Abstract
In this chapter we study Markovian discrete time random walk models. In one dimension the position of the walker after n steps approaches a Gaussian distribution. This can be checked in a computer experiment. A 3-dimensional random walk provides a simple statistical model for the configuration of a biopolymer (freely jointed chain). 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 (entropic elasticity). The random motion of a heavy particle in a bath of light particles (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.
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- 1.
Different steps are independent.
- 2.
A special case of the more general continuous time random walk with a waiting time distribution of P(τ)=δ(τ−Δt).
- 3.
General random walk processes are characterized by a distribution function P(R,R′). Here we consider only correlated processes for which P(R,R′)=P(R′−R).
- 4.
For a 1-dimensional polymer \(\overline{\cos\theta_{i}}=0\) and \(\overline{(\cos\theta_{i})^{2}}=1\). In two dimensions \(\overline{\cos\theta_{i}}=\frac{1}{\pi}\int_{0}^{\pi}\cos\theta\,d\theta=0\) and \(\overline{(\cos\theta_{i})^{2}}=\frac{1}{\pi}\int_{0}^{\pi}\cos^{2}\theta\, d\theta=\frac{1}{2}\). To include these cases the factor 3 in the exponent of (16.33) should be replaced by the dimension d.
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Scherer, P.O.J. (2013). Random Walk and Brownian Motion. In: Computational Physics. Graduate Texts in Physics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00401-3_16
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DOI: https://doi.org/10.1007/978-3-319-00401-3_16
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