Random Walks

Random processes

The topics in this lecture are covered in chapter 7 sections 1-5 of Giordano & Nakanishi

Up to now, only deterministic systems have been discussed. These are systems described by some set of mathematical rules, like, e.g., a set of differential equations, which lead to uniquely defined solutions dependant on the initial values. Now, for the remainder of this lecture series, we turn to a class of systems in which randomness plays an important role. Such systems are called random or stochastic systems. There are in principle two reasons why a system is random:

1. If it is a quantum system, it is inherently probabilistic. While the probabilities for the possible outcomes of an experiment may be calculable from first principles, the experiment itself will yield meaningful results only after sampling, such that the statistical property of the quantum mechanical measurement process kicks in.
2. The system may consist of very large (statistically large) numbers of degrees of freedom, which may be associated with particles, spins, or similar. Randomness can then emerge in various ways. First of all, while the exact manner of interaction of the d.o.f.'s among each other may be known and may be deterministic, it might be impossible to work this out in detail, due to either incomplete knowledge of all positions etc. or due to the sheer number of d.o.f.'s. On the other hand, the system may interact with a thermal reservoir or similar in a non-trivial manner, such that only a probabilistic description is possible. In both cases, solutions can be obtained best by resorting to statistical descriptions, using averages and probabilities.

A typical problem of this kind is diffusion. This describes such important processes as how a drop of milk spreads in a cup of tea. Assume that we start with a nice hot cup of tea and then delicately place a drop of milk in the middle (no, not just dropping it!). If we resist stirring the cup, this drop will cling together for some time before it slowly starts to dissolve until it is distributed over all the cup of tea. At a microscopic level, this process would be described in the following way: A drop of milk consists of a large number of milk particles (typically clusters of molecules). If we were able to watch one of them, we would see a complicated trajectory, going along a straight line for some time, before a collision with another milk or coffee particle changes its velocity vector.

But how can a useful description of this phenomenon be constructed? In principle, equations of motion could be written down and solved for a macroscopically large (of the order of 10²³) number of particles - in practice this is an impossible task well beyond the capabilities computers can reasonably be expected to have in our lifetimes. A second problem is that even if such a solution existed it is not clear how we could interpret the velocities and positions of so many particles in a meaningful way and extract useful information. For example, assume such a monster calculation would show that the milk completely mixes with the tea in, say, 20 minutes, how do we know how long the mixing takes in a cup twice the size? The point here is that such a calculation would over-flood us with unnecessary and unwanted information - what is really needed and wanted is a statistical description of the problem and a corresponding idea about the behaviour of the physical system.

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