= Stochastic thermodynamics
{wiki}
<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 $k_BT$. The surroundings remain a <thermal reservoir> at <temperature> $T$, while the observed coordinates follow a <stochastic process>.<Stochastic thermodynamics, fluctuation theorems, and molecular machines by Udo Seifert>[§§1.1–1.2 and 2.3, trajectory energetics of a driven colloidal particle]{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> $\delta W=\partial_\lambda U\,d\lambda$ on the particle. The coordinate change contributes $\partial_xU\circ dx=-\delta Q_{\rm bath}$, where the circle denotes the <Stratonovich integral> convention. The ordinary <chain rule> then gives the trajectory <first law of thermodynamics>, $dU=\delta W-\delta Q_{\rm bath}$. 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.<Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales by G. M. Wang and others>[Abstract: translated optical trap, recorded particle trajectories and finite-time fluctuation test]{ref} The <integral fluctuation theorem> recovers nonnegative mean <entropy production> through <Jensen's inequality>, while allowing individual negative values.
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