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

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 <?CDATA k(t) = k₀+a\,δ(t)?> k ( t ) = k 0 + a δ ( t ) , involving the Dirac delta function. The microscopic work performed on the oscillator is shown to be <?CDATA W = (a²/2m)\,q²-a q v?> W = ( a 2 / 2 m ) q 2 − a q v , 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 <?CDATA W = a²T/(2m\,k₀)?> ⟨ W ⟩ = a 2 T / ( 2 m k 0 ) 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.