Editing Polymer physics (section)

===Random walks in time===
The first example of a random walk is one in space, whereby a particle undergoes a random motion due to external forces in its surrounding medium.  A typical example would be a pollen grain in a beaker of water.  If one could somehow "dye" the path the pollen grain has taken, the path observed is defined as a random walk.

Consider a toy problem, of a train moving along a 1D track in the x-direction.  Suppose that the train moves either a distance of +''b'' or −''b'' (''b'' is the same for each step), depending on whether a coin lands heads or tails when flipped.  Lets start by considering the statistics of the steps the toy train takes (where ''S<sub>i</sub>'' is the ith step taken):

:<math>\langle S_{i} \rangle = 0</math> ; due to ''a priori'' equal probabilities
:<math>\langle S_{i} S_{j} \rangle = b^2 \delta_{ij}.</math>

The second quantity is known as the [[correlation function]].  The delta is the [[kronecker delta]] which tells us that if the indices ''i'' and ''j'' are different, then the result is 0, but if ''i'' = ''j'' then the kronecker delta is 1, so the correlation function returns a value of ''b''<sup>2</sup>.  This makes sense, because if ''i'' = ''j'' then we are considering the same step.  Rather trivially then it can be shown that the average displacement of the train on the x-axis is 0;

:<math>x = \sum_{i=1}^{N} S_i</math>
:<math>\langle x \rangle = \left\langle \sum_{i=1}^N S_i \right\rangle</math>
:<math>\langle x \rangle = \sum_{i=1}^N \langle S_i \rangle.</math>

As stated <math>\langle S_i \rangle = 0</math>, so the sum is still 0.  
It can also be shown, using the same method demonstrated above, to calculate the root mean square value of problem.  The result of this calculation is given below

:<math>x_\mathrm{rms} = \sqrt {\langle x^2 \rangle} = b \sqrt N. </math>

From the [[diffusion equation]] it can be shown that the distance a diffusing particle moves in a medium is proportional to the root of the time the system has been diffusing for, where the proportionality constant is the root of the diffusion constant.  The above relation, although cosmetically different reveals similar physics, where ''N'' is simply the number of steps moved (is loosely connected with time) and ''b'' is the characteristic step length.  As a consequence we can consider diffusion as a random walk process.