General Principles

Consider a set of particles interacting through a 2-body potential, $\Phi(r)$. The equations of motion can be written in the form

$${\d\bi{r}_i\over\d t} = \bi{v}_i$$ (4.20)

$${\d\bi{v}_i\over\d t} = -{1\over m_i}{\partial\over\partial\bi{r}_i} \sum_{j\not=i}\Phi\left(\left\vert\bi{r}_i - \bi{r}_j\right\vert\right).$$ (4.21)

In practice it is better to use the scalar force $F(r) = \partial\Phi/\partial r$ to avoid unnecessary numerical differentiation.

A common feature of such problems is that the time derivative of one variable only involves the other variable as with $\bi{r}$ and $\bi{v}$ in the above equations of motion. In such circumstances a leap-frog like method suggests itself as the most appropriate. Hence we write

$$\bi{r}^{n\phantom{+1}}_i = \bi{r}^{n-2}_i + 2\delta t \bi{v}^{n-1}_i$$ (4.22)

$$\bi{v}^{n+1}_i = \bi{v}^{n-1}_i + {2\delta t\over m} \sum_{j\not=i} F\left(\left\v... ...ht) {\bi{r}^n_i - \bi{r}^n_j\over\left\vert\bi{r}^n_i - \bi{r}^n_j\right\vert}.$$ (4.23)

This method has the advantage of simplicity as well as the merit of being properly conservative.

The temperature is defined from the kinetic energy of N particles via

$${\textstyle\frac32} k_{\rm B} T = {1\over N}\left<\sum_i {\m... ...tstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}m_i \bi{v}_i^2\right>$$ (4.24)

where the averages $<>$ are taken with respect to time.

As an example of another thermodynamic quantity consider the specific heat at constant volume $C_V$. This can be calculated by changing the total energy by multiplying all the velocities by a constant amount ( $\alpha\approx 1$) and running for some time to determine the temperature. Note that, as the temperature is defined in terms of a time average, multiplying all the velocities by $\alpha$ does not necessarily imply a simple change of temperature, $T\mapsto \alpha^2 T$. At a 1st order phase transition, for example, the system might equilibrate to the same temperature as before with the additional energy contributing to the latent heat.

When the specific heat is known for a range of temperatures it becomes possible, at least in principle, to calculate the entropy from the relationship

$$T\left.\partial S\over\partial T\right)_V = \left.\partial U\over\partial T\right)_V$$ (4.25)

The pressure is rather more tricky as it is defined in terms of the free energy, $F=U-TS$, using

$$p = -\left.\partial F\over\partial V\right)_T$$ (4.26)

and hence has a contribution from the entropy as well as the internal energy. Nevertheless methods exist for calculating this.

It is also possible to define modified equations of motion which, rather than conserving energy and volume, conserve temperature or pressure. These are useful for describing isothermal or isobaric processes.

Note that for a set of mutually attractive particles it may not be necessary to constrain the volume but for mutually repulsive particles it certainly is necessary.