Quantum Statistical Mechanical Formulations of Three-Body Problems via Non-Local Potentials

Authors: A. Maghari, V. H. Maleki

Abstract:

In this paper, we present a quantum statistical mechanical formulation from our recently analytical expressions for partial-wave transition matrix of a three-particle system. We report the quantum reactive cross sections for three-body scattering processes 1+(2,3)→1+(2,3) as well as recombination 1+(2,3)→1+(3,1) between one atom and a weakly-bound dimer. The analytical expressions of three-particle transition matrices and their corresponding cross-sections were obtained from the threedimensional Faddeev equations subjected to the rank-two non-local separable potentials of the generalized Yamaguchi form. The equilibrium quantum statistical mechanical properties such partition function and equation of state as well as non-equilibrium quantum statistical properties such as transport cross-sections and their corresponding transport collision integrals were formulated analytically. This leads to obtain the transport properties, such as viscosity and diffusion coefficient of a moderate dense gas.

Keywords: Statistical mechanics, Nonlocal separable potential, three-body interaction, Faddeev equations.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1338458

References:

[1] L. D. Faddeev” Scattering theory for a three-particle system”, Sov. Phys.-JETP, vol. 12, 1961, pp. 1014-1019.
[2] A. Maghari and N. Tahmasbi “Scattering properties for a solvable model with a three-dimensional separable potential of rank 2”, J. Phys. A: Math. Gen., vol. 38, 2005, pp. 4469-4481.
[3] A. Maghari and M. Dargahi “The solvable three-dimensional rank-two separable potential model: partial-wave scattering”, J. Phys. A: Math. Theoret. vol. 41, 2008, pp. 275306-17.
[4] A. Maghari and V. M. Maleki “Analytical Solution of Partial-Wave Faddeev Equations with Application to Scattering and Statistical Mechanical Properties”, Commun. Theor. Phys. vol. 64, 2015, pp. 22- 28.
[5] R. G. Newton, “Scattering theory of waves and particles”, Springer- Verlag, Berlin, 1982.
[6] R. F. Snider “Simple example illustrating the different parametrizations of the Moller operator” J. Chem. Phys. vol. 88, 1988, pp. 6438-6447.
[7] J. G. Muga and R. F. Snider “Solvable three-boson model with attractive delta -function interactions” Phys.Rev., vol. 57A, 1998, pp. 3317-3329.
[8] J. G. Muga and J. P. Palao “Solvable model for quantum wavepacket scattering in one dimension” J. Phys. A: Math. Gen. vol. 31, 1998, pp. 9519-34.
[9] N. Tahmasbi and A. Maghari “Scattering problem with nonlocal separable potential of rank-two: Application to statistical mechanics” Physica A, vol. 382, 2007, pp. 537-548.
[10] A. Maghari and M. Dargahi “Scattering via a separable potential with higher angular momenta: application to statistical mechanics”, J. Stat. Mech. 2008, P10007.
[11] Hirschfelder J. O., Curtiss C. F. and Bird R. B., 1954 Molecular Theory of Gases and Liquids (Wiley, New York).
[12] Chapman S. and Cowling T. G., 1970 The Mathematical Theory of Non- Uniform Gases 3rd ed (London: Cambridge University Press).