PHY.5:5.3 - Retain fluctuations after fast velocity has relaxed

For a one-dimensional Brownian particle in a uniform equilibrium bath, take mass m, drag coefficient gamma and temperature T, with no applied force. The underdamped account has position x, velocity v and thermal forcing. Let tau=m/gamma and D=k_B*T/gamma. With an initially equilibrated velocity, its velocity covariance is (k_B*T/m)*exp(-|t-s|/tau).

Integrating that covariance over the two times gives

E[(x(t)-x(0))^2] = 2*D*(t-tau*(1-exp(-t/tau))).

At times large compared with tau, the overdamped diffusion description gives 2*D*t. Its omitted contribution to this mean-square displacement is bounded by 2*D*tau. This calculation states which long-time consequence the reduction preserves.

Setting the mean velocity to zero and deleting the forcing instead gives no displacement spread. The unresolved bath continues to transfer random impulses after the velocity’s preparation has relaxed. For the displacement distribution, retain their diffusion effect.

Changed work. Change the bath to a spatially varying temperature and ask about entropy production. The uniform-bath calculation no longer answers the question. Celani and coauthors show a further distinction: under their smooth-temperature and small-inertia conditions, the overdamped position process has the appropriate limit, while the mean rate of entropy production retains an additional positive contribution absent from the naive overdamped expression. Return to the thermodynamic observable and its limiting calculation. Position accuracy alone cannot decide that use. Their 2012 paper states the preparation and the contribution.