The Generalized Stochastic Smoluchowski Equation

Abstract

We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy 17, 3205 (2015)]. We first neglect fluctuations and provide a macroscopic description of the system based on the deterministic mean field Smoluchowski equation. We then take fluctuations into account and provide a mesoscopic description of the system based on the stochastic mean field Smoluchowski equation. We establish the main properties of this equation and derive the Kramers escape rate formula, giving the lifetime of a metastable state, from the theory of instantons. We relate the properties of the generalized stochastic Smoluchowski equation to a principle of maximum dissipation of free energy. We also discuss the connection with the dynamical density functional theory of simple liquids.

Impact of kinetic and potential self-interactions on scalar dark matter

“…For an isothermal halo, which corresponds to α = 2, we obtain M⋆ M ∝ M −1/3 . This agrees with the result obtained by [72], which appeared during completion of this paper, where the author considers thermodynamical equilibrium of a solitonic core inside an isothermal halo of temperature T . This can also be understood from the continuity of the Newtonian potential Φ and the fact that in the isothermal halo the circular velocity v c is constant and Φ(r) ∝ ln(r), so that Φ ⋆ ≃ Φ(R) up to logarithmic corrections, and v c (R ⋆ ) = v c (R).…”

“…However, we also expect solitons to form and merge inside halos generated by gravitational collapse and stable configurations to survive, while the outer regions should again follow the standard NFW profile. The case of solitonic cores inside isothermal halos was considered in the recent analytical work [72], which appeared during completion of this paper. Here we do not assume thermodynamical equilibrium within an isothermal halo and build a simple matching to the outer NFW halo.…”

Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View

“…Thus, the main aim of this paper is to investigate the anisotropic fluctuation field of counterion density due to the coarse-grained dynamics of counterion density around a ground state distribution , using the stochastic density functional equation. The stochastic density functional theory is based on the so-called Dean–Kawasaki (DK) equation that describes the evolution of the instantaneous microscopic density field of overdamped Brownian particles [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ]. The stochastic density functional theory has been used as one of the most powerful tools for describing slowly fluctuating and/or intermittent phenomena [ 16 , 17 , 18 , 19 , 20 ], such as glassy dynamics, nucleation or pattern formation of colloidal particles, stochastic thermodynamics of colloidal suspensions, dielectric relaxation of Brownian dipoles, and even tumor growth.…”

“…The original DK equation includes nonlinear terms of dynamic origin due to the kinetic coefficient that is proportional to fluctuating field [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ]. While the nonlinearility of the original DK equation leads to the above successful descriptions of various phenomena [ 16 , 17 , 18 , 19 , 20 ], a more tractable form is required.…”

“…The original DK equation includes nonlinear terms of dynamic origin due to the kinetic coefficient that is proportional to fluctuating field [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ]. While the nonlinearility of the original DK equation leads to the above successful descriptions of various phenomena [ 16 , 17 , 18 , 19 , 20 ], a more tractable form is required. It has been recently demonstrated that the DK equation can be linearized with respect to when , and that the linear stochastic equation of density fluctuations is of great practical use [ 21 , 22 , 23 , 24 , 25 , 26 , 27 ].…”