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Progress of Theoretical Physics

Quantum Statistical Mechanics of a Many-Body System of Interacting Particles

Raj K. Pathria, M. P. Kawatra

Published Jun 1, 1962• 2 citations
Publisher

Abstract

The binary collision method of Lee and Yang is applied to investigate the equilibrium properties of a many-body system of particles interacting through a generalized two-body potential. It is found that, to the order a2/λ2 where a and λ represent respectively the range of the two-body interaction and the mean thermal wavelength of the particles, the final results can be written down explicitly in terms of the relevant phase shift for the S-waves. Ground state properties of a few systems of particles interacting through certain typical potentials and obeying Fermi-Dirac or Bose-Einstein statistics are studied.

DOI
10.1143/ptp.27.1085

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The binary collision method of Lee and Yang is applied to investigate the equilibrium properties of a many-body system of particles interacting through a generalized two-body potential. It is found that, to the order a2/λ2 where a and λ represent respectively the range of the two-body interaction and the mean thermal wavelength of the particles, the final results can be written down explicitly in terms of the relevant phase shift for the S-waves. Ground state properties of a few systems of particles interacting through certain typical potentials and obeying Fermi-Dirac or Bose-Einstein statistics are studied.
Following the binary collision approach to the many-body problem in Quantum Statistical Mechanics, as developed recently by Lee and Yang, the equation of state, to the order a2/λ2, is derived for a system of particles interacting through a square-well potential and obeying Bose-Einstein or Fermi-Dirac statistics. Here, λ is the mean thermal wavelength of the particles and a represents the range of the two-body interaction. Expressions are obtained for the second and the third virial coefficients of the system and also for its various thermodynamical properties, such as energy, specific heat and entropy, at finite but low temperatures.
The binary collision method of Lee and Yang is extended in order to investigate the statistical mechanics of a multicomponent mixture of interacting gases obeying Bose-Einstein or Fermi-Dirac statistics. First, the case of a mixture of two kinds of hard-sphere spinless bosons is studied and then the generalization is carried over to the case of a mixture of several components consisting of particles with arbitrary spin and statistics. An explicit derivation of the fugacity expansion, correct to the second order in the interaction parameters, is given. Expressions are derived for the second and third virial coefficients of the system and a discussion is given of the salient features of the results obtained. Finally, the case of a binary mixture, with one component consisting of bosons and the other of fermions, is considered and the phenomenon of Bose-Einstein condensation in this system is studied.
A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain UUU functions defined for the same system with Boltzmann statistics. It is then shown that these UUU functions can be evaluated in successive approximations in terms of a binary kernel BBB which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.
A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation ("two-body approximation") corresponds to the exact resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applications of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at Tc [ALMOST EQUAL TO] 2[pi]n/[m log(2[pi]/mg )], where n is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of [MICRO SIGN]/T ) and computed from the numerical solutions. For bosons in three spatial dimensions, we present evidence that the gas undergoes 3 a strongly interacting version of Bose-Einstein condensation at n[lamda]T [ALMOST EQUAL TO] 1.3, where n is the number density and [lamda]T is the thermal wavelength. Finally, we look at the ratio of shear viscosity to entropy density [eta]/s of the unitary quantum gas, which has a conjectured lower bound of [eta]/s [GREATER-THAN OR EQUAL TO] /4[pi]kb based on the AdS/CFT correspondence of a strongly coupled Yang-Mills theory.