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Einstein gravity of a diffusing fluid

We discuss Einstein gravity for a fluid consisting of particles interacting with an unidentified environment of some other particles whose dissipative effect is approximated by a diffusion. The environment is described by a time dependent cosmological term which is compensating the lack of the conservation law of the energy momentum of the diffusing fluid. We are interested in a homogeneous flat expanding Universe described by a scale factor $a$. For a fluid of massless particles at finite temperature we obtain explicit solutions of the diffusion equation which are in the form of a modified Juttner distribution with a time dependent temperature. At later time Universe evolution is described as a diffusion at zero temperature with no equilibration. We find solutions of the diffusion at zero temperature which can be treated as a continuation to a later time of the finite temperature solutions describing an early stage of the Universe. A conservation of the total energy momentum determines the cosmological term up to a constant. The resulting energy momentum inserted into Einstein equations gives a modified Friedmann equation. Solutions of the Friedmann equation depend on the initial value of the cosmological term. The large value of the cosmological constant implies an exponential expansion. If the initial conditions allow a power-like solution for a large time then it must be of the form $a\simeq τ$ (no deceleration, $τ$ is the cosmic time). The final stage of the Universe evolution is described by a non-relativistic diffusion of a cold dust.