Brownian motion

Brownian Motion: Langevin Equation

Online simulation: http://labs.minutelabs.io/Brownian-Motion/

Discrete space: random walk

Random walk on 1D lattice

Biased random walk, probability distribution is Binomial

Limits in time variable

• Discrete time and space: discrete time Master equation
• continuous time limit, but keeping space discrete, we get the continuous time Master equation.

A discrete space-time random walk has a standard deviation in position that is proportional to square root of number of steps:

$\frac{\sigma_x}{a}=\sqrt{n}=\sqrt{\frac{t}{\tau}}$

$\sigma_x=\frac{a}{\sqrt{\tau}}\sqrt{t}$

Clearly if we want $\sigma_x$ to stay finite for a finite $t$, we want $\frac{a}{\sqrt{\tau}}$ to stay finite, and we get $\sigma_x\propto\sqrt{t}$ in continuous limit. We also get non-differentiable paths as $\frac{a}{\tau}\rightarrow \infty$.

Random walk on 2D square lattice. Combinatorics get harder

Solving random walk diffusion on a finite domain with different boundary conditions

Polya's recurrence theorem for random walks

See also probability distribution for random walk (same as for polymer) [For example here or in Soft Matter Physics notes. The probability density at origin goes like $1/N^{d/2}$ (normalization of Gaussian). One can then sum over all possible lengths of time (i.e. $N$s) and get the expected number of times one returns to (a neighbourhood of) the origin (See Note 1 in Probability theory for why). For $d=1,2$, this is $\infty$, while it's finite for $d \geq 3$. This can be interpreted for a polymer as it being "dense" or "sparse", as summing over $N$ we are asking the question how many monomers of our very long polymer are close to a given point (say the origin)?

One can also find probability of ever coming back, and this can be related to the expected number of times to come back. This can also be derived heuristically for the asymptotic limit of large times.

First passage time: First passage time calculation using generating functions.. The generating functions also give the {survival probability}, which is the same as {probability of ever coming back}.

Mean first passage time for an ergodic Markov chain equals the reciprocal of the stationary probability. Intuition

Random walk in a graph

Continuous space

Continuous space-time limit from discrete random walk

Diffusion If continuous space and continuous time: Diffusion equation

Can also have continuous space, and discrete time, although not often used.

Phenomenological derivation of Diffusion equation Use Fick's laws of diffusion, and Einstein–Smoluchowski relation

Eistein's original derivation from Chapman-Kolmogorov equation, as Brownian motion is assumed to be a Markov process

Simulate on Matlab