I would like to know the probability of return to the initial point in three dimensional Brownian motion. Does someone know an expression for the diffusion constant? (Suggestions of books on this would also be welcome)
[Physics] the return probability for Brownian motion in three dimensions
brownian motiondiffusionfluid dynamics
Best Answer
For Brownian motion, Langevin equation, Fokker-Planck equations, Stochastic process.. from the viewpoint of physicists, the following are standard references:
For a free particle that starts from rest at $t=0$ and position $\mathbf{x}_0$, the probability density to find the particle at position $\mathbf{x}$ at time $t$ is $$p(\mathbf{x},t | \mathbf{x}_0, 0) = \frac{1}{(4\pi D t)^{3/2}} e^{-\frac{(\mathbf{x} - \mathbf{x}_0)^2}{4Dt}}$$ where $$D = \frac{k_BT}{6 \pi \eta a}$$ is the diffusion constant in three dimensions for a particle of radius $a$ immersed in a fluid of viscosity $\eta$ at temperature $T$. The probability density to find the particle at the same point that it started from, i.e., $\mathbf{x} = \mathbf{x}_0$, is therefore $$p(\mathbf{x}_0,t | \mathbf{x}_0, 0) = \frac{1}{(4\pi D t)^{3/2}}$$
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