Condensed Matter Physics, 2006, vol. 9, No. 1(45), p. 2336, English
DOI:10.5488/CMP.9.1.23
Title:
Classical statistical thermodynamics of a gas of charged particles in
magnetic field
Author(s):
 I.M.Dubrovskii
(Institute for Metal Physics 36 Vernadsky Str., Kyiv 03680, Ukraine)

We will demonstrate that the paradox of classical statistical
thermodynamics for a gas of charged particles in a magnetic field
(GMF) has not yet been explained. The paradox lies in the
statement that the average magnetic moment of a gas is zero,
whereas the timeaverage magnetic moment of each particle is
always negative. We consider a gas of charged particles moving in
a plane perpendicular to a uniform magnetic field. The density of
distribution of the ensemble describing statistical properties of
the GMF is derived starting from the basics, with due regard for
the specific character of dynamics of the charged particles in the
magnetic field. It is emphasized that neither the imposition of a
potential barrier restricting the existence region of the GMF, nor
the introduction of a background neutralizing charge occupying a
finite area, is a necessary condition for the stationary
equilibrium state of the GMF to exist. We show that the reason for
this fact is that the density of distribution of the ensemble is
dependent, besides the Hamiltonian, on another positive definite
integral of motion that is a linear combination of the Hamiltonian
and the angular momentum of the GMF. Basic thermodynamic relations
are deduced in terms of the new density of distribution, and it is
demonstrated that the GMF has a magnetic moment whose magnitude
and sign are determined by the external potential field.
Particularly, the GMF is diamagnetic in the absence of the
neutralizing background charge. Thus, the statement of the
Bohrvan Leeuwen theorem, deduced using the ordinary Gibbs density
of distribution depending on the Hamiltonian only, is wrong. It is
noted that a great deal of works on the theory of electronic
phenomena in magnetic field are based either on the same wrong
density of distribution or on the formula for average occupation
numbers depending on the energy of states, which follows from this
density of distribution within quantum theory. These theories
should be revised in view of the new form of the density of
distribution.
Key words: gas, charged particles, magnetic field,
density of distribution, phase space, integral of motion,
thermodynamics, magnetic moment
PACS: 05.20.Gg, 75.20.g
