Klimontovich's contributions to the kinetic theory of nonlinear Brownian motion and new developments

Diffusion coefficient scaling of a free Brownian particle with velocity-dependent damping

An analytical expression for the diffusion coefficient D of a free Brownian particle with velocity-dependent damping γ(v) is derived from the Green-Kubo formula. A special case of damping that decreases monotonically with velocity is considered. At high temperature T, the diffusion coefficient is found to exhibit two scaling types: (i) for a power-law decrease of damping with the particle's kinetic energy, γ(v)∝1/v^{2α}, it scales as D∝T^{α+1}; (ii) for a Gaussian function γ(v), it diverges at temperatures above a critical value T_{c} and behaves as D∝1/sqrt[T_{c}-T] at T slightly below T_{c}. At T>T_{c}, the particle trajectory contains long flight events, which are not observed at T<T_{c} in case (ii) and at all temperatures in case (i).

Fluctuation relations for diffusion that is thermally driven by a nonstationary bath

In the context of the Markovian dynamical evolution in a nonstationary thermal bath, we construct a family of fluctuation relations for the entropy production that are not verified by the work performed on the system. We exhibit fluctuation relations, which are global versions either of the generalized fluctuation-dissipation theorem around a nonequilibrium diffusion or of the usual fluctuation-dissipation theorem for energy resulting from a pulse of temperature.

Nonlinear dissipation effect in Brownian relaxation

In an ensemble of noninteracting Brownian particles, a finite systematic average velocity may temporarily develop, even if it is zero initially. The effect originates from a small nonlinear correction to the dissipative force, causing the equation for the first moment of velocity to couple to moments of higher order. The effect may be relevant when a complex system dissociates in a viscous medium under strongly nonequilibrium conditions.

Vortex formation by active agents as a model for Daphnia swarming

We propose a self-propelled particle model for the swarming of Daphnia that takes into account mutual repulsion and attraction to a center. Surprisingly, a vortex is formed only for an intermediate strength of propulsion. The phase diagram and the transitions between states with and without a vortex are analyzed, and the nature of the phase boundaries is discussed based on a linear stability analysis of the motion of individual swimmers. This allows us to identify various key parameters determining the characteristic features of the collective motion.