 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.

Stochastic thermodynamics is a branch of statistical mechanics that extends classical thermodynamics to systems that are small enough to be influenced by random fluctuations, particularly at the microscopic or nanoscale. It combines principles of thermodynamics with stochastic processes to describe the behavior of systems where thermal fluctuations play a significant role.