Diffusion’s induced transport in periodic channels and an inverse problem

A diffusion’s induced transport is defined for a linear model of a Fokker-Plank equation under periodic boundary conditions in one-dimensional geometry. The flow is generated by a diffusion and a periodic deriving force induced by a velocity potential. An inverse problem is suggested for evaluating the deriving force in terms of the response function associated with the flow. It is also shown that the inverse problem can be partially solved under some simplifying assumption. 1 Transport induced by a diffusion Recently, there is an increasing interest in transport process induced by an irreversible diffusion. This is motivated by some aspects of molecular dynamics and molecular motors [DEO]. The general paradigm of diffusion’s induced transport is a model of several components. Each component undergo an independent diffusion process, and the different components are coupled together by the poisson process [SKB], [PJAP]. In some cases the basic model is that of a diffusion of a single component, while the transport mechanism is induced by an time periodic deriving potential [KK]: ρt = (σρx +Ψxρ)x x ∈ [0, 1] , (1.1) (σx +Ψxρ)(0,t) = (σx +Ψxρ)(1,t) = 0 . (1.2) Here the deriving potential Ψ := Ψ(x, t) = Ψ(x, t + T ) is an appropriately defined function, and ρ ≥ 0 is the probability density of position of a ”test particle”. The model system (1.1), as the one studied in [KK], deals with no-flux boundary conditions (1.2). It is proved that the solution of (1.1,1.2) converges asymptotically, under general condition, to a periodic solution ρ(x, t) = ρ(x, t+ T ). The phenomenon of transport is introduced in a descriptive way: The period average of the asymptotic density ρ is shown to concentrate, under suitable conditions on Ψ, near one of the ends of the interval. The basic model introduced in [DEO], however, implies a geometry of periodic spatial structure. So, it seems reasonable to replace (1.2) by ρ(0, t) = ρ(1, t) , ρx(0, t) = ρx(1, t) . (1.3) 1 Department of Mathematics, Technion, Haifa, Israel. E.mail [email protected]