Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3

Put $E(t)=-\delta A/\delta \mathbf{x}(t)$. For a specified forward trajectory, the equation of motion determineswhereas its time reverse requiresAssume the action is invariant under time-reversal symmetry, the position is even and velocity is odd under time reversal, and the trajectory-to-noise Jacobian is identical in the two directions. The Onsager--Machlup path probability is thenwith $\zeta \dot{\mathbf{x}}$ replaced by $-\zeta \dot{\mathbf{x}}$ for ${\mathbb{P}}_B$. Sincetheir ratio is

The functional derivative of the action isFor the Hamiltonian functionand a time-independent Lagrangian,Therefore

Detailed balance says that each equilibrium transition is balanced by its time reverse:With the Boltzmann distribution $P_{\mathrm{eq}}\propto e^{-\beta H}$ and $\Delta H=H_2-H_1$, this requiresParts a and b instead give $\exp [-(2\zeta /{\sigma }^2)\Delta H]$. Equality for every pair of endpoints yields the fluctuation-dissipation relation for a Langevin particle

With an external force $\mathbf{F}$, the required forward and backward noise histories becomeRepeating the difference of squares and using $2\zeta /{\sigma }^2=\beta$ givesThe second term is $\beta$ times the work performed by the external force.

The first law of thermodynamics, with $\Delta Q$ defined as heat lost by the particle to the bath, isThus $W-\Delta H=\Delta Q$, and part d becomes the local detailed balance identityThe logarithmic path-probability ratio is the bath's entropy production $\Delta S_{\mathrm{bath}}=\Delta Q/T$ in units of $k_B$.

If $\nabla \times \mathbf{F}=0$ on a simply connected region containing the bounded trajectory, then the Poincare lemma gives a scalar potential $U$ with $\mathbf{F}=-\nabla U$. The work iswhich remains bounded when $\mathbf{x}$ and $\mathbf{F}$ remain bounded. Since $H$ is also bounded, $\Delta Q=W-\Delta H$ cannot grow linearly with the observation time.

Thus sustained linear heat dissipation in this setting requires a nonconservative force, and, under the stated simply connectedness assumption,Such a force can perform nonzero work on repeated bounded cycles. On a multiply connected domain, a curl-free force can have nonzero circulation, so the topological assumption is essential.

For a specified field trajectory, the required normalized noise isand time reversal changes only $\dot{\mathbf{p}}$ to $-\dot{\mathbf{p}}$. Assuming equal additive-noise Jacobians, the difference of the two Onsager--Machlup actions givesThe functional chain rule identifies the first term as $-\beta \Delta F$, soThe forcing performs generalized work $W_Y=\int \mathbf{Y}\cdot \dot{\mathbf{p}}$, and $W_Y-\Delta F$ is the heat dissipated into the bath. The formula is therefore the field-theory form of local detailed balance and quantifies nonequilibrium entropy production.