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 $k(t)=k_0+a\delta (t)$, involving the Dirac delta function. The microscopic work performed on the oscillator is shown to be $W=(a^2\mathrm{/}2m)q^2-aqv$, 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 $⟨W⟩=a^{2}T\mathrm{/}(2m{k}_0)$ and the distribution $f(W)$ has the form of the product of exponential and modified Bessel functions. The distribution is asymmetric and diverges as $W\to 0$. The system’s response for $t>0$ is evaluated for specific models.