Editing Polymer physics (section)

===Random walks in space===
{{main|Ideal chain}}
Random walks in space can be thought of as snapshots of the path taken by a random walker in time.  One such example is the spatial configuration of long chain polymers.

There are two types of random walk in space: ''[[Self-avoiding walk|self-avoiding random walks]]'', where the links of the polymer chain interact and do not overlap in space, and ''pure random'' walks, where the links of the polymer chain are non-interacting and links are free to lie on top of one another.  The former type is most applicable to physical systems, but their solutions are harder to get at from first principles.

By considering a freely jointed, non-interacting polymer chain, the end-to-end vector is 
:<math>\mathbf{R} = \sum_{i=1}^{N} \mathbf r_i</math>
where '''r'''<sub>''i''</sub> is the vector position of the ''i''-th link in the chain.  
As a result of the [[central limit theorem]], if ''N'' ≫ 1 then we expect a [[Gaussian distribution]] for the end-to-end vector.  We can also make statements of the statistics of the links themselves;

* <math>\langle \mathbf{r}_{i} \rangle = 0</math> ; by the isotropy of space
* <math>\langle \mathbf{r}_{i} \cdot \mathbf{r}_{j} \rangle = 3 b^2 \delta_{ij}</math> ; all the links in the chain are uncorrelated with one another

Using the statistics of the individual links, it is easily shown that 
:<math>\langle \mathbf R \rangle = 0</math>
:<math>\langle \mathbf R \cdot \mathbf R \rangle = 3Nb^2</math>.
Notice this last result is the same as that found for random walks in time.

Assuming, as stated, that that distribution of end-to-end vectors for a very large number of identical polymer chains is gaussian, the probability distribution has the following form

:<math>P = \frac{1}{\left (\frac{2 \pi N b^2}{3} \right )^{3/2}} \exp \left(\frac {- 3\mathbf R \cdot \mathbf R}{2Nb^2}\right).</math>

What use is this to us?  Recall that according to the principle of equally likely ''a priori'' probabilities, the number of microstates, Ω, at some physical value is directly proportional to the probability distribution at that physical value, ''viz'';

:<math>\Omega \left ( \mathbf{R} \right ) = c P\left ( \mathbf{R} \right )</math>

where ''c'' is an arbitrary proportionality constant.  Given our distribution function, there is a maxima corresponding to '''R''' = '''0'''.  Physically this amounts to there being more microstates which have an end-to-end vector of 0 than any other microstate.  Now by considering

:<math>S \left ( \mathbf {R} \right ) = k_B \ln \Omega {\left ( \mathbf R \right) } </math>
:<math>\Delta S \left( \mathbf {R} \right ) = S \left( \mathbf {R} \right ) - S \left (0 \right )</math>
:<math>\Delta F = - T \Delta S \left ( \mathbf {R} \right )</math>

where ''F'' is the [[Helmholtz free energy]], and it can be shown that

:<math>\Delta F = k_B T \frac {3R^2}{2Nb^2} = \frac {1}{2} K R^2 \quad ; K = \frac {3 k_B T}{Nb^2}.</math>

which has the same form as the [[potential energy]] of a spring, obeying [[Hooke's law]].

This result is known as the ''entropic spring result'' and amounts to saying that upon stretching a polymer chain you are doing work on the system to drag it away from its (preferred) equilibrium state.  An example of this is a common elastic band, composed of long chain (rubber) polymers.  By stretching the elastic band you are doing work on the system and the band behaves like a conventional spring, except that unlike the case with a metal spring, all of the work done appears immediately as thermal energy, much as in the thermodynamically similar case of compressing an ideal gas in a piston.

It might at first be astonishing that the work done in stretching the polymer chain can be related entirely to the change in entropy of the system as a result of the stretching. However, this is typical of systems that do not store any energy as potential energy, such as ideal gases. That such systems are entirely driven by entropy changes at a given temperature, can be seen whenever it is the case that are allowed to do work on the surroundings (such as when an elastic band does work on the environment by contracting, or an ideal gas does work on the environment by expanding). Because the free energy change in such cases derives entirely from entropy change rather than internal (potential) energy conversion, in both cases the work done can be drawn entirely from thermal energy in the polymer, with 100% efficiency of conversion of thermal energy to work. In both the ideal gas and the polymer, this is made possible by a material entropy increase from contraction that makes up for the loss of entropy from absorption of the thermal energy, and cooling of the material.