 Solution

ID: past-exam-of-the-field-theories-and-collective-phenomena-in-condensed-matter-course-of-the-university-of-oxford/2025/1/d/solution

Y (t) := y (t) - y (0) = \int_{0}^{t} η (s) d s .

⟨ y (t) - y (0)⟩ = 0.

⟨ Y (t)^{2} ⟩ = \int_{0}^{t} d s \int_{0}^{t} d s^{'} ⟨ η (s) η (s^{'})⟩ = Γ \int_{0}^{t} d s \int_{0}^{t} d s^{'} δ (s - s^{'}) = Γ t .

Thus

⟨[y (t) - y (0)]^{2} ⟩ = Γ t

, which is also the variance because the displacement has zero expected value. More generally its covariance is

⟨ Y (s) Y (t)⟩ = Γ min (s, t)

. Equivalently

Y (t) = Γ W_{t}

D = Γ/2

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