Stochastic thermodynamics

Stochastic thermodynamics assigns work, heat and entropy production to individual fluctuating sample trajectories. It is useful for a small system, such as a trapped colloidal particle, whose exchanged energy is comparable to the thermal scale $k_{B}T$. The surroundings remain a thermal reservoir at temperature $T$, while the observed coordinates follow a stochastic process.^[ref]

For a one-dimensional optical trap with potential energy $U(x,\lambda )=k(x-\lambda {)}^2/2$, moving its centre $\lambda$ does incremental work $\delta W={\partial }_{\lambda }Ud\lambda$ on the particle. The coordinate change contributes ${\partial }_{x}U\circ dx=-\delta Q_{\mathrm{bath}}$, where the circle denotes the Stratonovich integral convention. The ordinary chain rule then gives the trajectory first law of thermodynamics, $dU=\delta W-\delta Q_{\mathrm{bath}}$. A particle ahead of the moving trap centre can give a negative instantaneous work increment, even during a protocol with positive mean dissipated work.

In the optical-trap experiment of G. M. Wang and collaborators, a recorded colloidal particle trajectory allowed short-time entropy-consuming realizations to be compared with entropy-producing ones. Repeating the same driving protocol measures a probability distribution rather than only a mean response.^[ref] The integral fluctuation theorem recovers nonnegative mean entropy production through Jensen's inequality, while allowing individual negative values.