  In an equilibrium chemical reaction mixture, the number of molecules present
obeys a Poisson distribution. We ask when the same is true of the steady state
of a nonequilibrium reaction network and obtain an essentially complete answer.
In particular, we show that networks with certain topological features must
have a Poisson distribution, whatever the reaction rates. Such driven systems
also obey an analog of the fluctuation-dissipation theorem. Our results may be
relevant to biological systems and to the larger question of how equilibrium
concepts might apply to nonequilibrium systems.
