Omsk State University,

55a Mira Ave.,

644077 Omsk,

RUSSIA

E-mail: mikheyev@phys.omsu.omskreg.ru, shirokov@univer.omsk.su

**Method of coadjoint orbits in thermodynamics of non-compact
manifolds**

**Abstract:**

In presented work we take into consideration the main problem of thermodynamics
of homogeneous spaces (quite synonymous to that quantum statistic mechanics)
- evaluation of heat kernel and corresponding statistic sum or partition
function. To solve that problem for given space one must integrate heat
kernel or Bloch differential equation with specific boundary condition
and then perform integration over the volume of the space. There is no
applicable algorithm of doing it for arbitrary non-compact space because
of sufficient divergences caused by infinite volume.

We suggest the method for solution of the stated problem for non-compact Lie group (non-compact homogeneous space with group structure) which is based on the framework of non-commutative integration, originating in theory of coadjoint orbits. That algorithm allows to find heat kernel and partition function avoiding inconveniences connected with infinite volume of the space. The method is illustrated with non-trivial examples.