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 $k(t)=k_0+a\,δ(t)$, involving the Dirac delta function. The microscopic work performed on the oscillator is shown to be $W=(a^2/2m)\,q^2-a q v$, 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 $\langle W \rangle=a^2T/(2m\,k_0)$ 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.

Figures (4)

• Figure 1: The work distribution function (\ref{['df']}) for several values of $|\alpha|=|a|/(m\,\omega_0)$.
• Figure 2: The response function $\varphi(t)$, defined by Eq. (\ref{['response_def']}), for the Markovian overdamped oscillator, described by the generalized Langevin equation with the kernel $K(t)=\gamma\,\delta(t)$, for $\beta=\gamma/\omega_0=5.0$ and several positive (left) and negative (right) values of the perturbation intensity $\alpha=a/(m\omega_0)$.
• Figure 3: The response function $\varphi(t)$ given by (\ref{['response_explicit']}), for several positive (left) and negative (right) values of the perturbation intensity $\alpha=a/(m\omega_0)$.
• Figure 4: The integration contour $\Gamma$ in Eq. (\ref{['w8']}).