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Physical Review

Quantum Statistical Mechanics of a Many-Body System with Several Components

Raj K. Pathria, M. P. Kawatra

Published Jan 15, 1963• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.129.944

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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.
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 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.
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.
We use the Lee-Yang cluster expansion method to study quantum-statistical properties of a mixture of interconvertible atoms and dimers, where the dimers form in a two-body bound state of the atoms. We point out an infinite series of cluster diagrams whose summation leads to the Bose-Einstein condensation of the dimers below a critical temperature. Our theory captures some important features of a cold atom-dimer mixture such as interconversion of atoms and dimers and properties of the mixture at the unitarity limit.
Multicomponent quantum gases are ideal platforms to study fundamental phenomena arising from the mutual interaction between different constituents. Particularly, due to the repulsive interactions between two species, the system may exhibit a phase separation. We develop a mean-field-based theory for a two-component Bose mixture, which is equivalent to the Hartree-Fock-Bogoliubov approximation, and derive analytical expressions for the phase boundary and miscibility. The majority of existing theories, which are valid only for weakly interacting Bose gases, predict that the phase boundary is determined by the criterion ${g}_{ab}\ensuremath{\leqslant}\sqrt{{g}_{aa}{g}_{bb}}$ (where gab{g}_{ab}gab​ is a coupling constant between the components aaa and bbb). We show that in the Bose-Einstein condensation phase ($T\ensuremath{\leqslant}{T}_{c}$) the system may remain in a stable and miscible phase also for larger values of gab{g}_{ab}gab​, depending on the gas parameter $\ensuremath{\gamma}$ and temperature.