Probability-free relativistic kinetic theory of classical systems of charged particles

In the complete system of equations of evolution of the classical system of charges and the electromagnetic field generated by them, the field variables are excluded. An exact closed relativistic non-Hamiltonian system of nonlocal kinetic equations, that describes the evolution of a system of charges in terms of their microscopic distribution functions, is obtained. The solutions of this system of equations are non-invariant with respect to time reversal, and also have the property of hereditarity.


Introduction
Currently, theoretical studies of the thermodynamic and kinetic properties of condensed systems, gases, and plasma use mainly the ideas of equilibrium and non-equilibrium statistical mechanics. Classical statistical mechanics is based on two core principles.
• The system of interacting particles is characterized by its Hamiltonian ‫(ܪ‬ ‫,)ݐ(‬ ‫ݍ‬ ‫,))ݐ(‬ depending on the generalized coordinates ‫ݍ‬ ‫)ݐ(‬ and momenta ‫‬ ‫)ݐ(‬ , ݅ = 1,2, … ܰ (ܰ is the number of degrees of freedom of the system). The evolution of this system in time is described by the Hamilton system of equations.
• Probabilistic measures (microcanonical, canonical and grand canonical distributions) are defined in the phase space of the system. The choice of probabilistic measures depends on the external conditions imposed on the system. Thus, in the framework of statistical mechanics, the thermodynamics of a many-particle system is uniquely determined by its Hamiltonian, i.e. has a purely mechanical origin.
The calculation of the thermodynamic functions of systems is reduced to solving of three problems.
• The choice of inter-particle potentials. It is usually assumed that this problem is outside the scope of statistical mechanics. As interatomic potentials, some model potentials are used, chosen from considerations of simplicity and more or less feasible physical reasons.
• The choice of a probabilistic measure in the phase space of the system corresponding to the external conditions imposed on the system. For example, for a given temperature ܶ, volume ܸ and the number of particles ܰ of the system, the canonical distribution is used.
• Calculation of partition functions of systems in the thermodynamic limit, or uncoupling of BBGKY chains with the inevitable participation of additional hypotheses such as the principle of correlations It should be noted that the solution to the third of these problems meets enormous mathematical difficulties, which were got over only for a few extremely simplified models, very remote from "reality" [1].
However, besides to the purely mathematical problems of constructing the microscopic thermodynamics of many-body systems, there are fundamental questions, the answers to which can unlikely be found in the framework of statistical mechanics.
1. What is the nature of thermodynamic equilibrium? What is the difference between the equilibrium and nonequilibrium states of a system in the framework of its microscopic dynamics? What is the real physical mechanism irreversibly leading an initially nonequilibrium isolated system to a state of thermodynamic equilibrium? Within the framework of the statistical approach it is assumed that the equilibrium state is the most probable state. However, this assumption is nothing more than a hypothesis accepted as an axiom [2,3].
2. Is it possible a consistent combination of the concepts of deterministic classical mechanics with probabilistic concepts introduced into the kinetic theory of matter by Maxwell, Boltzmann and Gibbs? Note that at the end of the 19th century not only atomism, but also Boltzmann's probabilistic approach was severy criticized by a group of physicists and mathematicians (Mach, Duhem, Poincaré, Kirchhoff, Ostwald, Helm, etc.) [4,5]. In fact, the Loschmidt and Zermelo paradoxes that have not found conclusive solution are internal contradictions in Boltzmann's kinetic theory [6,7].
Nevertheless, at present, most of the efforts of researchers are aimed at developing probabilistic methods in the theory of many-body problems. It is assumed that the abundance of the number of degrees of freedom of many-body systems and the instability of phase trajectories of Hamiltonian systems (dynamic chaos) [8,9,10] are sufficient foundation for applying probabilistic concepts in classical mechanics.
3. Theoretically, the situation changed radically after the work of Kac [11,12], in which a quite convincing counterexample to statistical mechanics was found, namely, an exactly solved dynamic ring model. It is shown that the irreversible behavior of this dynamic model is a consequence of the incorrect use of seemingly feasible probabilistic hypotheses such as the assumption of molecular chaos. Despite this, the concept of probability continues to be used as an apparatus in statistical mechanics, primarily due to the lack of a constructive alternative concept.
4. Classical mechanics is in obvious contradiction with the thermodynamic behavior of real systems, which raises quite reasonable doubts about the fundamental possibility of microscopic foundation of thermodynamics in the framework of classical mechanics [4,5,6,7]. Therefore, it remains unclear are such fundamental problems as the correct explanation of the mechanisms leading to phase transitions, a consistent explanation of the nature of the second law of thermodynamics, and its reducibility (or irreducibility) to the more fundamental laws of physics solvable within the framework of existing theories [13,14].
Thus, statistical mechanics does not reveal the real physical mechanism of the thermodynamic behavior of real many-body systems, and existing attempts to explain the irreversibility using the concept of probability in combination with classical Hamiltonian dynamics seem to be exceptionable.
In addition, in the framework of statistical mechanics, the question of describing the interactions between the structural units of a substance (atoms, molecules, ions, free radicals, and others) remains open. In particular, even the idea of the existence of these structural units is rather limited, especially in the case of such species of condensed matter, as soft matter, disordered systems, electrolytes, etc. Finally, even the provisory distinguished structural units of matter are not some "rigid" formations with the fixed structures, but complex systems with interacting internal degrees of freedom.
It is known that all "real" interactions between particles of matter are of electromagnetic origin. Therefore, strictly speaking, instead of a system of "hard" particles with model fixed interactions between them, an extended system consisting of charged particles and the electromagnetic field generated by these In this regard, it is of interest to study the classical dynamics of a closed system of point charges interacting through the electromagnetic field created by them. Such a system consists of two subsystems: charges and an electromagnetic field. The evolution of this extended system is described by the Maxwell equations for the electromagnetic field and the equations of dynamics of charged particles.
The purpose of this paper is as follows.
1. Construction of the classical microscopic kinetic theory of systems consisting of classical charged point particles and the electromagnetic field created by them. For this, it is necessary to exclude field variables in the dynamics equations of this system.
2. Derivation of a complete system of equations describing the dynamics of particles with excluded field variables. Qualitative analysis of the properties of solutions of the equations of evolution for a system of interacting charges.

Microscopic distribution functions and equations of their evolution
Consider an isolated classical system consisting of a finite number of point charges interacting with each other through an electromagnetic field. The temporal evolution of this system as a whole is described by the coupled equations of motion of particles and Maxwell's equations. Of interest is the dynamics of the particle subsystem.

Microscopic distribution functions
We consider the dynamics of a system of particles in terms of microscopic distribution functions where ‫܀‬ ௦ ఈ ‫)ݐ(‬ and ‫۾‬ ௦ ఈ ‫)ݐ(‬ are the coordinates and momentum of the ‫ݏ‬ -th particle at the time instant ‫,ݐ‬ ܰ ఈ is the total number of particles of the ߙ -th type. Note that by definition, microscopic distribution functions are not probabilistic functions. Sums over ‫ݏ‬ of arbitrary "one-particle functions" ߰ ఈ ൫‫܀‬ ఈ ‫,)ݐ(‬ ‫۾‬ ఈ (‫)ݐ‬൯ are expressed through microscopic distribution functions using the identity In particular, for each type of particle their instantaneous microscopic particle densities ݊ ఈ ‫,ܚ(‬ ‫,)ݐ‬ charge densities ߩ ఈ ‫,ܚ(‬ ‫,)ݐ‬ flows ‫ܞ‬ ఈ ‫,ܚ(‬ ‫)ݐ‬ and electric currents ‫ܒ‬ ఈ ‫,ܚ(‬ ‫)ݐ‬ in the system are functionals of microscopic distribution functions and have the following form: where ܿ is speed of light, ‫‬ = ‫,|ܘ|‬ ݉ ఈ is mass of ߙ-th type particle.
Here we have used the relation between the momentum of particle ‫ܘ‬ ఈ and its velocity ‫ܞ‬ ఈ

Kinematic and dynamic contributions to the evolution of microscopic distribution functions
Differentiating the function (1) with respect to time, we find where ቀ and ቀ The quantities (9) and (10) are the kinematic and dynamic contributions to the rate of change of the microscopic distribution function of the particle system, respectively.
The expression for the kinematic contribution ቀ ப ഀ ‫)‪,௧‬ܘ,ܚ(‬ ப௧ ቁ ଵ using the identity (2) is reduced to the form: where ‫܀(۳‬ ௦ ఈ ‫,)ݐ(‬ ‫)ݐ‬ and ‫܀(۶‬ ௦ ఈ ‫,)ݐ(‬ ‫)ݐ‬ are strengths of electric and magnetic fields at a point ‫܀‬ ௦ ఈ ‫)ݐ(‬ at time ‫.ݐ‬ This equation expresses the instantaneous force ‫̇۾‬௦ ఈ ‫)ݐ(‬ acting on a particle as a function of its coordinates ‫܀‬ ௦ ఈ ‫)ݐ(‬ and velocity ‫̇܀‬௦ ఈ ‫.)ݐ(‬ Substituting (12) into (10) allows us to reduce the sum over ‫ݏ‬ to the form needed for use the identity (2). As a result, we have: Substituting this expression into the formula (10), we obtain the following representation for the dynamic part of the rate of change of the microscopic distribution function: In the absence of external fields, the electromagnetic field contained in this formula can be expressed through the microscopic distribution functions of the system.

The relationship of electromagnetic fields with microscopic distribution functions
As it is known [15], scalar ‫,ܚ(߮‬ ‫)ݐ‬ and vector ‫,ܚ(ۯ‬ ‫)ݐ‬ potentials of the electromagnetic field at time instant ‫ݐ‬ depend on the spatial distribution of the charges ‫,′ܚ(ߩ‬ ‫)′ݐ‬ and the currents ‫,′ܚ(ܒ‬ ‫)′ݐ‬ of the system at all previous time instants ‫′ݐ‬ ≤ ‫ݐ‬ according to Lienard-Wiechert relations We find the total charge density and current density at the point ‫′ܚ‬ at time instant ‫′ݐ‬ = ‫ݐ‬ − ‫|‪ᇱ‬ܚିܚ|‬ taking into account the formulas (4) and (6) We pass from the potentials of the electromagnetic field to the strengths of the electric ‫,′ܚ(۳‬ ‫)ݐ‬ and magnetic ‫,′ܚ(۶‬ ‫)ݐ‬ fields. As a result, we obtain the following relationships expressing the strengths of the electric and magnetic fields in terms of microscopic particle distribution functions: ‫,′ܚ(۶‬ ‫)ݐ‬ = ∇ ‫ܚ‬ᇱ × ‫,′ܚ(ۯ‬ ‫)ݐ‬ These expressions we use to calculate the dynamic contribution to the rate of change of the microscopic distribution functions (14).

Dynamic contribution to the evolution of the distribution function
Let us separate the electric and magnetic components of the dynamic contribution to the rate of change of the microscopic distribution function (14) ቀ where ቀ and ቀ Substitutions the expression (20) into (23) and (21) and respectively.

Complete system of relativistic kinetic equations
The equations of dynamics of a system of particles interacting with each other through an electromagnetic field are obtained by substituting the formulas (11), (23) and (24) into the equation (8) and have the form: where ߙ = 1,2, … , ‫,ܯ‬ ‫ܯ‬ is the number of types of particles contained in the system). This exact closed system of equations with excluded field variables describes the classical dynamics of a system of particles

Does a Hamiltonian of a system of interacting atoms exist?
Since all substances consist of charged particles, the interaction energy of a system of atoms has an electromagnetic origin. Let us discuss the question of the possibility of representing the interaction energy of atoms through interatomic potentials, which depend on the simultaneous coordinates of all particles in the system.

Electromagnetic energy of a system of resting charges
The expression for the energy of the static electric field of resting charges has the form [15]: The first term on the right side of this equation is the sum of the energies of the static fields of all particles. This energy is independent of particle positions. The second term is the Coulomb energy of interactions between particles: This energy is that part of the energy of a static field, which depends on the positions of particles in space ‫܀‬ ௦ . Thus, the Coulomb energy of interactions of resting charges is a part of the total energy of a static field, which depends on the distribution of these charges in space.

Electromagnetic energy of a system of moving particles
The energy conservation law for the classical system of charges in the general case has the following form [15]: (31) is the functional of the scalar ‫,ܚ(߮‬ ‫)ݐ‬ and the vector ‫,ܚ(ۯ‬ ‫)ݐ‬ potentials of the electromagnetic field at the same moment in time ‫.ݐ‬ However, the potentials of the electromagnetic field at time ‫ݐ‬ in accordance with the formulas (18) and (19) depend on the positions of the charges and the distribution of currents in space in all earlier time instants ‫′ݐ‬ ≤ ‫.ݐ‬ Therefore, the interaction between atoms cannot be represented as an expression depending on the simultaneous positions ‫܀‬ ௦ ‫)ݐ(‬ of all particles of the system.

Self-contradictoriness of statistical mechanics
Thus, the potential energy of a system of moving interacting particles (in particular, atoms) does not exist. Consequently, the Hamiltonian, depending on the simultaneous coordinates and momenta of the particles of the system, does not exist too. Note that the system of equations of motion for microscopic distribution functions, taking into account their construction, is relativistic. In this regard, it is appropriate to note that in the 1960s it was proved that the relativistic Hamiltonian formalism leads to the absence of interaction between particles [16,17,18,19,20]. Thus, the relativistically invariant Hamiltonian of the system of interacting particles does not exist, and the system of equations (27) is non-Hamiltonian.
At the same time, the assumption of the existence of the Hamiltonian of a system of particles is one of