PACE REPORT

The concentrations are time-dependent since they vary in time due to diffusion, characterised by diffusion constants D_i, and chemical reactions, characterised by reaction functions F_i(c(x, t), where c = (c₁, …, c_M). The equations of motion governing the dynamics is then the ordinary reaction-diffusion equations plus a term B_i(c_i(x, t)) capturing flows across the system border in the case of an open system,

          .                                  (16)

The term B_i(c_i(x, t)) typically is in the form of diffusion-controlled flows. We assume that the in- and out-flow is directly connected to the whole system. For example, in a two-dimensional system, this means that there is a flow across the “surface” of the system in the direction of a third dimension. We can view this as if the reaction volume everywhere is in contact with a reservoir having a constant concentration c_{i, res} which results in an inflow

         .                                                                    (17)

with b_i being a diffusion constant. (A negative value of that expression reflects an outflow.)

The equations of motion for the resolution dependent concentrations are derived from Eq. (16) by applying the resolution operator, see Eq. (10), to both sides of the equation,

                           (18)

Since the reaction terms typically are non-linear the resolution operator need to remain in front of the reaction function F_i.