Displacement

Displacement is the change in position between two points.

Table of contents
- Mean displacement Displacement

Mean displacement ( $⟨ Δ r (t)⟩ = ⟨ r (t) - r (0)⟩$ )

The mean displacement is the expected value of a particle's displacement over a time interval. It describes systematic transport, while the mean squared displacement also measures fluctuations. For one component with finite second moment,

⟨(Δ x)^{2} ⟩ = Var (Δ x) + ⟨ Δ x ⟩^{2} .

For constant drift speed

u

and ordinary diffusion of coefficient

D

, the mean displacement is

u t

and its centered variance is

2 D t

, so the uncentered mean squared displacement is

u^{2} t^{2} + 2 D t

. With fixed initial velocity

v_{0}

and free inertial Langevin dynamics, the mean displacement is

v_{0} τ_{v} (1 - e^{- t / τ_{v}})

, where

τ_{v}

is the velocity relaxation time. Its fluctuation variance is given by the integrated Ornstein-Uhlenbeck displacement.

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