Characteristic solutions of the chain of Vlasov equations

TL;DR

This paper introduces a novel method for deriving exact solutions of the infinite Vlasov chain by transforming equations into a solvable form, with applications to quantum systems and quantum dots.

Contribution

The paper presents a new characteristic transformation method that reduces Vlasov chain equations to the first Vlasov form, enabling exact solutions via Schrödinger equation techniques.

Findings

Successfully applied to a time-dependent quantum system with inverse temperature.

Derived solutions applicable to quantum dot systems.

Provided an algorithm for constructing solutions for any Vlasov chain equation.

Abstract

A new method has been presented of constructing a class of exact solutions of an infinite self-linking chain of the Vlasov equations for distribution functions of kinematic quantities of all orders. Using the characteristic transformation of variables proposed in this paper, any equation from the Vlasov chain can be reduced to the mathematical form of the first Vlasov equation. Since the solution of the first Vlasov equation can be found by the solution of the Schr\"odinger equation, the authors have proposed an algorithm for constructing characteristic solutions for an arbitrary equation from the Vlasov chain. The proposed method of construction of exact solutions has been successfully implemented on an example of time-dependent quantum system with thermodynamic parameter in the form of inverse temperature. These found exact solutions are also applicable to quantum dot systems.