A Schrödinger bridge is the most probable time-dependent probability distribution that connects an initial probability distribution $w_i$ to a final one $w_f$. The problem has been solved and widely used for the case of simple Brownian evolution (noninteracting particles). It is closely related to the problem of entropy-regularized Wasserstein optimal transport. In this article, we generalize the problem to systems of interacting particles. We derive some equations for the forward and backward single-particle “wave functions,” which allows computation of the most probable evolution of the single-particle probability between the initial and final distributions.

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• Brownian motion
• Nonequilibrium statistical mechanics
• Optimization problems