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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Statistical mechanics of nonrelativistic charged particles in a constant magnetic field

Guy B. Standen, David J. Toms

Published Nov 1, 1999• 18 citations
Publisher

Abstract

The statistical mechanics of a system of nonrelativistic charged particles in a constant magnetic field is discussed. The spatial dimension D is arbitrary, with D>or=3 assumed. Calculations are presented from first principles using the effective action method. For D>or=5 the system has a phase transition with a Bose condensate. We show how the effective action method deals in a very natural way with the condensate, and study its role in the magnetization of the gas. For large values of the magnetic field we show how the magnetized gas in D spatial dimensions behaves like the free Bose gas in D-2 spatial dimensions. Even though for D=3 the magnetized gas does not have a phase transition for any nonzero value of the magnetic field, we show how the specific heat starts to resemble the result for the free gas as the magnetic field is reduced. A number of analytical approximations for the magnetization and specific heat are given, and compared with numerical results. In this way we are able to study in precise detail how the B-->0 limit of the magnetized gas is achieved.

DOI
10.1103/physreve.60.5275

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The statistical mechanics of a system of nonrelativistic charged particles in a constant magnetic field is discussed. The spatial dimension D is arbitrary, with D>or=3 assumed. Calculations are presented from first principles using the effective action method. For D>or=5 the system has a phase transition with a Bose condensate. We show how the effective action method deals in a very natural way with the condensate, and study its role in the magnetization of the gas. For large values of the magnetic field we show how the magnetized gas in D spatial dimensions behaves like the free Bose gas in D-2 spatial dimensions. Even though for D=3 the magnetized gas does not have a phase transition for any nonzero value of the magnetic field, we show how the specific heat starts to resemble the result for the free gas as the magnetic field is reduced. A number of analytical approximations for the magnetization and specific heat are given, and compared with numerical results. In this way we are able to study in precise detail how the B-->0 limit of the magnetized gas is achieved.
The statistical mechanics of a system of non-relativistic charged particles in a constant magnetic field is discussed. The spatial dimension DDD is arbitrary with D≥3D\geq 3D≥3 assumed. Calculations are presented from first principles using the effective action method. For D≥5D\geq 5D≥5 the system has a phase transition with a Bose condensate. We show how the effective action method method deals with in a very natural way with the condensate and study it's rôle in the magnetization of the gas. For large values of the magnetic field we show how the magnetized gas in DDD spatial dimensions behaves like the free Bose gas in (D−2)(D-2)(D−2) spatial dimensions. Even though for D=3 the magnetized gas does not have a phase transition for any non-zero value of the magnetic field, we show how the specific heat starts to resemble the result for the free gas as the magnetic field is reduced. A number of analytical approximations for the magnetization and specific heat are given and compared with numerical results. In this way we are able to study in precise detail how the B→0B\to 0B→0 limit of the magnetized gas is achieved.
We consider the problem of Bose-Einstein condensation for a system of nonrelativistic spin-0 bosons in a space of arbitrary dimension D. A general static homogeneous magnetic field is imposed. The effective action approach and \ensuremath{\zeta}-function regularization are used. If D=2\ensuremath{\delta} or 2\ensuremath{\delta}+1, a constant magnetic field is characterized by \ensuremath{\delta} independent components. If p\ensuremath{\le}\ensuremath{\delta} of these components are nonzero, the condition for Bose-Einstein condensation to occur is D-2p\ensuremath{\ge}3. This means that if D=2\ensuremath{\delta}, then Bose-Einstein condensation never occurs for p=\ensuremath{\delta}-1 or \ensuremath{\delta}. If D=2\ensuremath{\delta}+1, Bose-Einstein condensation never occurs for p=\ensuremath{\delta}. For D-2p\ensuremath{\ge}3, Bose-Einstein condensation does occur, and we show how it may be interpreted as symmetry breaking to give a ground state which is not constant.
A simple pedagogical introduction to the effective action method of quantum field theory is given at a level suitable for beginning postgraduate students. It is shown how to obtain the effective potential at zero temperature from a regularized zero-point energy. The results are applicable to curved as well as to flat space. The generalization to finite temperatures is also given. It is shown how to obtain high temperature expansions of the thermodynamic potential for the neutral free Bose gas and the charged Bose gas in both the relativistic and nonrelativistic limits. The results are obtained for an arbitrary spatial dimension and in curved space. Results are also obtained for the self-interacting relativistic gas in three spatial dimensions. A detailed discussion of how the formalism may be applied to study Bose-Einstein condensation is given. The interpretation of Bose-Einstein condensation as symmetry breaking is discussed. Application is given to the study of charged bosons, both relativistic and nonrelativistic, in a constant magnetic field. The Meissner effect is obtained for the nonrelativistic gas in three spatial dimensions. The final application is to the study of nonrelativistic bosons in a harmonic oscillator confing potential trap. A number of analytical approaches to this are discussed.
The condensation of the spinless ideal charged Bose gas in the presence of a magnetic field is revisited. The conventional approach is extended to include the macroscopic occupation of excited kinetic states lying in the lowest Landau level, which plays an essential role in the case of large magnetic fields. In that limit, signatures of two diffuse phase transitions (crossovers) appear in the specific heat. In particular, at temperatures lower than the cyclotron frequency, the system behaves as an effectively one-dimensional free boson system, with the specific heat equal to 1/2Nk(B) and a gradual condensation at lower temperatures.
The thermodynamic properties of a Bose gas of particles with nonzero spin and magnetic moment in states with and without a Bose-Einstein condensate are studied as a function of the external magnetic field. It is shown that there exists a temperature interval where a transition into a state with a Bose-Einstein condensate occurs in a Bose gas with constant temperature and particle number density when the magnetic field is increasing and reaches a critical magnitude. The temperature and magnetic field derivatives of the specific heat at constant volume and the magnetic susceptibility undergo a jump at such a transition.