  Brownian dynamics is a popular fine-grained method for simulating systems of
interacting particles, such as chemical reactions. Though the method is simple
to simulate, it is generally assumed that the dynamics is impossible to solve
exactly and analytically, aside from some trivial systems. We here give the
first exact analytical solution to a non-trivial Brownian dynamics system: the
reaction $A+B\xrightleftharpoons[]{}C$ in equilibrium in one-dimensional
periodic space. The solution is a function of the particles' diffusion
coefficients, radii, length of space and unbinding distance.

Statistical Mechanics