Random Walk
Let us assume that the walker performed N steps. This constitutes one realization of a random walk. We may now be interested in computing properties of such a walk. From just one realization we can not draw any conclusion since the walk may be atypical. We need to generate many walks, calculate for every walk the desired property and then average over the results. The point which we want to make is that the generation of the samples, i.e., all the realizations of random walks are generated independently. Let Ai be the observable property computed for the i-th realization of a random walk. We define the average, or expectation value for the observable A, denoted by <A>, as the arithmetic mean over all Ai

<A> = SAi,

just as we did for the random numbers to calculate the first moment of the distribution. The distribution of the random walks is uniform.

One property we may want to compute is the end-to-end distance Re(N) of a walk of length N. The end-to-end distance is the Euclidean distance between the starting point and the end point of a walk. Here again the average means that we take all walks that we generated into account and all have the same statistical weight. But what is the error in the quantity. For this we need to compute the fluctuations or the expected deviation from the mean value

D(Re(N),n) = <Re(N)2> - <Re(N)>2.

The error depends on the number of observations (n) and the length of the walk (N). We may be inclined to think that the error tends to zero if we make the length of the walks longer and longer, suggesting that the length N of the walk has the same meaning as the volume of a system. Recall that for an infinite system a single observation suffices to obtain the observable.

Note that the error for the end-to-end distance is computed as the second central moment. Our notation of the angular bracket for the average quantities then implies that there is distribution for the values of the end-to-end distance and the radius of gyration.

For the simple random walk it is easy to show that the end-to-end distance scales as

Re(N) ~ Nn ~ N1/2
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