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

Table of contents
- Solution d

Solution

With an external force

F

, the required forward and backward noise histories become

f_{F} = E + ζ \dot{x} - F, f_{B} = E - ζ \dot{x} - F .

Repeating the difference of squares and using

2 ζ / σ^{2} = β

gives

\frac{P _{F}}{P _{B}} = exp [- β Δ H + β \int_{t_{1}}^{t_{2}} F (x) \cdot \dot{x} d t] .

The second term is

β

times the work performed by the external force.

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