**3. Equation of collisionless plasma motion considering dissipation. Anisotropic case**

To describe nonequilibrium thermodynamic processes in continuous media in a linear approximation, the Hungarian physicist Gyarmati formulated a variational principle that combines the principle of the least dissipation of Onsager's energy and the principle of the least production of Prigogine's entropy. To obtain the equation of motion that takes

into account dissipation, we introduce the entropy production function *<sup>σ</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> i*¼1 *JiXi*, as

well as the scattering potentials <sup>Ψ</sup> <sup>¼</sup> <sup>1</sup> 2 P*n i*, *k Li*,*kXiXk* and <sup>Φ</sup> <sup>¼</sup> <sup>1</sup> 2 P*n i*, *k Ri*,*kJiJk*, expressed in

terms of thermodynamic forces *Xi* (gradients of temperature, pressure, potential, field strength, and so on), and fluxes*Ji* corresponding to the observed transfer processes. If we now construct a function, *L* ¼ Ψ þ Φ � *σ*, then, as shown in [7], thermodynamic nonequilibrium processes near a steady state develop in such a way that the integral of over the volume occupied by the medium under study is minimal

$$\underset{\mathfrak{U}}{\text{d}\mathfrak{U}}\mathfrak{U} = \underset{\mathfrak{U}}{\text{[}[\Psi + \Phi - \sigma]d\mathfrak{U} = \min]}{\text{ $\mathfrak{U}$ }}$$

In this formulation, the Gyarmati principle is similar to Hamilton's principle in mechanics and the variation of this integral is equal to zero. Following the general provisions of [7, 11], we represent the tensor pressure of positively charged particles of an ionized gas as a sum of two parts. One part *P* \$ depends on the state and corresponds to the equilibrium part, the other part *P* \$ *<sup>d</sup>* depends on the rate of change of this state and corresponds to the nonequilibrium part, that is

$$
\stackrel{\leftrightarrow^i}{\dot{P}}\_{\Sigma}^i = \stackrel{\leftrightarrow^i}{\dot{P}}\_d^i + \stackrel{\leftrightarrow^i}{\dot{P}}\_d^i \tag{1}
$$

the subscript }*i*} denotes the ionic component of the equilibrium and nonequilibrium parts of the plasma pressure tensor. From the general provisions on the form of the explicit dependence of pressure *P* \$*i <sup>d</sup>*, it follows that it should depend on the macroscopic velocity of the medium *V <sup>i</sup>* and on the physical reasons causing the appearance of the nonequilibrium part of the pressure (for example, for viscous media with Brownian particles, this is taken into account by introducing the corresponding coefficients of viscosity and a random Langevin source). In our case, viscosity in the usual sense is absent, and the nonequilibrium part of the equation should be proportional to the flows of charged particles, which also corresponds to the general concept of pressure transfer through electromagnetic interaction, and also takes into account the discreteness of the ionized medium (its atomic-molecular structure [2]). With this approach, the ionic component of the pressure tensor *P* \$*i <sup>d</sup>* is similar to a Langevin source. According to what has been said, we represent the nonequilibrium part of the pressure in the form

$$
\stackrel{\rightharpoonup}{P}\_d^i = -m\_i \left( \stackrel{\rightharpoonup}{V}\_k^i \cdot \stackrel{\rightharpoonup}{J}\_n \right) \stackrel{\rightharpoonup}{I} \tag{2}
$$

where *I* \$ is the unit tensor, and for the equilibrium part we write out the standard representation of this part of the pressure [12]

$$\left(\stackrel{\dashrightarrow i}{P}\right)\_{kn} = p\_{\varPi}^{i}\stackrel{\dashrightarrow}{e}\_{k}\stackrel{\dashrightarrow}{e}\_{n} + p\_{\bot}^{i}\left(\delta\_{kn} - \stackrel{\dashrightarrow}{e}\_{k}\stackrel{\dashrightarrow}{e}\_{n}\right), \stackrel{\dashrightarrow}{e}\_{1} = \frac{\stackrel{\dashrightarrow}{H}}{H}.\tag{3}$$

Spatial heterogeneity and concentration n are taken into account in the explicit form of the flow *J i*.

We represent the Gyarmati principle in the form [13],

$$\delta \int\_{\mathfrak{U}} (\sigma\_d - \Psi\_d) d\mathfrak{V} = \mathbf{0} \tag{4}$$

where *<sup>σ</sup><sup>d</sup>* <sup>¼</sup> <sup>P</sup>*<sup>f</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>J</sup> jX <sup>j</sup>* and <sup>Ψ</sup>*<sup>d</sup>* <sup>¼</sup> <sup>1</sup> 2 P*<sup>f</sup> <sup>j</sup>*,*k*¼<sup>1</sup>*LjkX jXk*. The integral in (4) is taken over the entire volume ℧ occupied by the plasma. Since in a collisionless plasma there are no chemical reactions and sources of death and production of particles, and the interaction of currents leads to dissipative phenomena, then according to the general principles of construction *σ<sup>d</sup>* and Ψ*<sup>d</sup>* [7] we have for the positive plasma component

$$\sigma\_d^i = -\overset{\leftharpoonup}{P}\_d^i : \left(\overset{\leftharpoonup}{\nabla} \cdot \overset{\leftharpoonup}{V}\right), \Psi\_d^i = \frac{1}{2} m\_i \left(\overset{\leftharpoonup}{V} \cdot \overset{\leftharpoonup}{J}\right) \left(\overset{\leftharpoonup}{\nabla} \cdot \overset{\leftharpoonup}{V}\_i\right) = -\frac{1}{2} \overset{\leftharpoonup^i}{P}\_d : \left(\overset{\leftharpoonup}{\nabla} \cdot \overset{\leftharpoonup^i}{V}\right).$$

Considering *σ<sup>d</sup>* and Ψ*<sup>d</sup>* values and on the basis of (4), we obtain

$$\delta \int\_{\mathfrak{D}} \left( -\stackrel{\leftarrow}{P}\_d : \stackrel{\leftarrow}{\nabla} \cdot \stackrel{\leftarrow}{V}^i + \frac{1}{2} \stackrel{\leftarrow}{P}\_d : \stackrel{\leftarrow}{\nabla} \cdot \stackrel{\leftarrow}{V}^i \right) d\mathfrak{D} = -\frac{1}{2} \delta \int\_{\mathfrak{D}} \stackrel{\leftarrow}{P}\_d : \left( \stackrel{\leftarrow}{\nabla} \cdot \stackrel{\leftarrow}{V}^i \right) d\mathfrak{D}.\tag{5}$$

To calculate the integrand, we use the equation of balance of translational kinetic energy [7]

$$\rho\_i \frac{d}{dt} \frac{\left(\stackrel{\leftarrow i}{\mathbf{V}} \stackrel{\leftarrow}{\mathbf{V}}\right)}{2} + \stackrel{\leftarrow}{\mathbf{V}} \cdot \left(\stackrel{\leftarrow i}{\mathbf{P}}\_{\Sigma} \cdot \stackrel{\leftarrow}{\mathbf{V}}\right) = \rho\_i \left(\stackrel{\leftarrow i}{\mathbf{V}} \cdot \stackrel{\leftarrow i}{\mathbf{F}}\_{\text{ext}}\right) + \stackrel{\leftarrow}{P}\_{\Sigma} : \left(\stackrel{\leftarrow}{\mathbf{V}} \cdot \stackrel{\leftarrow}{\mathbf{V}}\right) \tag{6}$$

where *P* \$*i* <sup>Σ</sup> is total pressure, determined by (1), *F i ext* is external and internal forces per mass unit, *ρ<sup>i</sup>* is ion component density. If we now express *P* \$*i <sup>d</sup>* : ∇ � *V <sup>i</sup>* � � in (5) on the basis of (6), we obtain

$$-\frac{1}{2}\delta\underbrace{\int\_{\mathfrak{U}}^{\cdot i} \left(\rho\_i \frac{d\overleftarrow{\dot{V}}^i}{dt} + \text{Di}\dot{\nu}\overleftarrow{\dot{P}}^i - m\_i\overleftarrow{\dot{V}}^i\overleftarrow{\dot{\nabla}}^i\cdot\overrightarrow{\dot{J}}^i - \rho\_i\overleftarrow{\dot{F}}^i\_{\text{ext}}\right)d\mathfrak{V} = -\delta\underbrace{\int\_{\mathfrak{U}}^{\cdot} \mathbf{L}\_i d\mathbf{\mathcal{D}}}\_{\mathfrak{U}} = \mathbf{0},$$

where *Li* <sup>¼</sup> <sup>1</sup> 2*V i ρi dV i dt* þ *DiνP* \$*i* � *miV i* ∇ *i* � *J i* � *ρiF i ext* � � is Lagrange density, which satisfy the general equation

*Application of Onsager and Prigozhin Variational Principles of Nonequilibrium… DOI: http://dx.doi.org/10.5772/intechopen.103116*

$$\frac{\partial L}{\partial \mathbf{V}\_{\beta}} - \sum\_{a=1}^{3} \frac{\partial}{\partial \mathbf{X}\_{a}} \frac{\partial L}{\partial \left(\partial \mathbf{V}\_{\beta}/\partial \mathbf{X}\_{a}\right)} = \mathbf{0},\tag{7}$$

which is also valid for electronic component. Substituting the value *Li* into Eq. (7) and performing differentiation, we obtain the equation of motion for the ionic component "*i*"

$$\rho\_i \frac{d\overleftarrow{\dot{V}}^i}{dt} = -D\dot{\nu}\overleftarrow{P}^i + \rho\_i \overleftarrow{\dot{F}}^i\_{\text{ext}} + 2m\_i \left(\overleftarrow{\dot{V}}^i \cdot \overrightarrow{\nabla}^i \overrightarrow{J}^i\right) \tag{8}$$

where *Diν* the operator denotes tensor divergence. Repeating the same procedure for the plasma negative component, which is near the thermodynamic equilibrium (*Te* ≈*Ti* ), a similar equation may be obtained for electronic *e* component. Adding the obtained equation for electrons to (8) and considering *V e* ≈*V i* ¼ *V* , *F i ext* ≈ *F e ext* ¼ *F ext*, and *me* þ *mi* ¼ *mi*ð Þ 1 þ *me=mi* ≈ *mi* ¼ *m*, *ρ<sup>i</sup>* ¼ *mini* ≈*ρ* we obtain the following equation of motion:

$$\rho \frac{d\overleftarrow{\boldsymbol{V}}}{dt} = -D\dot{\boldsymbol{n}}\overleftarrow{\boldsymbol{P}}^{i} + \rho \overleftarrow{\boldsymbol{F}}\_{\text{ext}}^{i} + 2m\dot{\boldsymbol{V}} \left[ \overleftarrow{\boldsymbol{\nabla}} \cdot \overrightarrow{\boldsymbol{J}}^{i} + \frac{m\_{\epsilon}}{m\_{i}} \frac{\partial}{\partial t} (\boldsymbol{n}\_{\epsilon} - \boldsymbol{n}\_{i}) \right] \tag{9}$$

where *P* \$ ¼ *P* \$*e* þ *P* \$*i* ¼ *P* \$*e*,*<sup>i</sup>* <sup>⊥</sup> þ *P* \$*e*,*<sup>i</sup> II* . In (9) ∇ � *J i* value is expressed through *<sup>∂</sup> <sup>∂</sup>t*ð Þ *ne* � *ni* , considering the violation of quasi-neutrality and condition 1> > *me=mi*.

Taking into account the structure of a physically infinitesimal element of the medium, it must be remembered that it has linear dimensions of the order of the Debye radius, within which the condition of quasineutrality, due to fluctuations, can be violated. This is of fundamental importance, since it is the fluctuations that determine the character of the development of possible instabilities in the plasma. Therefore, in the last expression, the partial derivative of the difference between the concentrations of the electronic and ionic components is multiplied by a small value ð Þ *me=mi* ≈ ð Þ *me=m* .

Since in (9) the total flux is determined through the sum of fluxes of positively charged particles *J* ≈ *J i* <sup>¼</sup> <sup>P</sup> *k J i <sup>k</sup>*, then after simple ones associated with calculating the corresponding divergences in the drift approximation for fluxes [1, 16] (see appendix), we have

$$
\stackrel{\leftarrow}{J}\_1 = \frac{cn}{eH}rot\left[\frac{p\_\perp}{n}\stackrel{\leftarrow}{H}\right], \stackrel{\leftarrow}{div}\stackrel{\leftarrow}{J}\_1 = 0,\tag{10}
$$

$$
\stackrel{\leftarrow}{U}\_{2} = nc\frac{\left[\stackrel{\leftarrow}{E}, \stackrel{\leftarrow}{H}\right]}{H^{2}}, \\
d i \stackrel{\leftarrow}{J}\_{2} = \frac{2}{m\nu\_{\perp}^{2}} \left(\stackrel{\leftarrow}{E}\stackrel{\leftarrow}{j}\_{m}\right) - \frac{2}{m\nu\_{\perp}^{2}} \left(\stackrel{\leftarrow}{E}\stackrel{\leftarrow}{j}\_{g\Gamma}\right), \tag{11}
$$

$$
\stackrel{\leftarrow}{J}\_{\mathfrak{J}} = n \frac{mc\nu\_{\perp}^{2}}{2eH^{3}} \left[ \stackrel{\leftarrow}{H}, \stackrel{\leftarrow}{\nabla}H \right], \\
d\stackrel{\leftarrow}{i\nu}\_{\mathfrak{J}} = \frac{2}{m\nu\nu\_{\perp}^{2}} \left( \stackrel{\leftarrow}{F}\_{m}\stackrel{\leftarrow}{j}\_{m} \right), \tag{12}
$$

$$
\stackrel{\leftarrow}{J}\_{4} = \frac{nc}{eH^{2}} \left[ \stackrel{\leftarrow}{H}, \stackrel{\leftarrow}{\nabla} \frac{p\_{\perp}}{n} \right], \\
d\dot{\boldsymbol{\omega}} \stackrel{\leftarrow}{J}\_{4} = \frac{2}{me\nu\_{\perp}^{2}} \left( \stackrel{\leftarrow}{F}\_{m} \stackrel{\leftarrow}{j}\_{m} \right) - \frac{2}{me\nu\_{\perp}^{2}} \left( \stackrel{\leftarrow}{F}\_{m} \stackrel{\leftarrow}{j}\_{gr} \right), \tag{13}
$$

$$\stackrel{\leftarrow}{J}\_{\mathfrak{F}} = n \frac{m c \nu\_{\rm II}^{2}}{e H^{2} \mathcal{R}^{2}} \left[ \stackrel{\leftarrow}{R}, \stackrel{\leftarrow}{H} \right], \\ d \stackrel{\leftarrow}{\nu}\_{\mathfrak{F}} = \frac{2}{m c \nu\_{\perp}^{2}} \left[ \stackrel{\leftarrow}{F\_{c}} \stackrel{\leftarrow}{j}\_{m} \right] + e \frac{\nu\_{\perp}^{2}}{\nu\_{\rm II}^{2}} \left( \stackrel{\leftarrow}{E} \stackrel{\leftarrow}{j}\_{c} \right) + \frac{\nu\_{\perp}^{2}}{\nu\_{\rm II}^{2}} \left( \stackrel{\leftarrow}{F\_{m}} \stackrel{\leftarrow}{j}\_{c} \right) \right], \quad \text{(14)}$$

$$
\stackrel{\leftarrow}{H}\stackrel{\leftarrow}{\epsilon} = \frac{nc}{eH^2} \left[ \stackrel{\leftarrow}{H}, \stackrel{\leftarrow}{\nabla} \left( \frac{p\_{\text{II}}}{n} \right) \right],
\text{di}
\stackrel{\leftarrow}{J}\_6 = -\frac{2}{me\nu\_\perp^2} \left[ 2\epsilon \left( \stackrel{\leftarrow}{E} \stackrel{\leftarrow}{j}\_m \right) + 2\left( \stackrel{\leftarrow}{F}\_m \stackrel{\leftarrow}{j}\_m \right) - 2\left( \stackrel{\leftarrow}{E} \stackrel{\leftarrow}{j}\_{gr} \right) - 2\left( \stackrel{\leftarrow}{F}\_m \stackrel{\leftarrow}{j}\_{\varepsilon} \right) \right],
\tag{15}
$$

$$
\stackrel{\leftarrow}{J}\_{\heartsuit} = n\nu\_{\text{II}}\stackrel{\leftarrow}{e}\_{\text{1}}, \stackrel{\leftarrow}{div}\stackrel{\leftarrow}{J}\_{\heartsuit} = \frac{\mathbf{1}}{m\nu\_{\text{II}}^2} \left(\stackrel{\leftarrow}{F}\_{\text{M}}\stackrel{\leftarrow}{j}\_{\text{II}}\right) + \frac{\mathbf{1}}{m\nu\_{\text{II}}^2} \left(\stackrel{\leftarrow}{E}\stackrel{\leftarrow}{j}\_{\text{II}}\right). \tag{16}
$$

Flows *J* 2,3 arise due to electric and gradient drifts. Accounting for fluxes *J* 4,6 is associated with the interdependence of magnetic pressure and plasma pressure observed in the quasi-hydrodynamic approximation, since the pressure of charged particles in the absence of collisions is transferred by currents. In addition, the fluxes *J* 4,6 also take into account thermal diffusion, which is associated with the temperature gradient (*p*<sup>⊥</sup> � *T*<sup>⊥</sup> and *p*<sup>⊥</sup> � *T*<sup>⊥</sup> ). The flow *J* <sup>5</sup> is associated with centrifugal forces due to the curvature of the lines of force, *J* <sup>7</sup> - the flow of charged particles along the line of force. Opposite the corresponding values of the fluxes, their divergences are presented, in the derivation of which the invariance *n=H* and *μ* (first adiabatic invariant) with the accuracy of the drift approximation were taken into account and the following designations were adopted [16]:

$$\begin{aligned} \stackrel{\leftarrow}{j}\_{gr} &= \stackrel{\leftarrow}{H^{\sharp}} \mu \left[ \stackrel{\leftarrow}{H}, \stackrel{\leftarrow}{\nabla} H \right] \text{ is gradient drift current;} \stackrel{\leftarrow}{j}\_{m} = -\stackrel{\leftarrow}{H} \mu \text{rot}\stackrel{\leftarrow}{H} \text{is magneticizing current;} \\ \stackrel{\leftarrow}{j}\_{c} &= \stackrel{\leftarrow}{k^{2}} \left[ \stackrel{\leftarrow}{R}, \stackrel{\leftarrow}{H} \right] \text{ is centrifugal drift;} \stackrel{\leftarrow}{F}\_{m} = -\stackrel{\leftarrow}{\nabla} H \text{ is magnetic force;} \\ \stackrel{\leftarrow}{\nabla} &= \stackrel{\leftarrow}{m^{2}} \stackrel{\leftarrow}{\nabla} \quad \text{\\_} \ \stackrel{\leftarrow}{\nabla} \stackrel{\leftarrow}{H} \text{ is } \text{\\_} \text{ -- effective differential or-valued variable in i.e.} \end{aligned}$$

*F <sup>c</sup>* <sup>¼</sup> *<sup>m</sup>ν*<sup>2</sup> *II <sup>R</sup>*<sup>2</sup> *R* <sup>¼</sup> <sup>2</sup> *<sup>ε</sup>II <sup>H</sup>* ∇ <sup>⊥</sup>*H* is a force, affecting a charged particle in inhomogeneous magnetic field (centrifugal). It is clear that in this case the divergence of the flow of particles *J* <sup>1</sup> is equal to the divergence of the flow of leading centers, since in an ionized medium the motion of non-interacting particles differs from the motion of leading centers only by vortex terms, therefore *diν J* <sup>1</sup> ¼ 0. In addition, in deriving (10), the change in the average kinetic energy along the magnetic field line was neglected. Let us consider the second term in square brackets of (9), associated with the violation of the quasi-stationarity condition. Fluctuational charge separation in plasma leads to the appearance of an alternating electric field, which is responsible for the onset of polarization drift, which, in turn, leads to the formation of a drift polarization current *j p*. The magnitude of the drift current arising from the separation of charges is proportional to the rate of change in the electric field strength. This allows us to consider it as a displacement current that occurs during the polarization of dielectrics. Having carried out the appropriate calculations, and without limiting the generality of the proposed approach, we consider a special case when an alternating electric field is perpendicular to the magnetic field, we obtain (see Appendix)

$$\frac{\partial}{\partial t}(n\_{\epsilon} - n\_{i}) = -\frac{4H^{2}}{\left(H^{2} + 4\pi mmc^{2}\right)} \frac{\overleftarrow{F}\_{m}\overleftarrow{j}\_{\begin{subarray}{c}p\\ \end{subarray}}}{em\nu\_{\bot}^{2}}.\tag{17}$$

Substituting divergence values (9), calculated from the corresponding fluxes (10-16), and expression (17) into the equation of motion (9), we obtain

*Application of Onsager and Prigozhin Variational Principles of Nonequilibrium… DOI: http://dx.doi.org/10.5772/intechopen.103116*

$$\begin{split} \frac{d\vec{V}}{dt} &= -\frac{1}{\rho} \text{Di}\vec{\nu}\vec{P} + \overleftarrow{F}\_{\text{ext}} + \frac{2\vec{V}}{n\mu H} \left[ \overleftarrow{E} \left( \overleftarrow{j}\_{gr} - \overleftarrow{j}\_{m} + \frac{\nu\_{\perp}^{2}}{\nu\_{\text{II}}^{2}} \overleftarrow{j}\_{c} + \frac{\nu\_{\perp}^{2}}{\nu\_{\text{II}}^{2}} \overleftarrow{j}\_{\text{II}} \right) + \\ &+ \frac{\nu\_{\perp}^{2}}{e\nu\_{\text{II}}^{2}} \overleftarrow{F}\_{m} \left( \overleftarrow{j}\_{c} + \overleftarrow{j}\_{\text{II}} \right) + \frac{1}{e} \left( \overleftarrow{F}\_{c} \overleftarrow{j}\_{m} \right) \right] + \bar{\varepsilon} \frac{2V}{n\mu H} \frac{2H^{2}}{H^{2} + 4\pi nmc^{2}} \left( \overleftarrow{F}\_{m} \overleftarrow{j}\_{p} \right), \bar{\varepsilon} = (m\_{\epsilon}/m\_{i})<1. \end{split} \tag{18}$$

Since the derived equation uses macroscopic quantities *n*, *P* \$ , *E* , *H* ,*V* as the main parameters, there is no need for additional assumptions about the form of the distribution function associated with the termination of the chain of moments and the transition to hydrodynamic equations from the Vlasov kinetic equation. However, the most important thing in Eq. (18) is that it takes into account small dissipative and fluctuation processes arising due to the interaction of drift currents with inhomogeneous electric and magnetic fields. The reason for the smallness of the fluctuations taken into account in (18) is the condition of applicability of the leading center approximation and is a consequence of the perturbation theory, which is valid up to the constancy of the first adiabatic invariant ð Þ *μ* ¼ *const* and therefore allows the parameters to vary within this accuracy. At the same time, it is known that fluctuations in plasma are responsible for the appearance of local currents, which are determined by space-time inhomogeneities in the distribution of the field and plasma. In turn, the interaction of these currents with forces, also associated with inhomogeneities in the spatial distribution of magnetic and electric fields, determines the further development of the resulting fluctuations, as well as the nature of the possible instability.

Eq. (18) under the assumption of quasineutrality (*ne* ¼ *ni*) and infinite conductivity along the field line is greatly simplified (*E II* ¼ 0). In addition, if we consider a closed axially symmetric system, then the inhomogeneity in the plasma distribution along the drift trajectory may be absent and the current intensity *j II* proportional to this inhomogeneity tends to zero. Finally, instead of (18), we obtain a simplified, but not changing the physical essence, equation

$$\frac{d\vec{V}}{dt} = -\frac{1}{\rho} \text{Di}\vec{\nu}\vec{P} + \overleftarrow{F}\_{\text{ext}} + \frac{2\vec{\dot{V}}}{n\mu H} \left[ \frac{\nu\_{\perp}^{2}}{e\nu\_{\text{II}}^{2}} \left( \overleftarrow{F}\_{m} \cdot \overleftarrow{j}\_{\text{c}} \right) + \frac{1}{e} \left( \overleftarrow{F}\_{\text{c}} \cdot \overleftarrow{j}\_{m} \right) \right] = \frac{1}{\rho} d\boldsymbol{i}\nu\overleftarrow{P} + \overleftarrow{F}\_{\text{ext}} + \overleftarrow{f}\_{\text{di}} \left( \overleftarrow{F}\_{\text{c}} \cdot \overleftarrow{j} \right) \tag{19}$$

Eqs. (18) and (19) differ from generally used equations of motion by the third term in the right part, which describes dissipative interaction of drift currents with *eE* , *F <sup>m</sup>*, *F <sup>c</sup>*, forces. This additional part evidently take into account magnetization of physically infinitesimal element of a continuum, since besides the dependence on drift current *j gr*, *j <sup>c</sup>*, *j II* and *j <sup>m</sup>*, it is proportional to 1*=μ*. We should note, that in the case with axial-symmetrical plasma system, currents *j <sup>m</sup>* and *j <sup>c</sup>* constantly flow in it. Nevertheless, they do not break freezing-in, as *j <sup>m</sup>* and *j <sup>c</sup>* are directed along the azimuth and *F <sup>m</sup>*⊥ *j <sup>c</sup>*, *F <sup>c</sup>*⊥ *j <sup>m</sup>*. At the same time the appearance of fluctuations may cause azimuthal inhomogeneity and, consequently, coincidence of *F <sup>m</sup>* and *j <sup>c</sup>*, *F <sup>c</sup>* and *j <sup>m</sup>* components. Moreover, *F ext* � *j* , *H* h i force in the Eqs. (18) and (19) is expressed

through drift current explicit values, not through *rotH* , which is within the framework of general conception of this chapter: consideration of current corpuscular structure.
