## 1. Introduction

The fundamental problem of statistical mechanics is obtaining an ensemble average of physical quantities that are described by phase functions (classical physics) or operators (quantum physics). In classical statistical mechanics the ensemble density of distribution is defined in the phase space of the system. In quantum statistical mechanics the space of functions that describe microscopic states of the system play a role similar to the classical phase space. The probability density of the system detection in the phase space must be normalized. It depends on external parameters that determine the macroscopic state of the system.

An in-depth study of the statistical mechanics foundations was presented in the works of A.Y. Khinchin (Khinchin, 1949, 1960). For classical statistical mechanics an invariant set was introduced. It would be mapped into itself by transforming with the Hamilton equations. The phase point of the isolated system remains during the process of the motion at the invariant set at all times. If the system is in the stationary equilibrium state, this invariant set has a finite measure. The Ergodic hypothesis asserts that in this case the probability dP (R) to detect this system at any point R of the phase space is:

where

A hypersurface in a hyperspace is a set with zero measure. Therefore the invariant set is determined as a thin layer that nearly envelops the hypersurface in the phase space. The determining equations of this hypersurface are the equalities that fix the values of controllable motion integrals. A controllable motion integral is a phase function, the value of which does not vary with the motion of the system and can be measured. An isolated system universally has the Hamiltonian that does not depend on the time explicitly, and is the controllable motion integral. A fixed value of the Hamiltonian is the energy of the system. The kinetic energy of majority of systems is a positive definite quadric form of all momenta. It determines a closed hypersurface in the subspace of momenta of the phase space. If motions of all particles are finite, the hypersurface of the fixed energy is closed and the layer that envelops it has the finite measure. Then this hypersurface can determine the invariant set of the system. A finiteness of motions of particles as a rule is provided by enclosing the system in an envelope that reflects particles without changing their energy, if the system is considered as isolated. It is common in statistical mechanics to consider the layer enveloping the energy hypersurface as the invariant set. But A.Y. Khinchin (Khinchin, 1949) shows that other controllable integrals of the system, if they exist, must be taken into account. In the general case an isolated system can have another two vector controllable integrals. That is the total momentum of the system, and the total angular momentum relative to the system’s mass centre. The total momentum is a sum of all momenta of particles. If the volume of the system is bounded by an external field or an envelope, the total momentum does not conserve. In the absence of external fields the total momentum conservation cannot make particle motions finite. Therefore the total momentum cannot be a controllable motion integral that determines the invariant set.

The angular momentum is another case. A vector of angular momentum relative to the mass center always is conserved in an isolated system. If this vector is nonzero, a condition should exist that provides a limitation of a gas expansion area. For example, nebulas do not collapse because they rotate, and do not scatter because of the gravitation. In the system of charged particles in a uniform magnetic field the conservation of the angular momentum provides a limitation of a gas expansion area (confinement of plasma). If a gas system is enclosed into envelope, and total system has nonzero angular momentum, the vector of the angular momentum should be conserved. However an envelope can have the non-ideal form and surface. That is the cause of the failure to consider the angular momentum of the gas as a controllable motion integral (Fowler, & Guggenheim, 1939). But if the cylindrical envelope rotates and the gas rotates with the same angular velocity deviations of the angular momentum of the gas from the fixed value as the result of reflections of particles from the envelope should be small and symmetric with respect to a sign. These fluctuations are akin to energy fluctuations for a system that is in equilibrium with a thermostat. Therefore the angular momentum conservation in specific cases can determine the invariant set and the thermodynamical natures of the system together with the energy conservation. Taking into account all controllable motion integrals is the necessary condition of the validity of the Ergodic hypothesis (Khinchin, 1949).

There is a contradiction in physics at the present time. Firstly, it has been proven that in the equilibrium state a system spin can exist only if the system is rigid and can rotate as a whole (Landau, & Lifshitz, E.M., 1980a). Therefore a gas, which supposed not be able to rotate as a whole, cannot have any angular momentum and spin. Based on this reasoning R.P. Feynman proves that an electron gas cannot have diamagnetism (the Bohr – van Leeuwen theorem) (Feynman, Leighton, & Sands, 1964). On the other hand, it is well known that density of a gas in a rotating centrifuge is non-uniform. This effect is used for the separation of isotopes (Cohen, 1951). The experiment by R. Tolman, described in the book (Pohl, 1960), is a proof of the existence of the electron gas angular momentum. In this experiment a coil was rotated and then sharply stopped. An electrical potential was observed that generated a moment of force, which decreased to zero the angular momentum of electron gas.

The contradiction described above requires creation of statistical mechanics for non-rigid systems taking into account the nonzero angular momentum conservation. This statistical mechanics differs from common one in many respects. If the angular momentum relative to the axis that passes through the mass centre conserves, the system is spatially inhomogeneous. This means that passage to the thermodynamical limit makes no sense, a spatial part of the system is not a subsystem that similar to the total system, specific quantities such as densities or susceptibilities have no physical meaning.

The microcanonical distribution is seldom used directly when the computations and the justifications of thermodynamics are done. The more usable Gibbs distribution can be deduced from microcanonical one (Krutkov, 1933; Zubarev, 1974). The Gibbs assembly describes a system that is in equilibrium with environment. These systems do not have motion integrals because they are non-isolated. All elements of the Gibbs assembly must have equal values of parameters that are determined by the equilibrium conditions. In usual thermodynamics this parameters are the temperature and the chemical potential. The physical interpretation of these parameters is getting by statistical mechanics. A rotating system can be in equilibrium only with rotating environment. The equilibrium condition in this case is apparent. That is equality of the both angular velocities of the system and of the environment. The Gibbs assembly density of distribution and thermodynamical functions in the case of a rotating classical system will be obtained in the second section of this work. It was done (Landau, & Lifshitz E.M., 1980a) but an object, to which this distribution is applied, is incomprehensible, because an angular velocity of an equilibrium gas has not been determined.

In quantum statistical mechanics the invariant set is the linear manifold of the microscopic states of the system in which the commutative operators that correspond to the controllable motion integrals have fixed eigenvalues. The phase volume of system in this case is the dimension of the manifold, if this dimension is limited. It directly determines the number of distinguishable microstates of the system that are accessible and equiprobable. The role of the angular momentum conservation in quantum statistical mechanics is similar to one in classical statistical mechanics. The method of computing this phase volume will be also proposed in the second section of this work. The Gibbs assembly density of distribution and thermodynamical functions in the case of a rotating quantum system also will be obtained.

In the third section of this work statistical mechanics of an electron gas in a magnetic field is considered. This question was investigated by many during the last century. Many hundreds experimental and theoretical works were summarized in the treatises (Lifshits, I.M. et al., 1973; Shoenberg, 1984). However, together with successful theoretical explanations of many experimental effects some paradoxes and discrepancies with observed facts remain unaccounted.

“Finally, it is shown that the presence of free electrons, contrary to the generally adopted opinion, will not give rise to any magnetic properties of the metals”. This sentence ends a short report on the presentation “Electron Theory of Metals” by N. Bohr, given at the meeting of the Philosophical Society at Cambridge. It was well-known that a charged particle in a uniform magnetic field moves in a circular orbit with fixed centre in such a way that the time average value of the magnetic moment, generated by this motion, is directed opposite to the magnetic field and equal to the derivative of the kinetic energy with respect to the magnetic field. N. Bohr computed the magnetic moment of an electron gas by statistical mechanics with the density of distribution that is determined only by a Hamiltonian. Zero result of this theory (Bohr – van Leeuwen theorem) is the first paradox. Many attempts of derivation and explanation of this were summarized in the treatise (van Vleck, 1965). The most widespread explanation was that the magnetization generated by the electrons moving far from the bound is cancelled by the near-boundary electrons that reflect from the bound. But this explanation is not correct because, when formulae are derived in statistical mechanics, any peculiarities of the near-boundary states shall not be taken into account. Another paradox of the common theory went unnoticed. It is well known that a uniform magnetic field restricts an expanse of a charged particles gas in the plane perpendicular to the field. But from common statistical mechanics it follows that the gas uniformly fills all of the bounded area. The diamagnetism of some metals also was left non-explained.

L.D. Landau (Landau, 1930) explained the diamagnetism of metals as a quantum effect. He solved the quantum problem of an electron in a uniform magnetic field. The cross-section of the envelope perpendicular to the magnetic field is a rectangle with the sides

Here

where

It is suggested that the Fermi level is filled. In this case the magnetic moment does not depend not only on the Plank constant, but also on the number of the electrons. Therefore the fundamental formula for the thermodynamical potential is incorrect.

In the third section of this work the diamagnetism of an electron gas is investigated with taking into account the conservation of zero value of the total angular momentum in classical and quantum statistical mechanics. The paradoxes described above are eliminated; however many other theories should be reconsidered.

## 2. Statistical mechanics of rotating gas

For the computation of average values of macroscopic quantities it is necessary to derive a formula of the phase volume as a function of macroscopic parameters. This function is called “structural function” by Khinchin (Khinchin, 1949) and “number of accessible states (or complexions)” by Fowler (Fowler, & Guggenheim, 1939). It determines the normalizing factor in the probability density of the microcanonical distribution (1). In usual theory this function is essential to the derivation of formulae that connect statistical physics with thermodynamics.

The system that will be considered is a collection of

For integration characteristic functions over a phase space the method by Krutkov will be used. The main idea of this method is to make the Laplace transformation of the

Let us write several equalities with a characteristic function. If the system can be divided into two independent subsystems described by non-overlapping groups of phase variables, so that

Here the multiplier

where

Here

### 2.1. Classical statistical thermodynamics of rotating gas

The formula for average values, when the conservation of the angular momentum is taken into account, has the form:

In these formulae the axis

where

Let us consider equilibrium of a gas with a rotating rigid body. The rigid body can be determined as the body in which the rotatory degree of freedom can not transfer energy and angular momentum to the internal degrees of freedom. This possibility arises when this body is a cylindrical rotating envelope with non-ideal surface filled by a gas. The state of the gas is characterized by two parameters: the temperature

The total system can be considered as motionless if it will be described in the rotating reference frame, when the right part of the equality (11) is zero. The hollow cylinder is the envelope, the thermostat, and it keeps the gas spin. It should be named “termospinstat”. The potential energy of the centrifugal force

where

The formulae of the isotopes separation (Cohen, 1951) can be obtained from the distribution (12). If

where

### 2.2. Quantum statistical thermodynamics of rotating gas

The characteristic function of the invariant set that takes into account conservation of the angular momentum in quantum statistical mechanics can be presented as a set of diagonal elements of the operator:

in the space of microstates of the system. Here

Here the following variables are entered:

Thus, integrals are rearranged into integrals along contours which enclose the origin of coordinates. If

Let us describe a quantum gas in a termospinstat with the temperature

Dependence of the wave function on the time should be

where

Let us compute the thermodynamical potential

If this result is compared with that of Eq. (14), it can be shown that amendments differ only by coefficients.

## 3. Statistical mechanics of electron gas in magnetic field

The review of the current status of this theory is in the paper (Vagner et al., 2006). There are some inaccuracies in this problem consideration besides disregard of the angular momentum conservation. To clarify the problem, in the first subsection we consider formulations of the one-particle problem in classical and quantum mechanics and its simplest application to the statistical mechanics. For simplicity, we will restrict ourselves to the case of a two-dimensional gas on a plane perpendicular to the uniform magnetic field

### 3.1. Two-dimensional electron ideal gas in uniform magnetic field

This problem traditionally is considered in quasiclassical theory (Lifshitz, I.M. et al., 1973; Shoenberg, 1984). Some corrections will be inserted in this consideration in the section 3.1.1. In this section classical statistic mechanics of ideal gas will be discussed. In the next section the new correction will be obtained from the consistent quantum theory.

#### 3.1.1. Classical statistical mechanics of ideal gas in magnetic field

The Hamiltonian of an electron in a magnetic field has the form:

where

This Hamiltonian does not have the translation symmetry. This symmetry, seemingly, should be, if the magnetic field is uniform at an unlimited plane. But a uniform magnetic field at unlimited plane is impossible because an electrical current that generates it according to Maxwell equation should envelope a part of this plane. It is asserted (Landau, & Lifshitz, E.M. 1980b; Vagner et al., 2006) that the Hamiltonian (21) with the vector potential (22) would be converted by gauge transformation

and will have the translation symmetry in the direction of the axis

An isolated electron has three motion integrals. Those are the angular momentum relative to the centre of area and two coordinates of the centre electron orbit:

Two motion integrals that have the physical importance would be created from it: energy

Here

Here

The first determination is valid for any negative charged particle with and without an external magnetic field. The second equality is valid when the vector potential has the form (22), or when the equality

Going to consideration of the ideal gas with electron-elektron collisions, let us suppose that the interaction does by a central force. Then total energy and angular momentum are conserved. It is generally believed that the area is filled by uniform and motionless positive charged background that neutralizes the electrostatic interaction. This assumption is inconsistently. If electron gas is in equilibrium with motionless background, its angular momentum should be equal to zero. But then it should be nonuniform as is evident from the foregoing consideration. It should be regarded more comprehensively. Let us go to classical statistic mechanics for a gas of charged particles in magnetic field. The characteristic function of the total system (gas and background) is:

Here the indexes

The Hamiltonian

Let us substitute Hamiltonian (30) to the formula (29) and take into account formula

The integration over

#### 3.1.2. Quantum problem of electron in magnetic field at bounded area

The quasiclassical description of an electron in a magnetic field would not give the correct picture of the probability density distribution and the current density. It would not also describe the alternation of the energy spectrum when a perturbation does the classical motion nonperiodical. But that problem in quantum mechanics also is considered insufficiently. As suggested in the paper (Vagner et al., 2006) the density of the probability current of the wave function

The eigenfunctions

This condition retains the greatest possible symmetry. The current lines in this case are concentric circumferences. The density of the current can be zero only at separate circumferences. Therefore the magnetization is nonuniform. The magnitudes of the eigenfunctions should have the axial symmetry. The localization of the electron cannot coincide with any classical orbit because the uncertainties of values of the orbit centre coordinates (24) should satisfy to Heisenberg uncertainty relation.

The Hamiltonian is as follows:

The operators

Here

Let us study the degeneration eliminating by the boundary condition (33) in the absence of other perturbation. The polynomial

Here

where

when

In the section 3.2 it will be shown that for statistical mechanics of the electron gas in the magnetic field the Hamiltonian

The degenerate levels are transformed in zonule. It follows from formula (39) that the zonule upper edge is determined by minimum value of

where

These results can be described as energy spectrum breakdown into two bands. A spectrum lower part is denoted as a magnetic band, and the upper one will be denoted as a conduction band. Bands are not separated by a gap or sharp boundary, but far from transitive area the density of states and wave functions differ substantially. Fine structure of the density of states in the lower part of magnetic band represents the narrow zonule separated by gaps. The total width of the allowed zonule and gap is equal to

### 3.2. Statistical mechanics of electron gas in uniform magnetic field with regard for electrostatic interaction

The quantum-mechanical average value of the magnetic moment in the ordinary eigenstate of the Hamiltonian

It is a negative quantity because the positive term that proportional to

Here

When this result is compared with the formula (42) it is apparent that the thermodynamical potential

Let us obtain the density of distribution for an electron gas that is at equilibrium with thermostat, which is described by classical mechanics. The conservation of the zero value of the angular momentum also will be taken into account. The characteristic function of this system is:

This formula is obtained on a basis of the properties of characteristic functions that was described in the formulae (6 – 8), and the quantum characteristic function (see formula (15)). Here

by using the formula (8). Let us generalize the Krutkov method (Krutkov, 1933; Zubarev, 1974) for this case. We calculate the Laplace transformation with respect the total energy

where

where

This integration can be performed by the saddle-point method because

Here the first multiplier is the common formula of the statistical operator for the quantum Gibbs distribute (Landau, & Lifshitz, E.M., 1980a; Zubarev, 1974). The second multiplier would be computed by the Darwin – Fowler method (Fowler, & Guggenheim, 1939) like as in the formula (16) and describes the conservation of the particle number. If the grand canonical ensemble is considered, then the statistical operator of particle number

would be obtained from this multiplier by the Krutkov method. The last multiplier in formula (49) cannot be computed by those methods because it does not have any large-scale parameter. This multiplier imposes constraints on ensembles that the total angular momentum equal to zero. If Hamiltonian and an operator that should be averaged have the commutative term that is proportional to the total angular momentum operator, this term should be eliminated when the averaging is performed. That is the reason for the change

The model that will be considered below is described by the Hamiltonian:

Here

The electron density in the magnetic field should be distributed in such a way as to shield the external potential

Here

The quadratic term of the residual potential is of chief interest. Therefore the minimization would be performed in the quadratic approximation. The test function has the form:

and as result we obtain:

where

where

The Coulomb interaction and the electron density inhomogeneous that are neglected commonly decrease the frequency

The electron gas always interacts with electromagnetic field that in the ordinary circumstance has zero temperature. This leads to the fact that the gas passes to the ground state by the spontaneous photon irradiation. The role of statistical mechanics is that it imposes a constraint on a value of the angular momentum in the ground state. The energy of the ground state with zero angular momentum would be computed by the spectrum of the one-particle states that was described in the previous subsection. The energy levels, the degeneracy multiplicity, and the boundary of the magnetic band are determined only the orbital motion. Then the energy of the magnetic band is:

The number of states in the magnetic band equals to:

The quantity

Here the first term is the energy of the electron gas in the absence of a magnetic field. The second term that makes the main addend in the magnetic moment is the product of the Fermi energy of the electron gas

where

The magnetic moment is:

The first term equals to zero because

The orbital diamagnetic susceptibility decreases with increasing of the magnetic field. This function also has a spikes that is caused by the addend

## 4. Conclusion

The fundamental theory of statistical mechanics requires taking into account the law of the angular momentum conservation. The fulfilment of this requirement does not introduce any essential alterations into statistical thermodynamics, when the angular momentum of the system equal zero and the system Hamiltonian is a positive definite quadric form of all momenta. An equilibrium isolated system would have nonzero angular momentum only if an attraction of particles can resist centrifugal forces, as it is in nebulas. A gas can be in equilibrium with a rotating envelope that is a termospinstat. The condition of this equilibrium is the equality of the average value of sum of particles angular velocities to the angular velocity of the envelope. The Gibbs density of distribution and the thermodynamical functions are generalized for this case. If a system has the angular momentum equal to zero, the conservation of this value is important only when the Hamiltonian or/and an averaged quantity depend on the angular momentum. The problem of an electron gas in a uniform magnetic field is considered with taking into account the conservation of the zero value of the angular momentum. This consideration eliminates the paradoxical statement of the conventional theory that diamagnetic moment of the gas equals zero in classical as well as quantum physics (the Bohr – van Leeuwen theorem). The new formulae for the magnetic moment of the electron gas are obtained. It also leads to the effect of confinement of two-dimensional gas of charged particles by magnetic field. This results in effect of a non-uniform density of a gas, which decreases with distance from a center according both to classical as well as quantum theory. Then the model of noninteracting charged particles does not have areas of application. Many theories should be reconsidered, if they are founded on this model and on the statistical mechanics which does not take into account the angular momentum conservation law.