Abstract
We investigate a one-dimensional linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint, it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process.
Highlights
Unbiased velocity randomization by a jump process or by Brownian motion combined with acceleration by the force field produced by a confining potential can lead to convergence to an invariant probability measure, if the confinement is strong enough to balance the dispersive effect of velocity randomization
In this work a related type of particle dynamics is considered, where confinement is achieved by a biased velocity jump process, where the bias replaces the confining acceleration field
In the presence of a spatial chemo-attractant gradient, this stochastic process is biased upwards the gradient, E. coli is too small to reliably measure the gradient along its length. An explanation for this phenomenon is that E. coli is able to measure gradients in time along its path and increases its tumbling frequency, if it experiences decreasing chemo-attractant concentrations
Summary
Unbiased velocity randomization by a jump process or by Brownian motion combined with acceleration by the force field produced by a confining potential can lead to convergence to an invariant probability measure, if the confinement is strong enough to balance the dispersive effect of velocity randomization. For kinetic transport models of this kind, convergence to confined equilibria has been studied extensively leading to strong convergence results with algebraic [10, 14] and later exponential [17, 30] convergence rates This is strongly related to the corresponding macroscopic description by Fokker-Planck equations of drift-diffusion type [4]. A second, more realistic macroscopic limit is carried out in the second part of Section 4, where smallness of the parameter χ is assumed, and length and time is rescaled diffusively Both macroscopic limits produce drift-diffusion equations, whose diffusivities and convection velocities are different, but with the same qualitative behavior. This brings us back to the beginning of the introduction, since it shows that the macroscopic behavior created by biased velocity jumps is the same as for unbiased jumps combined with a confining potential
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