FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 5, PAGES 203-219

**Exact master equations describing reduced dynamics of the Wigner
function**

J. Kupsch

O. G. Smolyanov

Abstract

View as HTML
View as gif image

Master equations of different types describe the evolution (reduced
dynamics) of a subsystem of a larger system generated by the
dynamic of the latter system.
Since in some cases the (exact) master equations are relatively
complicated, there exist numerous approximations for such equations
which are also called master equations.

In the paper, one develops an exact master equation describing the
reduced dynamics of the Wigner function for quantum systems obtained
by a quantization of a Hamiltonian system with
a quadratic Hamilton function.
First, one considers an exact master equation for first integrals of
ordinary differential equations in infinite-dimensional, locally
convex spaces.
After this, one applies the obtained results to develop an exact
master equation corresponding to a Liouville type equation (which
is the equations for first integrals of the (system of) Hamilton
equation(s)); the latter master equation is called a master
Liouville equation, it is a linear first-order differential
equation with respect to a function of real variable taking
values in a space of functions on the phase space.
If the Hamilton equation generating the Liouville equation is linear,
then the vector fields that define the first-order linear differential
operators in the master Liouville equations are also linear, which in
turn implies that for a Gaussian reference state the Fourier
transform of a solution of the master Liouville equation also
satisfies a linear differential equation.

Location: http://mech.math.msu.su/~fpm/eng/k06/k065/k06516h.htm

Last modified: February 21, 2007