Thermodynamic relation for the systems with inhomogeneous distribution of particles

For the system with inhomogeneous distribution of macroscopic parameters we obtain thermodynamic relation which depends on the spatial point(coordinate). In our approach, to obtain such a relation we use the basic ideas of the method of nonequilibrium statistical operator combined with the Hubbard–Stratonovich transformation. First of all, we define the thermodynamic relation for the system with homogeneous distribution of particles. Possible behavior peculiarities of systems with different character of interaction in nonequilibrium case are predicted. By saddle-point method we find the dominant contributions to the partition function and obtain all thermodynamic parameters of the systems with different character of interaction. The formations of saddle state in all systems of interacting particles at different temperatures and particle distributions have the same physical nature and therefore they can be described in the same way. We consider the systems with attractive and repulsive interactions as well as self-gravitating systems.