Channel transport multiple particles

Exact solutions. It is possible to obtain some exact results for mean residence times even for channels with large numbers of particles although the results are typically cumbersome [90, 91]. Here, we briefly sketch the main points of the derivation for the case of single-file transport in a uniform channel in equilibrium with a solution of particles [90]. Most generally, the system of multiple particles in a channel is described by the multi-particle probability function P(x,t y) that the vector of particles positions is x at time t, starting from the initial vector y [53, 90, 92]. The crucial insight is that because the particles cannot bypass each other, the initial order of the particles is conserved if y < y for any two particles at the initial time, it implies that x < for all future times. That is, the parts of the phase space accessible to these particles are bounded by the planes defined by the condition = x in the vector space x. This implies a reflective boundary condition at the x = plane for any two different particles m and n,... 

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