**1. Introduction**

444 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

Zebbiche T. & Youbi Z., (2005b). Supersonic Two-Dimensional Minimum Length Nozzle

Zebbiche T. & Youbi Z. (2006), Supersonic Plug Nozzle Design at High Temperature.

*Exhibit, 9-12 Jan. 2006, ISBN 978-1-56347-893-2, Reno Nevada, Hilton, USA.* Zebbiche T., (2010a). Supersonic Axisymetric Minimum Length Conception at High

Zuker R. D. & Bilbarz O. (2002). *Fundamentals of Gas Dynamics*, John Wiley & Sons. ISBN 0-

Vol. 63, N° 04-05, PP. 171-192, May-June 2010, ISBN 0007-084X, 2010. Zebbiche T., (2010b). *Tuyères Supersoniques à Haute Température*. Editions Universitaires

Germany.

471-05967-6, New York, USA

*26-29 Sep. 2005, ISBN 978-3-8322-7492-4, Friendrichshafen, Germany.*

Conception. Application for Air. *German Aerospace Congress 2005, DGLR-2005-0257,* 

Application for Air, *AIAA Paper 2006-0592, 44*th *AIAA Aerospace Sciences Meeting and* 

Temperature with Application for Air. *Journal of British Interplanetary Society (JBIS)*,

Européennes. ISBN 978-613-1-50997-1, Dudweiler Landstrabe, Sarrebruck,

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 d*P* **R**

to detect this system at any point **R** of the phase space is:

$$\mathrm{d}P(\mathbf{R}) = \frac{\rho\_{\varepsilon}\left(\mathbf{R}\right)\mathrm{d}\Gamma}{\Omega\_{\varepsilon}\left[\left(2\pi\hbar\right)^{\mathrm{\mathcal{N}}}N!\right]}\tag{1}$$

where - the measure (phase volume) of the invariant set ; **R** - the characteristic function of the invariant set, which is equal to one if the point **R** belongs to this set, and is equal to zero in all other points of the phase space; =1 d= d d *<sup>N</sup> i i <sup>i</sup>* **p r** - the phase space volume element. The number of distinguishable states in a phase space volume element d is <sup>1</sup> <sup>3</sup> 2 ! *<sup>N</sup> <sup>N</sup>* . The system that will be under consideration is a collection of *<sup>N</sup>* structureless particles. The averaged value of a phase function *F***R** is *FF P* d **R R** . Here the integral goes over all phase space . This is microcanonical distribution. A characteristic function often would be presented as *f z* **R** , where *f* **R** is a phase function and *z* is it's fixed value.

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

Statistical Mechanics That Takes into Account Angular

been determined.

facts remain unaccounted.

Momentum Conservation Law - Theory and Application 447

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

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

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

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

"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

moment of force, which decreased to zero the angular momentum of electron gas.

quantities such as densities or susceptibilities have no physical meaning.

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

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

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

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

controllable motion integral that determines the invariant set.

validity of the Ergodic hypothesis (Khinchin, 1949).

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

Statistical Mechanics That Takes into Account Angular

integration.

independently, then

2

however many other theories should be reconsidered.

**2. Statistical mechanics of rotating gas** 

method with using the large parameter *N* .

*D B*

Momentum Conservation Law - Theory and Application 449

When 0 *T* , this sum can be computed without to change the summation to the

<sup>2</sup> <sup>2</sup> 2 8 *<sup>D</sup>*

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

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;

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 *N* structureless particles. If forces of interaction between particles manifest themselves only at distances considerably smaller than the average distance between particles, the interaction energy of particles is essential only in a small fraction of the phase volume. Therefore, the interaction of particles can be neglected or be taken into account as a perturbation in calculating the phase volume and the average values (Uhlenbeck & Ford, 1963). Otherwise, if particles interact by a long-range force, this interaction needs to be considered using a mean field method. This is a model of an ideal gas under an external field. Meanwhile, this external field can be also a periodical crystal field. In the commonly considered cases the Hamiltonian and other phase functions of the system can be presented as the sum of identical terms, each of which depends on the coordinates and

momenta of a single particle. Such phase function is said to be a summatory function.

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

 function with respect the value of the summatory function. Then the product of the *N* like exponents from the terms of the summatory function would be integrated over variables of the phase space. The inverse transformation would be made by the saddle-point

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 12 12 , **RR R** , and the determining functions possess the values

the fundamental formula for the thermodynamical potential is incorrect.

*n B*

2 22

*mS e H S*

*m*

 

ln 1 exp *<sup>n</sup>*

*k T* 

. (4)

. (5)

0

*eHS k T*

 

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 nonexplained.

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 2*Lx* and 2*Ly* . The solutions are determined by three motion integrals. The first is energy that takes the values <sup>2</sup> 12 2 *np <sup>с</sup> n p m* , where *<sup>с</sup>* is the cyclotron frequency *<sup>с</sup> eH m* , *e* is the charge and *m* is the mass of an electron, *H* is the magnetic induction, *n* is a positive integer or zero. The second is the *z* component of the momentum *p* . The third motion integral is the Cartesian coordinate of the centre of the classical orbit. It takes the values *yj x eHL j* , where 0, 1, 2,... *x y j eHL L* . The thermodynamical potential with this energy spectrum, when the spin degeneracy is taken into account, is:

$$\Delta\Omega = -k\_s T \int \mathrm{d}p \frac{L\_\varepsilon}{2\pi\hbar} \sum\_{v=0}^{\circ} \frac{eHS}{\pi\hbar} \ln\left[1 + \exp\left(\frac{\mu - \varepsilon\_{vp}}{k\_s T}\right)\right] \tag{2}$$

Here *<sup>B</sup> k* is the Boltzmann constant, *T* is the temperature, 4 *S LL <sup>x</sup> <sup>y</sup>* , is the chemical potential. If in this formula the summation over *n* is changed to the integration, the result <sup>0</sup> does not depend on *H* , and the magnetic moment M <sup>0</sup> *H* 0 . That agrees to the classical and paradoxical Bohr – van Leeuwen theorem. L.D. Landau uses the Euler – Maclaurin summation formula in the first order and obtains the amendment that depends on the magnetic field. In the limit 0 *T* the thermodynamical potential has appearance (Abrikosov, 1972):

$$
\Omega = \Omega\_0 + V \frac{e^2 H^2 p\_{\text{F}}}{24 \pi^2 \hbar m} \,'\,\tag{3}
$$

where 1 3 1 3 <sup>2</sup> 2 3 *<sup>F</sup> p m NV* , and is the Fermi energy, 4 *V LLL <sup>x</sup> <sup>y</sup> <sup>z</sup>* is the volume. This result cannot be correct because the magnetic moment does not depend on the Plank constant and thus it cannot be a quantum effect. This problem is simpler for a twodimensional gas. In this case the common formula of the thermodynamical potential has the form:

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-

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 2*Lx* and 2*Ly* . The solutions are determined by three motion integrals. The first is energy that takes the

the charge and *m* is the mass of an electron, *H* is the magnetic induction, *n* is a positive integer or zero. The second is the *z* component of the momentum *p* . The third motion integral is the Cartesian coordinate of the centre of the classical orbit. It takes the values

*L eHS kT p k T*

potential. If in this formula the summation over *n* is changed to the integration, the result <sup>0</sup> does not depend on *H* , and the magnetic moment M <sup>0</sup> *H* 0 . That agrees to the classical and paradoxical Bohr – van Leeuwen theorem. L.D. Landau uses the Euler – Maclaurin summation formula in the first order and obtains the amendment that depends on the magnetic field. In the limit 0 *T* the thermodynamical potential has appearance

*np z*

*n B*

2 2

<sup>0</sup> <sup>2</sup> 24 *<sup>F</sup> eHp <sup>V</sup> m*

volume. This result cannot be correct because the magnetic moment does not depend on the Plank constant and thus it cannot be a quantum effect. This problem is simpler for a twodimensional gas. In this case the common formula of the thermodynamical potential has the

, and

*<sup>с</sup>* is the cyclotron frequency *<sup>с</sup>*

 

 (2)

, (3)

is the Fermi energy, 4 *V LLL <sup>x</sup> <sup>y</sup> <sup>z</sup>* is the

. The thermodynamical potential with

*eH m* , *e* is

is the chemical

0 <sup>d</sup> ln 1 exp <sup>2</sup>

 

 

this energy spectrum, when the spin degeneracy is taken into account, is:

Here *<sup>B</sup> k* is the Boltzmann constant, *T* is the temperature, 4 *S LL <sup>x</sup> <sup>y</sup>* ,

explained.

*yj x* 

(Abrikosov, 1972):

form:

where 1 3 1 3 <sup>2</sup> 2 3 *<sup>F</sup> p m NV* 

 

values <sup>2</sup> 12 2

*np <sup>с</sup> n p m* , where

*eHL j* , where 0, 1, 2,... *x y j eHL L*

*B*

 

$$\Omega\_{2D} = -k\_s T \sum\_{\nu=0}^{\nu} \frac{eHS}{\pi \hbar} \ln \left[ 1 + \exp\left(\frac{\mu - \varepsilon\_\nu}{k\_s T}\right) \right]. \tag{4}$$

When 0 *T* , this sum can be computed without to change the summation to the integration.

$$
\Omega\_{2D} = -\frac{\zeta^2 mS}{2\pi\hbar^2} + \frac{e^2 H^2 S}{8\pi m} \,\, . \tag{5}
$$

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.
