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 Incoming links: Small-slope elastic membrane energy

Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 69 / 1 / a / Solution Created 2026-10-03 Updated 2026-10-14

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Put

D = d - 1

for the membrane's internal dimension and assume

σ > 0

. The small-slope expansion is

1 + ∣\nabla h ∣^{2} = 1 + \frac{1}{2} ∣\nabla h ∣^{2} + O (∣\nabla h ∣^{4}) .

(1)

The zeroth-order term is the projected area and is independent of the height field. Discarding it changes only an overall normalization of the partition function. Thus the leading fluctuation energy and its statistical weight are

H_{2} [h] = \frac{σ}{2} \int d^{D} x ∣\nabla h ∣^{2}, Z_{2} = \int D h exp [- \frac{β σ}{2} \int d^{D} x ∣\nabla h ∣^{2}] .

(2)

One obtains this Gaussian field theory by discretizing the height field on a microscopic mesh, integrating every height with its Boltzmann weight, and then writing the continuum measure. A microscopic cutoff is retained for quantities dominated by short wavelengths. Quadratic order here really means small slopes: a large rigid height shift is still free because the energy has no dependence on

h

itself.

The uniform height mode has zero energy. Fix its mean, impose a boundary condition, or divide out the integral over global translations to normalize the partition function. Height differences are independent of that choice. The approximation is the tension-only part of small-slope elastic membrane energy; there is no bending rigidity or confining potential in this part.

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