Brownian oscillator with time-dependent strength: a delta function protocol

Abstract

We consider a classical Brownian oscillator of mass m driven from an arbitrary initial state by varying the stiffness k(t) of the harmonic potential according to the protocol k(t)=k0+aδ(t) , involving the Dirac delta function. The microscopic work performed on the oscillator is shown to be W=(a2/2m)q2−aqv , where q and v are the coordinate and velocity in the initial state. If the initial distribution of q and v is the equilibrium one with temperature T, the average work is ⟨W⟩=a2T/(2mk0) and the distribution f(W) has the form of the product of exponential and modified Bessel functions. The distribution is asymmetric and diverges as W → 0. The system’s response for t > 0 is evaluated for specific models.