  Onsager's variational principle (OVP) 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
(SOMVP), we obtain the diffusion constant of a Brownian particle embedded in a
viscous fluid only 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 example of a non-equilibrium system.
Possible extensions of our formulation to internally driven active systems are
discussed.
