# Cell Volume as a function of Alpha

## Model

A simple check that the simulations are working is to see how the cell volume $V_j$ changes as $\alpha$ changes.

We expect that for large values of $\alpha$ energy costs for stretching and compression are high and so the cell will stay near the relaxed volume. Whereas when $\alpha$ decreases the spread in cell volume will get larger.

# Results

All results were done with $\gamma=10.0$ and a simulation space of 1,000 lattice points.

## 1 Cell Chain

The legend indicates the value of $\alpha$ for the respective run.

The relaxed volume size (black bar) was varied as well.

## 3 Cell Chain

The legend indicates the cell in the chain. They are numbered from the reflective boundary out.

The relaxed volume size (black bar) was varied as well.

## Observations

• As alpha increases the distributions become more sharply peaked about the relaxed cell volume.
• As alpha decreases the distributions become wider.
• As alpha gets smaller the symmetry in the distribution is broken by the fact that the volume cannot get smaller than 1.
• Relaxed cell volume does have an effect on the distribution.
• As $V_0$ increases the distribution becomes more spread out.
• This makes sense since changes in $V_j$ will now lead to smaller changes in the quantity $\left(V_j/V_0-1\right)^2$
• And so the energy cost for stretching/compression decreases.
• Making the relaxed volume large gives the benfit of a more symmetric distribution but at the cost of widening that distribution for all values of alpha.

It seems that having a small relaxed volume is more beneficial than a large one. With $V_0=50$ the distributions keep more of their desired shape even for small $\alpha$.

Written on March 31, 2015