**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 *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 method with using the large parameter *N* .

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 independently, then

Statistical Mechanics That Takes into Account Angular

particle is a stochastic quantity with an average value

The potential energy of the centrifugal force 2 2 2 *U mr cf*

average angular momentum of the gas 2 2

*B*

 

system of reference that rotates with the angular velocity

**R**

2! 2

*<sup>N</sup> <sup>z</sup>*

<sup>1</sup> 3 5 2

*<sup>h</sup> <sup>N</sup> k T*

velocity nonlinearly because the gas moment of inertia <sup>2</sup>

d 11 d exp 2 !Z 2 2

random variable with the average value *N*

the temperatures and

H

Momentum Conservation Law - Theory and Application 451

where 2 2 *Ur z* is potential energy that confines the particles in bounded volume, and the second term of the Hamiltonian is the potential energy of the centrifugal force that leads to a collapse of rotating nebula into disk. This Hamiltonian is not a summatory function.

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

introducing the statistical parameter that corresponds to the thermodynamical temperature it is necessary to deduce the canonical Gibbs distribution for a system that is in equilibrium with a thermostat. That can be done, for example, by a method developed by Krutkov (Krutkov, 1933; Zubarev,1974). The conditions of the equilibrium between the rotating envelope and the gas are apparent. Those are the equalities of the temperature and the angular velocity.Let us determine the angular velocity of a gas. An angular velocity of a

> 

*g* 

conditions of the equilibrium between the rotating envelope and the gas are the equality of

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".

of the gas particle in the rotating reference frame (Landau, & Lifshitz, E.M., 1980a). The

angular velocity. This function can be obtained from the Gibbs distribution for a gas in the

2 2 <sup>1</sup> <sup>3</sup> 2 2 <sup>2</sup>

*B i*

*l m <sup>N</sup> p p r U kT m r*

*m m F T*

 R

 

1 1 Z 2 ! exp <sup>d</sup>

*l m <sup>P</sup> p p r U N kT m r*

*G N zi ri i i i N B i <sup>N</sup> N N <sup>i</sup>*

*N zi ri i i i*

1 *N*

<sup>1</sup> , , 2 <sup>2</sup>

 

2 , , 2 2

**p r** (10)

*i i <sup>i</sup> m r Nmr L*

3 , , 2 0 1 1

2 2

 

> :

. The sum 1

*N i i* 

*<sup>g</sup>* . That is the angular velocity of a gas. The

. (11)

depends on the angular

*<sup>i</sup> I Nmr* 

2 2 2 2 2

, , 2 0 1 1

*N N <sup>i</sup>*

2 2 <sup>2</sup> <sup>1</sup> 1 exp exp , , 2 2 *N*

*B B*

*k T k T*

is the Gaussian

is added in the Hamiltonian

is the function of the

  (12)

*<sup>i</sup> <sup>i</sup> <sup>i</sup>*

*i zi ri i i N*

*l L p p Ur z m r m r*

1

. For

 2 2 2 2 2 2

*<sup>N</sup> <sup>i</sup>*

Therefore the Krutkov's method cannot be used for the subsequent computations.

gas is characterized by two parameters: the temperature *T* and the angular velocity

$$\begin{aligned} \boldsymbol{\Omega}\_{\boldsymbol{\Xi}} &= \boldsymbol{\Omega}\_{\boldsymbol{\Xi}\_{1}} \cdot \boldsymbol{\Omega}\_{\boldsymbol{\Xi}\_{2}\boldsymbol{\nu}} \quad \boldsymbol{\varrho}\_{\boldsymbol{\Xi}} (\mathbf{R}) = \boldsymbol{\varrho}\_{\boldsymbol{\Xi}\_{1}} (\mathbf{R}\_{\boldsymbol{\imath}}) \cdot \boldsymbol{\varrho}\_{\boldsymbol{\Omega}\_{2}} (\mathbf{R}\_{\boldsymbol{\imath}}), \\ \boldsymbol{\mathrm{d}P}(\mathbf{R}) = \boldsymbol{\mathrm{d}P}\_{\boldsymbol{\imath}}(\mathbf{R}\_{\boldsymbol{\imath}}) \cdot \boldsymbol{\mathrm{d}P}\_{\boldsymbol{\imath}}(\mathbf{R}\_{\boldsymbol{\imath}}) &= \left[ \boldsymbol{\varrho}\_{\boldsymbol{\varepsilon}\_{1}} \left( \mathbf{R}\_{\boldsymbol{\imath}} \right) \Big{\Big{/}} \boldsymbol{\Omega}\_{\boldsymbol{\imath}} \left( 2\pi \boldsymbol{\hbar} \right)^{\otimes \times 1} \right] \cdot \left[ \boldsymbol{\varrho}\_{\boldsymbol{\varepsilon}\_{2}} \left( \mathbf{R}\_{\boldsymbol{\imath}} \right) \Big{/} \boldsymbol{\Omega}\_{\boldsymbol{\imath}} \left( 2\pi \boldsymbol{\hbar} \right)^{\otimes \times 2} \right] \mathrm{d}\Gamma\_{\boldsymbol{\imath}} \mathrm{d}\Gamma\_{\boldsymbol{\imath}} \end{aligned} \tag{6}$$

Here the multiplier <sup>1</sup> *N*! is not taken into account because it cannot be introduced logically in classical statistical mechanics. Considering the fact that a density of distribution for a system is equal to the product of densities of distribution for subsystems, a conclusion is drawn in the treatise (Landau, & Lifshitz, E.M., 1980a) that the logarithm of the density of distribution should be an additive motion integral and, hence, it should be a linear combination of the additive controllable motion integrals, such as the energy, the momentum and the angular momentum. However, as it follows from the formula (6) this is incorrect for the microcanonical distribution, since the logarithm of the characteristic function is meaningless. A system in a thermostat does not have any motion integrals. If the invariant set is determined by some conservation laws, its characteristic function is

$$\varphi\_{\varepsilon}(\mathbf{R}) = \prod\_{i} \varphi\_{i}(\mathbf{R})\,. \tag{7}$$

where *<sup>i</sup>* **R** is the characteristic function that is determined by the conservation law number *i* . Let us denote a set, at which the phase function *A***R** is equal to *a* , by *<sup>a</sup> <sup>A</sup>* , its characteristic function by *<sup>a</sup> <sup>A</sup>* **R** , and its measure by *<sup>a</sup> <sup>A</sup>* . It is supposed that this measure is limited and is not equal to zero. Then

$$\begin{aligned} \int\_{\Gamma} \boldsymbol{\rho}\_{\boldsymbol{\lambda}}^{\boldsymbol{s}} \left( \mathbf{R} \right) \mathrm{d}\Gamma &= \boldsymbol{\Omega}\_{\boldsymbol{\lambda}'}^{\boldsymbol{s}} \int\_{\Gamma} \boldsymbol{\rho}\_{\boldsymbol{\lambda}}^{\boldsymbol{s}} \left( \mathbf{R} \right) f \Big( \mathbf{R}, A \left( \mathbf{R} \right) \right) \mathrm{d}\Gamma = \int\_{\Gamma} \boldsymbol{\rho}\_{\boldsymbol{\lambda}}^{\boldsymbol{s}} \left( \mathbf{R} \right) f \left( \mathbf{R}, a \right) \mathrm{d}\Gamma, \\ \int\_{\boldsymbol{\lambda}} \int\_{\Gamma} \boldsymbol{\rho}\_{\boldsymbol{\lambda}}^{\boldsymbol{s}} \left( \mathbf{R} \right) f \left( a, \mathbf{R} \right) \mathrm{d}\Gamma \, \mathrm{d}a = \int\_{\boldsymbol{\lambda}} \int\_{\Gamma} \boldsymbol{\rho}\_{\boldsymbol{\lambda}}^{\boldsymbol{s}} \left( \mathbf{R} \right) f \left( A \left( \mathbf{R} \right), \mathbf{R} \right) \mathrm{d}\Gamma \, \mathrm{d}a = \int\_{\boldsymbol{\lambda}} f \left( A \left( \mathbf{R} \right), \mathbf{R} \right) \mathrm{d}\Gamma \end{aligned} \tag{8}$$

Here A is the range of values of the function *A***R** . The prevalent formula *<sup>a</sup> A a Aa* (Landau, & Lifshitz, E.M., 1980a; Uhlenbeck, & Ford, 1963) satisfies to the equalities (7) and (8) but does not satisfy to the equality (6). It is more frequently considered the separation on subsystems that conserves the total value of the function. If *AA A a* **RR R** 11 22 , then d *a ax x <sup>A</sup> A A <sup>A</sup> <sup>x</sup>* **1 2** , and the prevalent formula is correct.

#### **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:

$$\begin{split} \overline{F} &= \left[ \left( 2\pi\hbar \right)^{\otimes \mathsf{N}} N! \right]^{-1} \frac{\Delta E \Delta L}{\Omega} \int\_{\Gamma} F(\mathsf{R}) \delta \left( \mathbb{H}(\mathsf{p}, \mathsf{r}) - E \right) \delta \left( L\left( \mathsf{p}, \mathsf{r} \right) - \overline{L} \right) \mathrm{d}\Gamma \\ \Omega &= \Delta E \Delta \overline{L} \int\_{\Gamma} \delta \left( \mathbb{H}(\mathsf{p}, \mathsf{r}) - E \right) \delta \left( L\left( \mathsf{p}, \mathsf{r} \right) - \overline{L} \right) \mathrm{d}\Gamma \end{split} \tag{9}$$

In these formulae the axis **Z** is parallel to the angular momentum **L** . The angular momentum of the gas is <sup>1</sup> , , *<sup>N</sup> i i <sup>i</sup> L l* **pr p r** . The effective Hamiltonian of the gas with the fixed angular momentum in the cylindrical coordinates can be obtained from the usual formula by substitution 1 <sup>2</sup> *N <sup>i</sup> <sup>i</sup> lL l* with reduction of the quadratic form to the standard appearance. It is:

1 2

logically in classical statistical mechanics. Considering the fact that a density of distribution for a system is equal to the product of densities of distribution for subsystems, a conclusion is drawn in the treatise (Landau, & Lifshitz, E.M., 1980a) that the logarithm of the density of distribution should be an additive motion integral and, hence, it should be a linear combination of the additive controllable motion integrals, such as the energy, the momentum and the angular momentum. However, as it follows from the formula (6) this is incorrect for the microcanonical distribution, since the logarithm of the characteristic function is meaningless. A system in a thermostat does not have any motion integrals. If the

> *<sup>i</sup> i*

number *i* . Let us denote a set, at which the phase function *A***R** is equal to *a* , by *<sup>a</sup> <sup>A</sup>* , its

 

*<sup>i</sup>* **R** is the characteristic function that is determined by the conservation law

, dd , dd , d

*f A f a*

 

1 2

 

11 22 1 1 2 2 1 2

**RR R R <sup>R</sup>** (6)

3 3

is not taken into account because it cannot be introduced

1 2

**R R** , (7)

(8)

*A* 

 *a Aa*

*<sup>A</sup>* **R** , and its measure by *<sup>a</sup> <sup>A</sup>* . It is supposed that this measure is

 

**RRR**

d =d d = 2 2 dd *N N*

invariant set is determined by some conservation laws, its characteristic function is

**R RRR RR**

 

A A

*fa a fA a fA*

Here A is the range of values of the function *A***R** . The prevalent formula *<sup>a</sup>*

(Landau, & Lifshitz, E.M., 1980a; Uhlenbeck, & Ford, 1963) satisfies to the equalities (7) and (8) but does not satisfy to the equality (6). It is more frequently considered the separation on subsystems that conserves the total value of the function. If *AA A a* **RR R** 11 22 ,

The formula for average values, when the conservation of the angular momentum is taken

In these formulae the axis **Z** is parallel to the angular momentum **L** . The angular

fixed angular momentum in the cylindrical coordinates can be obtained from the usual

2 ! , ,d

H

**R pr pr**

(9)

*i i <sup>i</sup> L l* **pr p r** . The effective Hamiltonian of the gas with the

*<sup>i</sup> <sup>i</sup> lL l* with reduction of the quadratic form to the standard

 

**pr pr**

*E L EL L*

*N*

, ,d

*<sup>N</sup> E L F N F EL L*

**R R R RR R R**

d , , d , d,

*a aa a A AA A*

*<sup>x</sup>* **1 2** , and the prevalent formula is correct.

**2.1 Classical statistical thermodynamics of rotating gas** 

<sup>1</sup> <sup>3</sup>

H

momentum of the gas is <sup>1</sup> , , *<sup>N</sup>*

formula by substitution 1 <sup>2</sup>

1 2 1 2

, ,

*N*!

*a a A A*

*PP P*

Here the multiplier <sup>1</sup>

where

then d *a ax x <sup>A</sup> A A <sup>A</sup>*

appearance. It is:

 

into account, has the form:

characteristic function by *<sup>a</sup>*

limited and is not equal to zero. Then

$$\mathcal{KL}\left(\mathbf{p},\mathbf{r}\right) = \frac{1}{2m} \sum\_{i=2}^{N} \left( p\_{z,i}^{2} + p\_{r,i}^{2} + \frac{l\_{i}^{2}}{r\_{i}^{2}} + \mathcal{U}\left(\sqrt{r\_{i}^{2} + z\_{i}^{2}}\right) \right) + \frac{\overline{L}^{2}}{2m \sum\_{i=1}^{N} r\_{i}^{2}} \tag{10}$$

where 2 2 *Ur z* is potential energy that confines the particles in bounded volume, and the second term of the Hamiltonian is the potential energy of the centrifugal force that leads to a collapse of rotating nebula into disk. This Hamiltonian is not a summatory function.

Therefore the Krutkov's method cannot be used for the subsequent computations. 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 *T* and the angular velocity . For introducing the statistical parameter that corresponds to the thermodynamical temperature it is necessary to deduce the canonical Gibbs distribution for a system that is in equilibrium with a thermostat. That can be done, for example, by a method developed by Krutkov (Krutkov, 1933; Zubarev,1974). The conditions of the equilibrium between the rotating envelope and the gas are apparent. Those are the equalities of the temperature and the angular velocity.Let us determine the angular velocity of a gas. An angular velocity of a particle is a stochastic quantity with an average value . The sum 1 *N i i* is the Gaussian random variable with the average value *N <sup>g</sup>* . That is the angular velocity of a gas. The conditions of the equilibrium between the rotating envelope and the gas are the equality of the temperatures and

$$
\alpha \omicron\_{\%} = \alpha \bullet \tag{11}
$$

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 2 2 2 *U mr cf* is added in the Hamiltonian of the gas particle in the rotating reference frame (Landau, & Lifshitz, E.M., 1980a). The average angular momentum of the gas 2 2 1 *N i i <sup>i</sup> m r Nmr L* depends on the angular velocity nonlinearly because the gas moment of inertia <sup>2</sup> *<sup>i</sup> I Nmr* is the function of the angular velocity. This function can be obtained from the Gibbs distribution for a gas in the system of reference that rotates with the angular velocity:

$$\mathrm{d}P\_{c}\left(\mathbf{R}\right) = \frac{\mathrm{d}\Gamma}{\left(2\pi\hbar\right)^{\mathrm{NL}}N!Z\_{\mathrm{u}}}\exp\left[\frac{1}{k\_{s}T}\left(-\frac{1}{2m}\sum\_{i=1}^{N}\left(p\_{z,i}^{2} + p\_{r,i}^{2} + \frac{l\_{i}^{2}}{r\_{i}^{2}}\right) + \frac{o^{2}m}{2}\sum\_{i=1}^{N}r\_{i}^{2} - lU\_{o}\right)\right]$$

$$\mathrm{Z}\_{\mathrm{u}} = \left\{\left(2\pi\hbar\right)^{\mathrm{NL}}\left(N!\right)\right\}^{-1}\int\_{\Gamma} \exp\left[\frac{1}{k\_{s}T}\left(-\frac{1}{2m}\sum\_{i=1}^{N}\left(p\_{z,i}^{2} + p\_{r,i}^{2} + \frac{l\_{i}^{2}}{r\_{i}^{2}}\right) + \frac{o^{2}m}{2}\sum\_{i=1}^{N}r\_{i}^{2} - lU\_{o}\right)\right]d\Gamma = \left\{\mathrm{d}\Sigma\right\}$$

$$\left\{\left(2\pi\hbar\right)^{\mathrm{NL}}\left(N!\right)\right\}^{-1}\left\{\frac{h\_{z}}{o^{2}},\sqrt{\left(2\pi k\_{s}T\right)^{\mathrm{L}}m}\left[1 - \exp\left(-\frac{o^{2}m\mathcal{R}^{2}}{2k\_{s}T}\right)\right]\right\}^{\mathrm{NL}} = \exp\left[-\frac{1}{k\_{s}T}F\left(T, \frac{o^{2}}{2}\right)\right],$$

Statistical Mechanics That Takes into Account Angular

,,;,, 2 exp

 

 

the invariant set for *N* -particle system is:

*L* 0, 0 

velocity

where j *l* ,

 . When 1 

with the angular velocity

Schrödinger equation:

The boundary condition

Here the following variables are entered:

of this model of an ideal gas when *L* 0 .

Momentum Conservation Law - Theory and Application 453

by multiplier that describes the conservation of the particle number. Then the measure of

*i i il*

*xi yi zi* exp , exp , exp

Sp exp ddd

*x y z xy z x y z*

 . The lower operator or sign should be taken for the Bose statistics, and upper ones should be taken for the Fermi statistics. The expression of the measure of the invariant set in the case of quantum statistics is obtained from the apparent formula of the characteristic function (15). This expression is similar to the initial one in the Darwin – Fowler method (Fowler, & Guggenheim, 1939), where it was substantiated as a mathematical device . These contour integrals can be computed by the saddle-point method,

 

Thus, integrals are rearranged into integrals along contours which enclose the origin of coordinates. If **L** 0 then the axis **Z** is parallel to the angular momentum **L** and

into account the conservation of the angular momentum does not change statistic mechanics

Let us describe a quantum gas in a termospinstat with the temperature *T* and the angular

only in classical statistical mechanics (Landau, & Lifshitz, E.M., 1980a). Other method should be used for quantum theory. Wave functions in a system of reference that rotates

2 2

0 determines the energy spectrum:

*<sup>p</sup> l l*

is the null of the Bessel function J *<sup>l</sup> x* with number in the order of increasing

 

and j ,1 *l l* . This spectrum is quasicontinuous

 <sup>2</sup> <sup>2</sup> 2 <sup>2</sup> j , 2 2

R

<sup>ˆ</sup> <sup>1</sup> ˆ ˆ <sup>2</sup> *z r <sup>l</sup> i pp tm r* 

> 

*z*

*m m*

. The potential of the centrifugal force would be introduced in the Hamiltonian

2 2

2

 

 

*z*

 , , 

depend on the time, and should be determined from the

. (18)

and J *<sup>l</sup> x* is the Bessel function.

, (19)

 

*a a*

 

 

. It can be shown that taking

exp*i z pz l t*

 .

(16)

*b b*

 . (17)

<sup>1</sup> <sup>3</sup> 1 !1

 

*C NEL Ni Ei Li*

*N EL l*

2 1 ddd .

 

<sup>222</sup> <sup>3</sup> 000

 

when *L* 0 . The saddle-points determine the values of

Dependence of the wave function on the time should be

*l*

 

Then <sup>2</sup> exp J 2 *z z <sup>l</sup>*

 R

 , j *l l* , 2 

 *ip l t r m l p* 

where *<sup>z</sup> h* and R are the dimensions of the envelope, and 2 2 <sup>0</sup> *U Nm* R 2 is an appending constant that does the energy positive. Going to the thermodynamics (it rather can be entitled by "thermospindynamics") it is naturally to consider 2 2 as an external parameter and the moment of inertia 2 1 *N <sup>i</sup> <sup>i</sup> Im r* as a characteristic of the gas. Then

$$
\overline{I} = \text{N}m\overline{r^2} = -\hat{\text{c}}\text{F}/\hat{\text{c}}\sigma\,. \tag{13}
$$

The formulae of the isotopes separation (Cohen, 1951) can be obtained from the distribution (12). If <sup>2</sup> *m kT* R *<sup>B</sup>* the formula (12) can be presented as:

$$\begin{aligned} F &\approx F\_o + \frac{1}{2} N \sigma m \mathfrak{R}^2 - \frac{N}{24} \frac{\sigma^2 m^2 \mathbb{R}^4}{k\_s T} = F\_o + \overline{I}\_o \sigma - \frac{N}{24} \frac{\sigma^2 m^2 \mathbb{R}^4}{k\_s T} \\ F\_o &= -N k\_s T \ln \left[ \frac{e \pi \hbar \chi\_\* \mathbb{R}^2}{N} \left( \frac{m k\_s T}{2 \pi \hbar^2} \right)^{\Psi 2} \right], \quad \overline{I}\_o = \frac{N m \mathbb{R}^2}{2}, \end{aligned} \tag{14}$$

where *F*0 - is the free energy of the ideal gas that does not rotate. Hence it follows that the parameters *<sup>z</sup> h* , <sup>2</sup> R , and correspondingly *Pz* , *PS* should be introduced instead of the volume *V* and the pressure *P* . Other thermodynamical equations are changed also. The parameter of expansion in the formula (14) can be of the order of unity when 25 -1 *m T* 10 kg, 1 m, 100 K, 100 s R.

#### **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:

$$\hat{\phi}\left(E,\overline{L}\right) = \left(2\pi\right)^{-2} \int\_{\circ} \exp\left[\left(i\tau + \mathcal{O}\right)\left(\hat{\mathcal{K}} - E\right)\right] \mathrm{d}\tau \int\_{\circ} \exp\left[\left(i\alpha + \eta\right)\left(\hat{L} - \overline{L}\right)\right] \mathrm{d}\alpha \tag{15}$$

 in the space of microstates of the system. Here 1 ˆ ˆ *<sup>N</sup> i i* H h is Hamiltonian of gas; 1 ˆ ˆ *<sup>N</sup> <sup>i</sup> <sup>i</sup> L l* is the operator of the total angular momentum of gas; *<sup>E</sup>* , *L* are values of these quantities for the considered macroscopic state; , are real numbers which will be defined below. As usually, let us assume that energies of one-particle states, and, hence, both eigenvalues of the operator Hˆ and gas energy *E* , are expressed by the dimensionless positive integers. This formula would be generalized by the transition to representation of secondary quantization. In this representation function of one-particle states are eigenfunctions of the one-particle Hamiltonian ˆ *i* h and angular momentum. These functions are numbered using index which consists of a pair of quantum numbers *t l*, , where *t* is energy and *l* is the angular momentum at the state . Let us suppose that only two quantum numbers determine the state. Both the production and annihilation operators are determined by the kind of statistics. The operator (15) should be replenished

constant that does the energy positive. Going to the thermodynamics (it rather can be

1 *N*

<sup>2</sup> *I Nmr F*

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

2 24 24

*N m N m F F Nm F I*

where *F*0 - is the free energy of the ideal gas that does not rotate. Hence it follows that the parameters *<sup>z</sup> h* , <sup>2</sup> R , and correspondingly *Pz* , *PS* should be introduced instead of the volume *V* and the pressure *P* . Other thermodynamical equations are changed also. The parameter of expansion in the formula (14) can be of the order of unity when 25 -1 *m T* 10 kg, 1 m, 100 K, 100 s

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

2 2

*<sup>i</sup> <sup>i</sup> L l* is the operator of the total angular momentum of gas; *<sup>E</sup>* , *L* are values of

defined below. As usually, let us assume that energies of one-particle states, and, hence,

positive integers. This formula would be generalized by the transition to representation of secondary quantization. In this representation function of one-particle states are

are numbered using index which consists of a pair of quantum numbers *t l*, , where

 *t* is energy and *l* is the angular momentum at the state . Let us suppose that only two quantum numbers determine the state. Both the production and annihilation operators are determined by the kind of statistics. The operator (15) should be replenished

*i* h H

 ,   

ˆ ˆ *<sup>N</sup> i i*

h

ˆ and gas energy *E* , are expressed by the dimensionless

 

 H

0 0 ˆ ˆ ˆ *E L*, 2 exp *i E* d exp *i LL* d

  ln , , 2 2

<sup>0</sup> *U Nm* R

 

*<sup>i</sup> <sup>i</sup> Im r* as a characteristic of the gas. Then

3 2 2 2

R R

2 24 2 24

R R

 

*B B*

*k T k T*

. (13)

 

2 is an appending

2 as an external

(14)

 

are real numbers which will be

and angular momentum. These functions

is Hamiltonian of gas;

(15)

where *<sup>z</sup> h* and R are the dimensions of the envelope, and 2 2

entitled by "thermospindynamics") it is naturally to consider 2

R *<sup>B</sup>* the formula (12) can be presented as:

2 0 0 0

R

1

*B*

.

2

these quantities for the considered macroscopic state;

eigenfunctions of the one-particle Hamiltonian ˆ

in the space of microstates of the system. Here 1

H

**2.2 Quantum statistical thermodynamics of rotating gas** 

0 0 2

*z B*

*e h mk T Nm F Nk T <sup>I</sup> N*

 

parameter and the moment of inertia 2

(12). If <sup>2</sup>

*m kT* 

R

elements of the operator:

1 ˆ ˆ *<sup>N</sup>*

both eigenvalues of the operator

by multiplier that describes the conservation of the particle number. Then the measure of the invariant set for *N* -particle system is:

$$C\left(N,E,\overline{L};\chi,\theta,\eta\right) = \left(2\pi\right)^{-\frac{2}{3}} \left[\int\int\limits\_{0}^{\pi} \exp\left[-N\left(i\phi+\chi\right)-E\left(i\tau+\theta\right)-\overline{L}\left(i\alpha+\eta\right)\right]\right] \times$$

$$\times \operatorname{Sp}\left\{\exp\left[\sum\_{\mathbf{v}} \left[\left(\left(i\phi+\chi\right)+\left(i\tau+\theta\right)\right)\varepsilon\left(\Psi\right)+\left(i\alpha+\eta\right)l\_{\ast}\left(\Psi\right)\right]\right] \middle| \begin{matrix} a\_{\mathbf{v}}^{+}a\_{\mathbf{v}}\\ b\_{\mathbf{v}}^{+}b\_{\mathbf{v}}\end{matrix}\right] \right\} \operatorname{d}\phi \mathbf{d}\,\tau \mathrm{d}\,\alpha = \left(\text{16}\right)\mathbf{v}$$

$$\left(2\pi\right)^{-\frac{3}{2}} \oint\oint \oint \mathbf{x}^{-\mathbb{N}-1} y^{-\mathbb{E}-1} \mathbf{z}^{-\mathbb{E}-1} \prod\_{\mathbf{v}} \left(\mathbf{1}\pm\mathbf{x} y^{\mathbf{v}(\mathbf{v})} z^{\left(\mathbf{v}\right)}\right)^{\mathrm{el}} \mathbf{dx} \mathrm{d}\mathbf{y} dz\right.$$

Here the following variables are entered:

$$\text{tr} = \exp(i\phi + \chi), \quad y = \exp(i\tau + \theta), \quad z = \exp(ia + \eta). \tag{17}$$

Thus, integrals are rearranged into integrals along contours which enclose the origin of coordinates. If **L** 0 then the axis **Z** is parallel to the angular momentum **L** and *L* 0, 0 . The lower operator or sign should be taken for the Bose statistics, and upper ones should be taken for the Fermi statistics. The expression of the measure of the invariant set in the case of quantum statistics is obtained from the apparent formula of the characteristic function (15). This expression is similar to the initial one in the Darwin – Fowler method (Fowler, & Guggenheim, 1939), where it was substantiated as a mathematical device . These contour integrals can be computed by the saddle-point method, when *L* 0 . The saddle-points determine the values of , , . It can be shown that taking into account the conservation of the angular momentum does not change statistic mechanics of this model of an ideal gas when *L* 0 .

Let us describe a quantum gas in a termospinstat with the temperature *T* and the angular velocity . The potential of the centrifugal force would be introduced in the Hamiltonian only in classical statistical mechanics (Landau, & Lifshitz, E.M., 1980a). Other method should be used for quantum theory. Wave functions in a system of reference that rotates with the angular velocity depend on the time, and should be determined from the Schrödinger equation:

$$i\hbar \frac{\partial \Psi}{\partial t} = \frac{1}{2m} \left( \hat{p}\_z^2 + \hat{p}\_r^2 + \frac{\hbar^2 \hat{l}^2}{r^2} \right) \Psi \,\,\,\tag{18}$$

Dependence of the wave function on the time should be exp*i z pz l t* . Then <sup>2</sup> exp J 2 *z z <sup>l</sup> ip l t r m l p* and J *<sup>l</sup> x* is the Bessel function. The boundary condition R0 determines the energy spectrum:

 <sup>2</sup> <sup>2</sup> 2 <sup>2</sup> j , 2 2 *z l <sup>p</sup> l l m m* R, (19)

where j *l* , is the null of the Bessel function J *<sup>l</sup> x* with number in the order of increasing . When 1 , j *l l* , 2 and j ,1 *l l* . This spectrum is quasicontinuous

Statistical Mechanics That Takes into Account Angular

vector potential of the magnetic field:

boundary.

Momentum Conservation Law - Theory and Application 455

<sup>2</sup>

1 1 , ,0 2 2

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 **A A** *f x*, , *y z* . If the

and will have the translation symmetry in the direction of the axis **X** in return for axial symmetry. That is strange assertion because the symmetry is the physical property of the system rather than of a method of it description. In fact the transformation to the Hamiltonian (23) in classical mechanics is result of the canonical transformation of the Hamilton variables with the generating function 2 *x y px p y eHxy* . Then *p p x x* 1 2 , *eHy p p y y* 1 2 , *eHx* , *x xy y* . Therefore the , *<sup>x</sup> <sup>y</sup> p p* (in fact , *<sup>x</sup> <sup>y</sup> p p* ) in the Landau Hamiltonian (23) are not the momentum components in the Cartesian coordinates and the absence of the *x* coordinate in this Hamiltonian does not lead to the momentum *x* component conservation. In quantum mechanics the unitary transformation with operator exp 2 *ieHxy* is equivalent to this canonical transformation. The boundary is created by the line of intersection of the plane with a solenoid that generates the magnetic field. Electrons, orbits of which transverse this boundary, will be extruded from the area, and the gas will evaporate. The more realistic problem is gas in the area with a reflecting

An isolated electron has three motion integrals. Those are the angular momentum relative to

, ,

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

*<sup>p</sup> <sup>x</sup> <sup>p</sup> <sup>y</sup> l xp yp X <sup>Y</sup>*

2 2

*eH eH* (24)

*y x*

the centre of area and two coordinates of the centre electron orbit:

*z yx*

*E* and squared centre electron orbit distance from the centre of area <sup>2</sup> *R* :

where *m* and *e* are the mass and the charge of an electron, **p** is momentum, and **A** is

h 1 2*m e* **p A** (21)

**A Hr** *yH xH* . (22)

<sup>2</sup> <sup>2</sup> 12 - *L xy m p eHy p* h , (23)

**3.1.1 Classical statistical mechanics of ideal gas in magnetic field**  The Hamiltonian of an electron in a magnetic field has the form:

function *f eHxy* 2 , then Hamiltonian will be in the Landau form :

because a distance between nearest-neighbor levels is proportional to <sup>2</sup> R . The lowest level has value 2 2 *m* R 2 when *l m* <sup>2</sup> R . Then the reference point of energy should be altered by this value. It conforms to the appearance of the centrifugal force potential in the classical system. Energies of states with 0 *l* are lower than the ones of states with equal *l* and 0 *l* . Then in the gas the part of particles with 0 *l* should be more than half, and as result a circular current of the probability density should exist. This describes rotating of the system. With increasing argument modulus of extremes of the Bessel functions decrease. If the values of energy and positive angular momentum are fixed the value of the null number in the rotating system should be lower than this value in the motionless system.

Therefore, the gas density increases with distance from the axis in rotating system. Let us compute the thermodynamical potential *k TB p l* ln 1 exp *pl B k T* of ideal rotating gas when exp 1 *k TB* . Nulls of Bessel functions would be approximated by formula j *l l* , 2 , but when 1 then j ,1 *l l* . The computation is performed by passing from summation over and *l* to integration over j and *l* . This approximation for non-rotating gas leads to the result that differs from the common result by the multiplier 4 . The result of computation for the rotating gas is:

$$\Omega = -V \frac{4\left(k\_B T\right)^{\circ 2} m^{\circ 2}}{\pi \left(2\pi\right)^{\circ 2} \hbar^3} \exp\left(\frac{\mu}{k\_B T}\right) \left| 1 - \frac{\alpha^2 \circledast \alpha^2}{3} \left(\frac{m}{2k\_B T}\right) + \frac{\alpha^4 \circledast \alpha^4}{10} \left(\frac{m}{2k\_B T}\right)^2 \right| \,. \tag{20}$$

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 **H** 0,0,*H* . As will be shown the magnetization of electron gas is nonuniform. We will suppose that the magnetization is small as against the uniform field, and will not regard effect of it. Then magnetic induction *H* is proportional to the external magnetic field strength by the coefficient <sup>0</sup> . Where it is needed, we imply the plane to be of finite "thickness" *z* , and, for example, the equations of electrodynamics are written for threedimensional space.

#### **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.

altered by this value. It conforms to the appearance of the centrifugal force potential in the classical system. Energies of states with 0 *l* are lower than the ones of states with equal *l* and 0 *l* . Then in the gas the part of particles with 0 *l* should be more than half, and as result a circular current of the probability density should exist. This describes rotating of the system. With increasing argument modulus of extremes of the Bessel functions decrease. If

approximation for non-rotating gas leads to the result that differs from the common result

. The result of computation for the rotating gas is:

exp 1 <sup>2</sup> 3 2 10 2

*kT m m m <sup>V</sup>*

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

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 **H** 0,0,*H* . As will be shown the magnetization of electron gas is nonuniform. We will suppose that the magnetization is small as against the uniform field, and will not regard effect of it. Then magnetic induction *H* is proportional to the external 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

5 2 <sup>2</sup> 3 2 2 2 4 4

*B BB*

*k T kT kT*

R

<sup>2</sup> R . Then the reference point of energy should be

 

<sup>0</sup> . Where it is needed, we imply the plane to be of finite

*z* , and, for example, the equations of electrodynamics are written for three-

*k TB* . Nulls of Bessel functions would be approximated

ln 1 exp

then j ,1 *l l* . The computation is

and *l* to integration over j and *l* . This

R R . (20)

*k T* of

 *pl B* 

and positive angular momentum are fixed the value of the null

in the rotating system should be lower than this value in the motionless system.

. The lowest level

because a distance between nearest-neighbor levels is proportional to <sup>2</sup>

Therefore, the gas density increases with distance from the axis in rotating system.

2 when *l m*

ideal rotating gas when exp 1

performed by passing from summation over

 

*B*

4

 

by formula j *l l* , 2 

Let us compute the thermodynamical potential *k TB p l*

, but when 1

3 2 3

**3.1 Two-dimensional electron ideal gas in uniform magnetic field** 

the new correction will be obtained from the consistent quantum theory.

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

has value 2 2 *m* R

the values of energy

by the multiplier 4

by coefficients.

strength by the coefficient

"thickness"

dimensional space.

number
