Wigner function of the 4-th rank

Abstract

The Lorentz-Abraham-Dirac differential motion equation is of the third order since it contains the second order acceleration v→¨, which stands for the radiation friction force. To achieve a rigorous and comprehensive assessment of the dynamics of such radiation-friction systems it is necessary to expand the phase space {r→,v→} and to introduce higher kinematic values, such as v→˙ and v→¨.Our study is dedicated to the problem of building of an expanded phase space {r→,v→,v→˙,v→¨}, within which the fourth Vlasov equation for the distribution function f(r→,v→,v→˙,v→¨,t) is considered. Since the second Vlasov equation for the distribution function f(r→,v→,t) in a particular case is shown to transform into the Moyal equation for the Wigner function W(r→,p→,t), the authors have elaborated and present a way to expand the Wigner function on the phase space {r→,v→,v→˙,v→¨} and also propose a new Moyal equation, which this function satisfies.