Effective dynamics of a particle diffusing in time-dependent confinement

A pointlike particle diffusing in a two-dimensional channel is considered. Its width h(x,t) varies not only with the longitudinal coordinate x, but also in time t. Reducing the transverse degree of freedom in the corresponding diffusion equation, we arrive at the effective one-dimensional dynamics along the channel, represented by the generalized Fick-Jacobs equation; its coefficients calculated recursively also depend on time. In contrast to the static confinement, the moving boundaries generate an effective force driving the particle in the channel, which can be explicitly expressed within our method. We analyze this force in two examples: deformation of the boundary advancing as the traveling wave and in the channel "breathing" with an amplitude varying in x.