Einstein gravity of a diffusing fluid

Abstract

We discuss Einstein gravity for a fluid consisting of particles interacting with an environment of some other particles. 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 dissipative fluid. The dissipation is approximated by a relativistic diffusion in the phase space. We are interested in a homogeneous isotropic flat expanding Universe described by a scale factor a. At an early stage the particles are massless. We obtain explicit solutions of the diffusion equation for a fluid of massless particles at finite temperature. The solution is of the form of a modified Jüttner 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 equation 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. The energy-momentum of the diffusing particles is defined by their phase space distribution. A conservation of the total energy-momentum determines the cosmological term up to a constant. The resulting energymomentum 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 ≃ 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.

Lévy Walk Dynamics in an External Constant Force Field in Non-Static Media

“…Moreover, two typical types of non-staticity of media are expansion and contraction. Diffusion processes taken place in chaotic systems and fluids turn out that the dynamics of the particles alter enormously due to the non-static medium, prompting the related researches [ 2 , 18 , 19 , 31 , 38 , 47 , 63 , 71 , 72 ]. The corresponding Fokker–Planck equation for random motion in non-static medium is presented in [ 72 ] on account of the generalized Chapman–Kolmogorov equation.…”

Lévy walk dynamics in non-static media

“…The expected resolution of the studied dynamics and the amplitude of the displacement of the media decide whether or not the non-static behaviors can be ignored. The expansion and contraction are two typical features of the media, which can be observed in many different areas, such as biology [26][27][28][29], cosmology [30][31][32], fluids [33]. In biology, the formation of pigmentation patterns depends on the growing concomitant tissues and organs.…”

“…For (a), we fix H = −0.1 and for (b) we take λ = 1. The solid lines are the theoretical results(33).…”

Diffusion in an expanding medium: Fokker-Planck equation, Green's function and first-passage properties

“…For instance, in developmental biology it is well-known that the formation of biological structures via diffusion-mediated processes can be significantly altered by the concomitant growth of tissues and organs [3][4][5][6]. Another example, taken from Cosmology, is the diffusion of cosmic rays in the expanding Universe [7][8][9]; moreover, the general problem of a fluid diffusing in the expanding Universe was addressed in [10], and this in fact could be considered a simplified model for the evolution of the Universe itself. All these facts highlight the necessity of developing a stochastic theory able to address the dynamics of ensembles of random walkers embedded in an expanding space by conveniently bridging the gap between the mesoscopic and the macroscopic level of description.…”