Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 2 / c

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- Solution c

Solution

Each net crossing of a periodic cell advances the unwrapped particle by

2 π

. The stationary mean velocity is therefore

v_{s} = 2 π J_{0} = \frac{2 π D ( 1 - e ^{- 2 π F_{0} / D} )}{\int _{M_{0}}^{M_{1}} u ( x ) d x} .

For

F_{0} > 0

the numerator is positive, so motion is on average toward increasing

x

, down the tilted potential.

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