The properties of the first equation of the Vlasov chain of equations

TL;DR

This paper rigorously derives the first Vlasov equation as a Schrödinger equation, linking probabilistic and deterministic descriptions of physical systems, and explores the physical meaning of wave function phases and their relation to electromagnetic equations.

Contribution

It provides a novel rigorous derivation of the Vlasov equation as a Schrödinger equation and interprets wave function phases in physical terms, connecting to electromagnetic phenomena.

Findings

Derivation of the Vlasov equation as a Schrödinger equation

Physical interpretation of wave function phase as a scalar potential

Connection between velocity potential vortices and the Pauli equation

Abstract

A mathematically rigorous derivation of the first Vlasov equation as a well-known Schr\"odinger equation for the probabilistic description of a system and families of the classic diffusion equations and heat conduction for the deterministic description of physical systems was inferred. A physical meaning of the phase of the wave function which is a scalar potential of the probabilistic flow velocity is demonstrated. Occurrence of the velocity potential vortex component leads to the Pauli equation for one of the spinar components. A scheme of the construction of the Schr\"odinger equation solving from the Vlasov equation solving and vice-versa is shown. A process of introduction of the potential to the Schr\"odinger equation and its interpretation are given. The analysis of the potential properties gives us the Maxwell equation, the equation of the kinematic point movement, and the movement of the medium within electromagnetic fields equation.