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The asymmetric simple exclusion process permits at most one particle per lattice site and biases jumps in one direction. A mean-field hydrodynamic limit has advective flux proportional to $c(1-c)$ because both an occupied departure site and vacant arrival site are required.

Particles on a countable lattice attempt jumps at the rates $p(x,y)$, and an attempt into an occupied site is suppressed. Thus each site contains at most one particle and the number of particles is conserved when finite. Here ${\eta }^{xy}$ interchanges the occupations at $x$ and $y$. Bounded nearest-neighbour rates give a well-defined Markov process. A symmetric jump kernel gives the symmetric simple exclusion process; an unequal left-right kernel gives the asymmetric simple exclusion process.

Under the independence closure, the mean rightward flux across the bond $(i,i+1)$ isThe mean equation is the discrete conservation law ${\dot{M}}_i=J_{i-1/2}-J_{i+1/2}$. Substitution of the rates from part c and Taylor expansion show that the exclusion factors cancel from the symmetric diffusive contribution but remain in the biased contribution:Therefore the mean-field asymmetric simple exclusion process limit isThe factor $1-c$ is the probability that the destination site is vacant.