From macroscopic irreversibility to microscopic reversibility via a nonlinear schrödinger‐type field equation

Abstract

The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrodinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrodinger-type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time-derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.