Second Law of Thermodynamics at Stopping Times

Events in mesoscopic systems often take place at first-passage times, as is for instance the case for a colloidal particle that escapes a metastable state. An interesting question is how much work an external agent has done on a particle when it escapes a metastable state. We develop a thermodynamic theory for processes in mesoscopic systems that terminate at stopping times, which generalize first-passage times. This theory implies a thermodynamic bound, reminiscent of the second law of thermodynamics, for the work exerted by an external protocol on a mesoscopic system at a stopping time. As an illustration, we use this law to bound the work required to stretch a polymer to a certain length or to let a particle escape from a metastable state.

Figure 1

Stretching a polymer to a certain length $\ell$ [panel (a)] or letting a particle escape from a metastable state [panel (b)]. Panel (a): an external agent (blue square) is connected with a spring (blue zigzag line) to one of the end points (green circles) of a polymer (grey zigzag line) and stretches the polymer until it reaches a length $\ell$, after which the polymer end point is attached to an anchor point (red object). Panel (b): a colloidal particle (full circle) escapes from a metastable state under the influence of an external protocol $\lambda (t)$ that changes the shape of the potential $\varphi (x;\lambda )$.

Figure 2

Simulation results for stretching a polymer [panel (a)] and the escape problem [panel (b)]. Panel (a): model parameters are $\ell =2.2$, $\mu =0.1$, $\beta ={\kappa }_p=1$, ${\kappa }_m=2$, ${\lambda }_i=0.2$, ${\lambda }_f=5$, and $\tau =1e+6$. The relaxation time ${\tau }_{\mathrm{rel}}=10/3$ and the mean first-passage time ${\tau }_{fp}\approx 1560$ are denoted by the vertical dotted lines. The black solid line equals zero and is a guide to the eye. Panel (b): model parameters are $\mu =0.1$, $\beta =x_{\max }=1$, ${\varphi }_{\max }=10$, and ${\tau }_{\mathrm{prot}}=4$. Markers are averages over $1e+4$ realizations of the process.