# Field Form of the Dynamics of Classical Many- and Few-Body Systems: From Microscopic Dynamics to Kinetics, Thermodynamics and Synergetics

## Abstract

**:**

## 1. Introduction

- molecular-kinetic mechanistic theory, in which probabilistic assumptions serve as such a source (Maxwell, Boltzmann, Gibbs);

- 1.
- equations of motion of particles immersed in a field;
- 2.
- equations of the dynamics of the field created by these particles.

- 1.
- Particles and the field are two interconnected subsystems, within each of which there are no interactions. In the general case, a subsystem of a Hamiltonian system is non-Hamiltonian [20]. Although trajectories in the phase space of a subsystem of particles certainly exist, but both the Liouville theorem on the conservation of phase volume and the Poincaré recurrence theorem for a subsystem of particles do not hold.
- 2.
- Due to the limited velocity of the field propagation, the instantaneous forces acting on each of the particles of the system are determined by the positions of all other particles at earlier times. Therefore, the dynamics of the system depends not only on its initial state, but also on its prehistory. Thus, the field character of interactions between particles leads to the phenomenon of heredity.

- Damping of oscillations of all atoms and transition of the system to the state of rest at large times $t\to \infty $. In this case, in the presence of an alternating external field, stationary forced oscillations arise in the system and a dynamic equilibrium is reached between the system of atoms and the external field. In essence, such a state is nothing but a thermodynamic equilibrium between atoms and the field they create.
- The amplitude of at least part of the oscillations increases indefinitely with time. This means the destruction of the lattice.

- 1.
- Finding interatomic potentials describing the interaction between resting atoms. The results of many years of intense efforts to calculate interatomic potentials are systematized in the papers [29,30,31,32,33,34]. However, the direct use of these results to calculate the thermodynamic and kinetic properties of matter within the framework of statistical mechanics, kinetic theory, and in approaches such as the molecular dynamics method encounters practically insurmountable obstacles. Therefore, in theoretical studies, instead of more or less real interatomic potentials, one has to restrict oneself to simple model potentials, which qualitatively correspond to intuitive physical concepts.
- 2.
- Calculation of the partition function of a system of particles interacting through a given interatomic potential. Exact solutions to this problem have been obtained only for the simplest one- and two-dimensional models.

## 2. Field-Theoretical Representation of Interatomic Interactions

#### 2.1. Rational-Algebraic Model of Interatomic Potentials

#### 2.2. Transition from Interatomic Potentials to Field Equations

- 1.
- Subsystem consisting of particles between which there is no direct interaction. The impact of some particles on others is carried out only through the field created by them.
- 2
- A subsystem consisting of an auxiliary composite field without direct self-action. The influence of the field at some points on the field at other points is carried out only through particles. Regardless of the number of particles in the system, the auxiliary field has infinitely many degrees of freedom.

#### 2.3. Green’S Functions of Elementary Fields and an Abundance of Interaction Retardations

- 1.
- A uniquely defined delay that corresponds to a wave propagating at the speed of light c$${\tau}_{1}=\frac{\left|\mathbf{r}-{\mathbf{r}}^{\prime}\right|}{c}.$$
- 2.
- An infinite set of delays$${\tau}_{2}\left(\xi \right)=\frac{\sqrt{{\xi}^{2}+{\left|\mathbf{r}-{\mathbf{r}}^{\prime}\right|}^{2}}}{c}\ge {\tau}_{1},\phantom{\rule{1.em}{0ex}}\left(0<\xi <\infty \right),$$

## 3. Qualitative Analysis of System Dynamics within the Framework of the Field Form of Interactions between Particles

#### 3.1. Two Body Problem

#### 3.2. Dynamics of a One-Dimensional Crystal and the Establishment of (Thermo) Dynamic Equilibrium

- 1.
- The amplitudes of all free oscillations tend to zero with time. In this case, the energy of the oscillating particles is transferred to the field through which the particles interact. In the absence of a boundary, the field vanishes to infinity, taking energy with it. All free vibrations stop. If the system of particles is placed in a box with impenetrable boundaries for the field, then the field returns to the particles as a force leading to forced stationary oscillations of the particles. This example illustrates a probability-free dynamic mechanism for establishing thermodynamic equilibrium in a system.
- 2.
- Amplitudes of at least part of oscillations of the crystal increase. In this case, the crystal structure is rearranged, the description of which inevitably requires going beyond the limits of the harmonic model. This phenomenon has signs of a synergistic effect.

#### 3.3. A Rather Amusing Example: Is Confinement Possible in Classical Relativistic Dynamics?

- The dynamic field corresponding to this static potential satisfies the Klein-Gordon equation and is therefore relativistic.
- This field is capable of ensuring the stability of a complex consisting of a finite number of particles within the framework of the non-relativistic approximation.

- When studying the oscillations of a two-particle system in the framework of the relativistic theory, as is known, the complexity of the roots of the characteristic equation leads to the impossibility of stationary oscillations and the loss of stability of the system.
- On the other hand, the infinite distance of particles from each other is hindered by the unlimited growth of the potential at $r\gg {\alpha}^{-1}$. Unfortunately, a qualitative analysis of the behavior of the system under the condition $r\gtrsim {\alpha}^{-1}$ encounters obvious fundamental difficulties.

## 4. Discussion and Conclusions

- 1.
- A rigorous microscopic substantiation of both thermodynamics and kinetic theory, based only on classical Newtonian mechanics, does not currently exist.
- 2.
- Interatomic interactions are of field origin. Therefore, any real system consists of particles and a field generated by these particles and transmitting interactions between these particles.
- 3.
- In the case of atoms at rest, the interaction between them can be described by interatomic potentials. But in the case of moving atoms, the interaction is described in terms of an auxiliary scalar relativistic field.
- 4.
- The auxiliary scalar field is a superposition of elementary fields, each of which is characterized by its own generally speaking complex mass and satisfies the Klein-Gordon equation. Parameters of elementary fields are uniquely expressed through the characteristics of static interatomic potentials.
- 5.
- Due to the finiteness of the masses of elementary fields, the propagation velocity of the Klein-Gordon fields can take on any values that are less than the speed of light. This leads to the fact that the delay of interactions between particles can reach arbitrarily large values.
- 6.
- Retardation of interactions between particles is a real physical mechanism leading to the irreversibility of the dynamics of both many-body and few-body systems. Thus, there is no need to use any probabilistic assumptions for the microscopic justification of both thermodynamics and kinetics.

- The development of a non-statistical dynamic mechanism of irreversible thermodynamic equilibrium in three-dimensional crystal structures is a generalization of our results [28] obtained for one-dimensional lattices.
- Development of a mathematical apparatus for the theoretical study of the processes of restructuring of the structure of microheterogeneous condensed systems.
- Search for methods for constructing microscopic thermodynamics and kinetics of small systems.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Qualitative representation of a static potential, which is the sum of two elementary potentials with complex parameters ${\mu}_{1}^{\pm}$ and ${\mu}_{2}^{\pm}$.

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Zakharov, A.Y.
Field Form of the Dynamics of Classical Many- and Few-Body Systems: From Microscopic Dynamics to Kinetics, Thermodynamics and Synergetics. *Quantum Rep.* **2022**, *4*, 533-543.
https://doi.org/10.3390/quantum4040038

**AMA Style**

Zakharov AY.
Field Form of the Dynamics of Classical Many- and Few-Body Systems: From Microscopic Dynamics to Kinetics, Thermodynamics and Synergetics. *Quantum Reports*. 2022; 4(4):533-543.
https://doi.org/10.3390/quantum4040038

**Chicago/Turabian Style**

Zakharov, Anatoly Yu.
2022. "Field Form of the Dynamics of Classical Many- and Few-Body Systems: From Microscopic Dynamics to Kinetics, Thermodynamics and Synergetics" *Quantum Reports* 4, no. 4: 533-543.
https://doi.org/10.3390/quantum4040038