1.
Foreword in
(p. 3 in (Rus) A. Vlasov Many-particle theory. Moscow-Leningrad, State publishing technical and theoretical literature (GITTL), 1950. 348pp. A. A. Vlasov, Many-particle theory and its application to plasma, Trans, Russian Gordon and Breach Science Publishers. Inc., New York, 1961.)
...
In the construction of the theory, three factors seemed to be the guiding ones from the point of view of methodology.
1) Abandonment of a strictly localized description of microparticles. The conception of a particle is a point conserving this property independently of any connection with the medium and other particles is only an approximate reflection of the reality. ...
2) A new approach to the concept of the closure of a physical system. ...
3) The attempt to construct a theory in which motion would be an inseparable property of the object, and would not be the result of the action of any "sources" (forces - in classical mechanics, the heat reservoir - in statistics, charges and currents - in electrodynamics). The effect of sources of this nature is that in essence the motion is introduced into the physical system from without. The idea in question is realized in the present theory by having each particle described by an extended function of the space distribution.
Section 7. The physical ideas of the theory
(p. 54 in A. A. Vlasov, Many-particle theory and its application to plasma, Trans, Russian Gordon and Breach Science Publishers. Inc., New York, 1961.)
1. Individual and collective properties.
...It appears to us that there is another point of view that is more fruitful and in any event more consistent, according to which the charge of the elementary particle is a characteristic that is essentially bound up with the presence of a system, a collective of particles.
2. Distant space-time connections.
...a necessary step in generalization must be the introduction of distant interactions, in which each particle interacts simultaneously with all the other particles of the collective. Moreover, it is not only the state of the system at a given moment of time that must be taken into account but also the effect of all its evolution in time over a finite or infinitely great interval of time. There should correspond to this, in the mathematical apparatus of the theory, a special integral-differential method of expressing the connections.
3. The problem of the relationship between micro and macro.
It seems to us, may be set forth and to certain extent explained if we express in our theory the direct connection of each particle with the collective as a whole. This problem cannot be solved by means of present statistical methods. Attempts to solve it in statistics are bound up with the introduction, in addition to paired interactions, of triple, quadruple, etc. interactions, which is of little effect, since even the problem of three bodies has not been solved in classical mechanics. The introduction of integral connections with the entire collective as a whole for each particle appears to be an unavoidable step if we are to go forward in the field in question.
4. Collective interactions.
Up to present, there have been no physical theories known in which the incorporation of each new particle into a system of particles would evoke changes in the interaction and behavior of particles initially present in the ensemble. And yet, the addition of a third particle to two others should change the law of the interactions of these two particles. Finding the most general quantitative expression of non-additive interactions of the collective type occurring for any particles is essential for expressing the role of the collective in the many-particle problem.
5. Motion in a collective of particles.
... in the theory being set forth, the factors reffered to are made concrete in the following manner.
- The idea that matter is unthinkable without motion is expressed in the very model of the particle. The laws of motion of this
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6. Generalization of the particle concept.
....Classical mechanics, statistical physics, electrodynamics, and in a certain case quantum mechanics as well, comprise the vaguely formulated presupposition that localization of particle occurs independently of the existence of connections of the particle with the surrounding medium and with the remaining particles. The basic defects such a localizing formulation of the particle come down to the following:
The point has no spatial dimensions.
The point does not contain within it any motions or the possibility of any internal development.
The localization of particles leads to difficulties in electrodynamics and statistical physics.
...the new conceptions concerning the particle liberate us from the well-known difficulties connected with the point localization of charged particles.
7. "Test corpuscles".
8. Closure of physical systems.
...
a) An essential cause of the limitation of the field within which a theory is applicable is the treatment of the concept of "closure" accepted in the theory.
b) There can be no universal theories (either "macro" or "micro") if only because it is impossible to give a universal definition of closure.
c) Since the concept of closure is different in different theories, abandonment of this concept implies connection of the system with the medium, the type of which is also variable.
9. Cauchy's problem.
States of a system that we consider initial states may in general be of two types. Cases are possible in which the initial state is the results of the development of this system itself or of external action, the result of which may be achieved by the natural evolution of the system. On the other hand, there are cases in which the initial state is the result of external actions on the system that are no longer reducible to the natural development of the system. Such actions may not be defined and described by the initial principles on which the equations of motion of the system under consideration are based. In the second case, the apparatus of the theory must contain something expressing the uniqueness of the initial moment of time as compared with all the other moments.
Although setting up Cauchy's problem sets apart a certain initial moment of time, we in solving this problem cannot, of course, explain all temporal processes possible in the given system, since those temporal processes taking place without this setting apart of the initial moment do not fall within our sphere of consideration. It is therefore of importance to know in what cases the initial moment can be isolated in solving Cauchy's problem and in what cases it cannot. In other words, it is necessary to give a formula for defining solutions no dependent on setting apart a privileged moment of time. These solutions should represent those processes whose natural evolution is not disturbed by physical actions on the system.
These two types of solutions are clearly represented in the theory being set forth. The solutions of the first type (with a unique initial moment of time) describe temporal processes in a many-particle system, characterized by the asymmetry of time that is inherent in irreversible processes. Solutions of the second type are symmetrical with respect to time, which is a property of reversible processes (see Chapter V).
The establishment and solution of Cauchy's problem have an additional feature as well, which reduces the complex of time processes covered by it. In a number of cases in Cauchy's problem, assumptions are made as to fixed values of the function sought at the limiting spatial surfaces over the entire time interval t from -в€ћ to +в€ћ. Hence, in addition to the initial conditions, conditions are also laid down as to the magnitude of certain surface integrals.
In the theory set forth, these conditions lead to a sharp limitation of the number of different types of temporal processes described by this theory. As will be seen later, the initial equations of the theory contain, on the one hand, solutions that include temperature effects and, on the other hand, the equations of classical mechanics, which has no parameter adequate to the temperature. We pass to the solutions of classical mechanics if we require that the surface integrals expressing the flow of particles out of a definite volume should vanish. This condition signifies elimination of the fluctuation emission of particles conditioned by temperature effect, which are included in the original equations. The transition to the classical pattern is connected with elimination of the temperature fluctuations. As a result, we obtain a pattern that does not contain any parameter adequate to the temperature.
The following general conclusions may be drawn from what has been said:
1. The solution of Cauchy's problem does not always cover all the temporal processes included in the initial equations of this or that theory. Establishing the problem sometimes involves postulating physical conditions that sharply limit the range of possible solutions.
2. It is therefore impossible to identify fulfillment of the general principle of causality with the solution of Cauchy's problem. The principle of causality as a philosophical assumption is not, of course, equivalent to particular methods of explaining temporal processes connected with the customary formulation of Cauchy's problem. The cause for some processes may consist not in the initial conditions for the function sought, but in other physical conditions that likewise determine it. Thus in the theory being set forth, for example, there arise, with continuous changes of the temperature and density of the medium, discontinuous and qualitatively new solutions (spatially periodic, nondamping solutions in the form of waves, etc.). From the mathematical point of view, these solutions are connected not with Cauchy's problem but with the problem as to the branching of the parameter entering into the equation. What Is involved is thus the transfer of the cause (for some phenomena) from the initial conditions for the function sought to the initial conditions for a parameter, which, however, radically alters both the physical and mathematical side of the question.
Herewith a definite step forward is taken towards a profounder knowledge of the physical significance of the principle of causality.
2.
p.6. Introduction
from Vlasov A.A. Statistical distribution functions. Moscow: Nauka, 1966. 355pp. (in Russian).
..."How would be solved the problem of existing and stability of spatial-limited collectives of particles which are confined by internal and external forces without support of walls and are characterized by temperature distribution of velocities? For example, gravitating and plasmatic family of particles. Classical methods of the statistical physics (for example, Gibb's method) is not suitable to solve this problem because the vessel with particles is required for. More over the volume of vessel and its shape must be specified from without. There is do important for the families being under consideration that both the shape and the amount of the size are not arbitrary, but these characteristics are defined by individual properties of systems and medium"....
Vlasov considers distribution functions and Cartan's supporting elements (see Cartan, E. (1934) Les espaces de Finsler, Actual. 79, Paris.)
Then on the p. 14 Vlasov considers the main issues of distribution function definition, and he states that "the number of the particles in a system can be absolutely arbitary".
3.
In his book Vlasov A.A. Nonlocal statistical mechanics. Moscow: Nauka, 1978. 264pp. (in Russian), he states clearly "the number of the particles in a system is expressed with the help of distribution function, and therefore, must not be only integral number".
