Diffusion in curved spacetimes

Abstract

Using simple kinematical arguments, we derive the Fokker-Planck equation for diffusion processes in curved spacetimes. In the case of pure Brownian motion, this equation coincides with Eckart's relativistic heat equation (albeit in a simpler form) and therefore provides a microscopic justification of his phenomenological heat-flux ansatz. Furthermore, it is easy to derive from it the small-time asymptotic expansion of the mean square displacement of Brownian motion in static spacetimes. Beyond general relativity itself, this result has potential applications in analogue gravitational systems.

Is Schrodinger Equation a Relativistic Effect?

“…The aforementioned results can be found in a theses [8] describing vortices in a superfluid if we interpret these outcomes as a general rule for the quantum particle as a spherical vortex entity. There is also a rich literature on the subject of fluid spacetime [9][10][11][12][13]. According to the findings of this theory the transformations of equations ( 11),( 12), (13) are the result of a mass field which changes the volume.…”

Tailoring diffusion in analog spacetimes

“…(See [11] for more details.) Here, thanks to the connection with Brownian motion established in [11], it is applied to any diffusion phenomenon in the presence of an analog gravitational field, and not just to heat dynamics in dissipative relativistic fluids.…”

“…We found that, if q ab = q ab (x) is static, the generalized diffusion equation reads [11] ∂ t p = κ q (Np).…”

“…Walking in these footsteps, we derived in [11] the equation governing the probability density of Brownian motion in the effective geometry…”

Mixtures of relativistic gases in gravitational fields: Combined Chapman-Enskog and Grad method and the Onsager relations

“…In equation (5), quantities denoted with a prime are evaluated with the momentum of the particles after a binary collision occurs, i.e., f …”