Random Walk and Brownian Motion

Abstract

Random walk processes are an important class of stochastic processes. They have many applications in physics, computer science, ecology, economics and other fields. A random walk is a sequence of successive random steps. In this chapter we study Markovian discrete time models. In one dimension the position of the walker after n steps approaches a Gaussian distribution, which does not depend on the distribution of the single steps. This follows from the central limit theorem and can be checked in a computer experiment. A 3-dimensional random walk provides a simple statistical model for the configuration of a biopolymer, the so called freely jointed chain model. In a computer experiment we generate random structures and calculate the gyration tensor, an experimentally observable quantity, which gives information on the shape of a polymer. Simulation of the dynamics is simplified if the fixed length segments of the freely jointed chain are replaced by Hookean springs. This is utilized in a computer experiment to study the dependence of the polymer extension on an applied external force (this effect is known as entropic elasticity). The random motion of a heavy particle in a bath of light particles, known as Brownian motion, can be described by Langevin dynamics, which replace the collisions with the light particles by an average friction force proportional to the velocity and a randomly fluctuating force with zero mean and infinitely short correlation time. In a computer experiment we study Brownian motion in a harmonic potential.

Problems

Problem 17.1 Random Walk in One Dimension

This program generates random walks with (a) fixed step length $\varDelta x=\pm 1$ or (b) step length equally distributed over the interval $-\sqrt{3}<\varDelta x<\sqrt{3}$. It also shows the variance, which for large number of walks approaches $\sigma =\sqrt{n}$. See also Fig. 17.2

Problem 17.2 Gyration Tensor

The program calculates random walks with M steps of length b. The bond vectors are generated from M random points $\mathbf {e}_{i}$ on the unit sphere as $\mathbf {b}_{i}=b\mathbf {e}_{i}$. End to end distance, center of gravity and gyration radius are calculated and can be averaged over numerous random structures. The gyration tensor (Sect. 17.3.2) is diagonalized and the ordered eigenvalues are averaged.

Problem 17.3 Brownian Motion in a Harmonic Potential

The program simulates a particle in a 1-dimensional harmonic potential

$$\begin{aligned} U(\mathbf {x})=\frac{f}{2}x^{2}-\kappa x \end{aligned}$$

(17.66)

where $\kappa$ is an external force . We use the improved Euler method (13.36). First the coordinate and the velocity at mid time are estimated

$$\begin{aligned} \mathbf {x}\left( t_{n}+\frac{\varDelta t}{2}\right) =\mathbf {x}(t_{n})+\mathbf {v}(t_{n})\frac{\varDelta t}{2} \end{aligned}$$

(17.67)

$$\begin{aligned} \mathbf {v}\left( t_{n}+\frac{\varDelta t}{2}\right) =\mathbf {v}(t_{n})-\gamma \mathbf {v}(t_{n})\frac{\varDelta t}{2}+\frac{\mathbf {F}_{n}}{m}\frac{\varDelta t}{2}-\frac{f}{m}\mathbf {x}(t_{n})\frac{\varDelta t}{2} \end{aligned}$$

(17.68)

where $\mathbf {F}_{n}$ is a random number obeying (17.65). Then the values at $t_{n+1}$ are calculated as

$$\begin{aligned} \mathbf {x}(t_{n}+\varDelta t)=\mathbf {x}(t_{n})+\mathbf {v}\left( t_{n}+\frac{\varDelta t}{2}\right) \varDelta t \end{aligned}$$

(17.69)

$$\begin{aligned} \mathbf {v}(t_{n}+\varDelta t)=\mathbf {v}(t_{n})-\gamma \mathbf {v}\left( t_{n}+\frac{\varDelta t}{2}\right) \varDelta t+\frac{\mathbf {F}_{n}}{m}\varDelta t-\frac{f}{m}\mathbf {x}\left( t_{n}+\frac{\varDelta t}{2}\right) \varDelta t. \end{aligned}$$

(17.70)

Problem 17.4 Force Extension Relation

The program simulates a chain of springs Sect. 17.3.3 with potential energy

$$\begin{aligned} U=\frac{f}{2}\sum \left( |\mathbf {b}_{i}|-b\right) ^{2}-\varvec{\kappa }\mathbf {R}_{M}. \end{aligned}$$

(17.71)

The force can be varied and the extension along the force direction is averaged over numerous time steps.