Classical statistical thermodynamics of a gas of charged particles in magnetic field

• Published in: Condensed Matter Physics
• Date: 2006
• Main Author: Dubrovskii, I.M.
• Format: Article
• Language: English
• Published: Інститут фізики конденсованих систем НАН України 2006
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/121280
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Classical statistical thermodynamics of a gas of charged particles in magnetic field / I.M. Dubrovskii // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 23-36. — Бібліогр.: 19 назв. — англ.

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 time-average 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 Bohr-van 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