An exact solution to Brownian dynamics of a reversible bimolecular reaction in one dimension

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.