, 1953 Summary In this chapter we discuss imaginary-time path integrals and the path-integral Monte Carlo method for the calculation of properties of quantum systems at non-zero temperature. We discuss how Fermi and Bose statistics enter, and how to generalize the fixed-node procedure to non-zero temperature. We then discuss an auxiliary-field method for the Hubbard model. The path-integral method can be used to perform ground-state calculations, allowing calculations of properties with less bias than the projector Monte Carlo method. We also discuss the problem of estimating real-time response functions using information from imaginary-time correlation functions. In previous chapters we described two QMC methods, namely variational QMC and projector (diffusion) QMC. Both of these methods are zero temperature, or, more properly, are formulated for single states. In this chapter we discuss the path-integral Monte Carlo (PIMC) method, which is explicitly formulated at non-zero temperature. Directly including temperature is important because many, if not most, measurements and practical applications involve significant thermal effects. One might think that to do calculations at a non-zero temperature we would have to explicitly sum over excited states. Such a summation would be difficult to accomplish once the temperature is above the energy gap, because there are so many possible many-body excitations. In addition, the properties for each excitation are more difficult to calculate than those for the ground state. As we will see, path-integral methods do not require an explicit sum over excitations. As an added bonus, they provide an interesting and enlightening window through which to view quantum systems. However, the sign problem, introduced in the previous chapter, is still present for fermion systems. The fixed-node approximation is used again. An advantage of PIMC is the absence of a trial wavefunction. As a result, quantum expectation values, including ones not involving the energy, can be computed directly. For the expert, the lack of an importance function may seem a disadvantage; without it one cannot push the simulation in a preferred direction.

Cambridge University Press eBooks•May 31, 2016•1 citation
Abstract We propose a new approach in obtaining the partition function of a system of several interacting particles (fermions) in external field within path integral Monte Carlo method based on direct averaging of the exchange contributions over the positive weight determined by non‐closed trajectories. This procedure is performed at fixed temperatures, ranging to rather low values. The complete partition function for each temperature is then obtained within an expanded ensemble procedure. We found a good agreement of data obtained by the new method with results of previously proposed path‐integral‐expanded‐ensemble Monte Carlo calculations [1, 2]. The new approach allows to reach observably lower temperatures compared to method [1, 2], thus providing a significant reduce of the sign problem. It yields an independent way to treat thermal properties of quantum systems, so the good agreement with previous data [1, 2] allows us to test and state the validity of both approaches. Simulations for systems of 2 up to 5 non‐interacting particles and particles interacting with Coulomb repulsion in 3D harmonic field were performed. The presented approach allows one to carry out calculations for low temperatures that makes it possible to extract data for the ground‐state energy and low‐temperature thermodynamics. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract Path integral Monte Carlo method in the expanded ensemble is used for calculation of the ratio of partition functions for different classes of permutations treating the problem of several interacting identical particles (fermions) in an external field. Wang‐Landau algorithm is used for adjustment of balancing factors. For systems consisting of greater than two number of particles we propose an advanced variant of our approach which implies calculation of the ratio of positive and negative contributions to the partition function. Densities and energies of the sequence of excited states starting from the ground state for a system of non interacting quantum particles are calculated in turn, one by one, by means of considering systems with artificially excluded lowest energy levels and further obtaining of the ”ground state” of each next system constructed in this way. The idea of evaluation of densities of excited states for a quantum system with interparticle interaction by evaluating density difference between systems of different number of noninteracting Fermi‐copies of the system of interest is realized in terms of cyclic expansion formalism for a simple 2D system of two spinless fermions interacting via Coulomb repulsion (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Expanded-ensemble Monte Carlo method with Wang-Landau algorithm was used for calculations of the ratio of partition functions for classes of permutations in the problem of several interacting quantum particles (fermions) in an external field. Simulations for systems consisting of 2 up to 7 interacting particles in harmonic or Coulombic field were performed. The presented approach allows one to carry out calculations for low enough temperatures that makes it possible to extract data for the ground-state energy and low-temperature thermodynamics.
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. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract A novel path integral Monte Carlo (PIMC) approach for correlated many‐particle systems with arbitrary pair interaction in continuous space at low temperatures is presented. It is based on a representation of the N ‐particle density operator in a basis of (anti‐)symmetrized N ‐particle states (configurations of occupation numbers). The path integral is transformed into a sum over trajectories with the same topology and, finally, the limit of M → ∞, where M is the number of high‐temperature factors, is analytically performed. This yields exact expressions for the thermodynamic quantities and allows to perform efficient simulations for fermions at low temperature and weak to moderate coupling. Our method is expected to be applicable to dense quantum plasmas in the regime of strong degeneracy where conventional PIMC fails due to the fermion sign problem (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The main difficulty for path integral Monte Carlo studies of Fermi systems results from the requirement of antisymmetrization of the density matrix and is known in literature as the 'sign problem'. To overcome this issue the new numerical version of the %{\large {\tiny {\Huge }}} Wigner approach to quantum mechanics for treatment thermodynamic properties of degenerate systems of fermions has been developed. The new path integral representation of quantum Wigner function in the phase space has been obtained for canonical ensemble. Explicit analytical expression of the Wigner function accounting for Fermi statistical effects by effective pair pseudopotential has been proposed. Derived pseudopotential depends on coordinates, momenta and degeneracy parameter of fermions and takes into account Pauli blocking of fermions in phase space. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. To test the developed approach calculations of the momentum distribution function of the degenerate ideal system of Fermi particles has been carried out in a good agreement with analytical Fermi distributions. Generalization of this approach for studies influence of interparticle interaction on momentum distribution functions of strongly coupled Fermi system is in progress.
It is shown from first principles that, in spite of the large interatomic forces, liquid 4 He should exhibit a transition analogous to the transition in an ideal Bose–Einstein gas. The exact partition function is written as an integral over trajectories, using the space-time approach to quantum mechanics. R.P. Feynman, 1953 Summary In this chapter we discuss imaginary-time path integrals and the path-integral Monte Carlo method for the calculation of properties of quantum systems at non-zero temperature. We discuss how Fermi and Bose statistics enter, and how to generalize the fixed-node procedure to non-zero temperature. We then discuss an auxiliary-field method for the Hubbard model. The path-integral method can be used to perform ground-state calculations, allowing calculations of properties with less bias than the projector Monte Carlo method. We also discuss the problem of estimating real-time response functions using information from imaginary-time correlation functions. In previous chapters we described two QMC methods, namely variational QMC and projector (diffusion) QMC. Both of these methods are zero temperature, or, more properly, are formulated for single states. In this chapter we discuss the path-integral Monte Carlo (PIMC) method, which is explicitly formulated at non-zero temperature. Directly including temperature is important because many, if not most, measurements and practical applications involve significant thermal effects. One might think that to do calculations at a non-zero temperature we would have to explicitly sum over excited states. Such a summation would be difficult to accomplish once the temperature is above the energy gap, because there are so many possible many-body excitations. In addition, the properties for each excitation are more difficult to calculate than those for the ground state. As we will see, path-integral methods do not require an explicit sum over excitations. As an added bonus, they provide an interesting and enlightening window through which to view quantum systems. However, the sign problem, introduced in the previous chapter, is still present for fermion systems. The fixed-node approximation is used again. An advantage of PIMC is the absence of a trial wavefunction. As a result, quantum expectation values, including ones not involving the energy, can be computed directly. For the expert, the lack of an importance function may seem a disadvantage; without it one cannot push the simulation in a preferred direction.