Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 1 / e

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- Solution e

Solution

For an admissible nonzero real trial function, integration by parts and the zero-flux boundary conditions reduce the Rayleigh quotient to

\frac{\int _{- L}^{L} y L y d x}{\int _{- L}^{L} w y ^{2} d x} = \frac{\int _{- L}^{L} p ( y ^{'} ) ^{2} d x}{\int _{- L}^{L} w y ^{2} d x} \geq 0.

A nonzero constant trial function attains zero. Positivity of

p

also shows that equality requires

y^{'} = 0

. Hence

λ_{0} = 0, f_{0} (x) = C, c_{eq} (x) = C e^{- β V (x)} .

If the conserved total amount is

M = \int_{- L}^{L} c d x

, the normalized stationary solution is

c_{eq} (x) = \frac{M e ^{- β V (x)}}{\int _{- L}^{L} e ^{- β V (s)} d s} .

Its diffusive flux vanishes everywhere. The Boltzmann distribution fixes the spatial profile, while the diffusivity affects the relaxation toward it. Boundedness of

V

makes the normalization finite on this finite interval.

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