**2. Statement of the problem**

Difficulties that arose from the very beginning after obtaining kinetic equations and introducing the terms collisional and collisionless plasma [8, 9] are associated in the physics of open systems with the concept of continuous medium. In this case, it becomes important to determine the physically infinitesimal scale corresponding to the point of the "continuous medium".

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

Indeed, the concept of continuous medium, depending on the chosen model for describing the behavior of an ionized gas (kinetic, diffusion, or hydrodynamic), implies the choice of a scale characterizing a physically infinitesimal point of a continuous medium ℓ *<sup>f</sup>* for which differential equations are written. However, in this case, information is lost inside these points, since the large number of particles filling their volume (*g*�<sup>1</sup> <sup>¼</sup> *<sup>n</sup>λ*<sup>3</sup> *<sup>D</sup>* > > 1, *g* is the plasma parameter, *λ<sup>D</sup>* is Debye radius) is not taken into account, which ultimately determines the internal openness of the chosen level of description [2]. Therefore, taking into account the structure due to the "artificial" introduction of an additional collision integral into the dissipative Vlasov equation when calculating the Landau collisionless damping coefficient leads to the appearance of dissipation and, as a consequence, to nonequilibrium. The need to take into account the structure of a physical "point" is one of the main provisions that determine the substantive part of fundamental works [2, 10, 11]. This position sets the direction of the search for the possibility of describing nonequilibrium processes on the kinetic and hydrodynamic scales from a single point of view, and will be used in this work.

It is known that the description of the dynamics of an ionized gas is also possible at the hydrodynamic level. Indeed, the kinetic method for some practical problems may turn out to be too detailed and mathematically complex. At the same time, without being interested in the motion and interaction of a large number of particles, one can significantly simplify the problem associated with the study of collective processes occurring in a plasma. Considering such macroscopic quantities as the average veloc-

ity of motion of a medium *V* , pressure *P*, density of particles n and currents *j* , and so on, postulating then the basic equations of hydrodynamics of continuous media, based on the laws of conservation of mass, momentum, energy and charge, together with Maxwell's equations, we can reduce the problem to the problems of magnetohydrodynamics (MHD). The system of MHD equations has the simplest form in the case of a one-fluid approximation for scalar (see, for example, [12, 13]) or tensor pressure (quasi-hydrodynamic approximation of Chu, Goldberger, and Lowe (ChGL) [14]).

At the same time, the Lorentz force acting on charged particles in a magnetic field twists them around the lines of force, preventing movement across the lines of force, and in this regard, the action of the field is similar to the effect of collisions, limiting the movement of the particle by the value of the Larmor radius. Consequently, the drift approximation shows how, in the absence of collisions, the order inherent in "collisional" continuous media and practically sufficient for describing the dynamics of a "collisionless" plasma at the hydrodynamic level is provided by a magnetic field. ("Practical sufficiency", from the point of view of the kinetic description, is achieved by neglecting the third moments in the equations, which corresponds to the not entirely justified neglect of the heat flux along the lines of force. Experimentally, this is realized in closed axial plasma systems or under real conditions, for example, in the region capture of the Earth's magnetosphere). Consequently, in a magnetized plasma, the role of the mean free path is played by the Larmor radius of ions *ρLi ρLi* > > *ρLe* , and the condition for the applicability of the continuous medium approximation takes the form *L*> > *ρLi* , where *L* is the characteristic size in the plasma. As for the frequency dependence, which makes it possible to consider a collisionless plasma as a continuous medium during the propagation of a wave process in it, it has the form: *ω*< <*ωLi* < <*ω*<sup>0</sup>*<sup>i</sup>* for a not too discharged ionized gas and a weak magnetic field (hot plasma) and *ω*< <*ω*<sup>0</sup>*<sup>i</sup>* < <*ωLi* for a magnetized plasma satisfying the drift approximation (cold plasma). Moreover, the possibility of describing the behavior of a

collisionless plasma using a pressure gradient is associated with the mechanism of pressure transfer not through collisions, but through the interaction of currents flowing in the plasma drift currents and magnetizing currents. In addition, a large role in the processes occurring in a collisionless plasma is played by self-consistent fields that bind particles and prevent them from scattering.

For physically small linear and time scales ℓ *<sup>f</sup>* and *τ <sup>f</sup>* , as well as the number of particles *N <sup>f</sup>* in the volume ℓ<sup>3</sup> *<sup>f</sup>* , the inequalities are valid *τ <sup>f</sup>* � ð Þ *λD=VT* < <*T*, ℓ *<sup>f</sup>* � *λ<sup>D</sup>* [8]. The first inequality makes it possible to use the "continuous medium" approximation, the second - to use the concept of "collisionless plasma", and the third notes the fact that the interaction of charged particles in an ionized medium has a collective character (*VT* is the thermal velocity of particles, *T* is the characteristic time).

However, magnetohydrodynamic equations (MHD equations) obtained in the drift approximation from the Vlasov equation through the moments of the distribution function do not take into account dissipative processes [3]. In other words, in this case, the structure of the physically small volume of the continuous medium is not taken into account, with respect to which the macroscopic equations are written. At the same time, the possibility of taking into account the drift approximation in the hydrodynamic consideration of the theory of magnetized plasma without any additional assumptions appears in the case of applying the variational principles of nonequilibrium thermodynamics of Prigogine and Onsager [5, 6], combined by Gyarmati [7]. Thus, in the mechanics of continuous media, it becomes possible to construct non-equilibrium models that describe the dynamics of continuous systems located near equilibrium (linear approximation). In turn, the construction of new models is an important section of continuum mechanics, and they are based on the search for additional relationships between the parameters that describe the state of the considered continuous medium.

With this in mind, the following provisions were the starting points for constructing a hydrodynamic model based on variational principles and drift equations [3, 7, 15]:


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