Statistical formulation of the Onsager-Machlup variational principle

Onsager's variational principle provides us with a systematic way to derive dynamical equations for various soft matter and active matter. By reformulating the Onsager-Machlup variational principle (OMVP), which is a time-global principle, we propose a new method to incorporate thermal fluctuations. To demonstrate the utility of the statistical formulation of OMVP, we obtain the diffusion constant of a Brownian particle embedded in a viscous fluid by maximizing the modified Onsager-Machlup integral for the surrounding fluid. We also apply our formulation to a Brownian particle in a steady shear flow, which is a typical nonequilibrium system. Possible extensions of our formulation to internally driven active systems are also discussed.