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 Incoming links: Zero-flux boundary condition

\frac{d}{d t} \int P d V = - \int \nabla \cdot J d V = - \int_{\partial V} J \cdot d S .

Assume the probability density function and its derivatives decay sufficiently rapidly at infinity, or impose a zero-flux boundary condition. The boundary integral then vanishes, so

\int P d V

is constant. Since it is initially one, the distribution remains normalized for all time.

A centered nearest-neighbour discretization of drift--diffusion is obtained from

k_{i}^{+} = \frac{D}{h ^{2}} + \frac{v ( x _{i} )}{2 h}, k_{i}^{-} = \frac{D}{h ^{2}} - \frac{v ( x _{i} )}{2 h} .

For sufficiently small

h

these rates are nonnegative. Taylor expansion of the mean equations gives

\partial_{t} c = D \partial_{x}^{2} c - \partial_{x} (v c) .

Absorption into the target to the left of the first compartment gives

c (0, t) = 0,

while the absence of outward jumps at the right endpoint gives the zero-flux boundary condition

v (L) c (L, t) - D \partial_{x} c (L, t) = 0.