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 Incoming links: Local detailed balance

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3 / e / Solution 2026-09-28

The first law of thermodynamics, with

Δ Q

defined as heat lost by the particle to the bath, is

Δ H = W - Δ Q .

Thus

W - Δ H = Δ Q

, and part d becomes the local detailed balance identity

\frac{P _{F}}{P _{B}} = e^{β Δ Q} .

The logarithmic path-probability ratio is the bath's entropy production

Δ S_{bath} = Δ Q / T

in units of

k_{B}

.

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3 / g / Solution 2026-09-28

For a specified field trajectory, the required normalized noise is

Λ_{F} = \frac{p ˙ + Γ δ F / δ p - Γ Y}{2 k _{B} T Γ},

and time reversal changes only

\dot{p}

to

- \dot{p}

. Assuming equal additive-noise Jacobians, the difference of the two Onsager--Machlup actions gives

lo g \frac{P _{F}}{P _{B}} = - β \int_{t_{1}}^{t_{2}} d t \int d r \dot{p} \cdot (\frac{δ F}{δ p} - Y) .

The functional chain rule identifies the first term as

- β Δ F

, so

\frac{P _{F} [ p ]}{P _{B} [ p ]} = exp [- β Δ F + β \int_{t_{1}}^{t_{2}} d t \int d r Y \cdot \dot{p}] .

The forcing performs generalized work

W_{Y} = \int Y \cdot \dot{p}

, and

W_{Y} - Δ 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.

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