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We derive a relation between the dissipation in a stochastic dynamics and theWasserstein distance. We show that the minimal amount of dissipation requiredto transform an initial state to a final state during a diffusion process isgiven by the Wasserstein distance between the two states, divided by the totaltime of the process. This relation implies a lower bound on the dissipation forany diffusion process in terms of its initial and final state. Using a lowerbound on the Wasserstein distance, we further show that we can give a lowerbound on the dissipation in terms of only the mean and convariance matrix ofthe initial and final state. We apply this result to derive the optimal forcesthat minimize the dissipation for given initial and final mean and covariance.

Thermodynamic interpretation of Wasserstein distance