Stochastic work, heat, and entropy

Consider an overdamped coordinate in a potential $U(x,\lambda)$, with constant mobility and an isothermal equilibrium bath satisfying $D=\mu k_BT$. Lambda is an externally prescribed control parameter. Define work W as energy supplied to the system by changing lambda, and heat $Q_{\mathrm{bath}}$ as energy delivered from the system to the bath. Then $\Delta U=W-Q_{\mathrm{bath}}$. This heat sign is opposite to the convention in which heat into the system is positive. Either convention is valid, but their first laws cannot be interchanged.

For conservative force $-\partial_xU$, the work increment is $\partial_\lambda U\,d\lambda$ and the bath heat increment is $-\partial_xU\circ dx$. The circle denotes a Stratonovich integral, ensuring the ordinary chain rule for U. The underlying Langevin equation can still be written in Itô form; with additive noise its drift is the same in both conventions. The distinction now concerns how the trajectory-dependent force is integrated against the noisy displacement, rather than a change of the physical dynamics.

If the displacement differential is interpreted in Itô form, the energy increment contains the additional term $D\partial_x^2U\,dt$. Dropping it would incorrectly violate the first law on average even at equilibrium. Equivalently, converting the Stratonovich heat to an Itô integral includes minus this same correction. A harmonic trap makes the issue visible: noise continually injects energy while drag removes it, and a constant equilibrium energy average results from their balance rather than from both effects individually vanishing.

At fixed lambda and with no additional force, heat to the bath equals the decrease in potential energy between the trajectory's endpoints. During an instantaneous parameter quench the coordinate has no time to change, so the potential-energy jump is work. During subsequent relaxation, endpoint energy differences alone do not specify the system's stochastic entropy, because they do not determine the probability densities. This separation is useful experimentally: positions and a calibrated potential may determine pathwise energy transfer, while entropy inference also needs ensemble or distribution information.

The stochastic system entropy can be represented by minus Boltzmann's constant times the logarithm of density relative to a fixed reference density. Its ensemble mean reproduces the corresponding Gibbs–Shannon entropy. A trajectory ending in a more probable state can have a negative system entropy change. Bath entropy can also be negative when a thermal fluctuation transfers heat into the system. Neither fact alone contradicts the second law, which constrains total entropy production on average under the complete model and preparation.

For regular thermally consistent Markov dynamics with mutually supported forward and reverse path ensembles, total production satisfies $\langle e^{-\Sigma/k_B}\rangle=1$. Jensen's inequality then gives nonnegative mean total production, while individual negative values remain possible. The system boundary term is essential; bath heat alone does not generally satisfy this same finite-time identity. Hidden reservoirs, feedback selection, singular support, or omitted degrees of freedom can require modified statements. The theorem therefore tests a fully specified trajectory ensemble, not an arbitrary heat histogram divorced from its preparation.