Rotational Brownian motion (astronomy) that the rotation axis of the binary is oriented at angle θ at time "t". The evolution equation for "F" is formula_17 If <Δξ>, "a", ρ and σ are constant in time, this becomes formula_18 where μ = cos θ and τ is the time in units of the relaxation time "t", where formula_19 The solution to this equation states that the expectation value of μ decays with time as formula_20 Hence, "t" is the time constant for the binary's orientation to be randomized by torques from field stars. Rotational Brownian motion was first discussed in the context of binary
Brownian motion moment of probability of displacement formula_2, is interpreted as mass diffusivity "D": Then the density of Brownian particles "ρ" at point "x" at time "t" satisfies the diffusion equation: Assuming that "N" particles start from the origin at the initial time "t"=0, the diffusion equation has the solution This expression (which is a normal distribution with the mean formula_12 and variance formula_13 usually called Brownian motion formula_14) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to