Classical Fluids

In order to perform statistical mechanical calculations for classical fluids, we must be able to compute phase space averages,

$$\langle A \rangle = \sum_{\xi} A(\xi) \frac{e^{-\beta E(\xi)}}{Z},$$ (376)

where $A(\xi)$ is the observable of interest evaluated in state $\xi$. In a classical system of particles, the index $\xi$ labels the microscopic phase point

$$\xi = (r_1, r_2, ..., r_n, p_1, p_2, ..., p_n) = (r^N, p^n),$$ (377)

i.e., the list of coordinates and momenta $(r^N, p^n)$. Therefore, the ensemble average is

$$\langle A \rangle = \int dr^N \int d p^N f(r^N, p^N) A(r^N, p^N),$$ (378)

where

$$f(r^N, p^N) = \frac{e^{-\beta E(r^N, p^N)}}{\int dr^N \int d p^N e^{-\beta E(r^N, p^N)}},$$ (379)

is the probability density at phase-point $(r^N,p^N)$, where the system has total energy

$$E(r^N, p^N)= V(r^N) + K(p^N),$$ (380)

with total kinetic energy

$$K(p^N) = \sum_{j=1}^N \frac{p_j^2}{2 m_j},$$ (381)

and total potential energy of interaction between particles $V(r^N)$.

The factorization of the total energy into kinetic and potential energy terms, allows one to factorize the probability distribution function as follows,

$$f(r^N, \rho^N) =\Phi(r^N) P(\rho^N),$$ (382)

where

$$P(p^N) = exp (-\beta K(p^N)) /\int d p^N exp(-\beta K(p^N)),$$ (383)

and

$$\Phi(r^N) = exp (-\beta V(r^N)) /\int d r^N exp(-\beta V(r^N)).$$ (384)

Further factorization of the momentum distribution is possible, since the total kinetic energy of the system $K(p^N)$ is the sum of the kinetic energy of individual particles

$$\Phi(p^N) = \prod_{j=1}^N \phi(p_j), \hspace{.3cm} \phi(p_j)= \frac{e^{-\beta p_j^2 /2 m_j}}{\int d p_j e^{-\beta p_j^2 /2 m_j}}.$$ (385)

Eqs. (384) and (385) thus define the probability distribution functions that are necessary to sample an ensemble of configurations of the fluid at thermal equilibrium.

Contrary to clusters of atoms or molecules, fluids with bulk properties usually involve an enormous number of particles $(i.e., \sim O(10^{23})$ particles where the fraction of particles near the walls of the container is negligibly small. However, computer memory and disk space requirements usually limits the capabilities of computer simulations to model sytems with less than $10^6---10^7$ particles. Contrary to macroscopic systems, these model systems have a significant fraction of particles near the walls of the container. Therefore, in order to simulate systems with bulk properties, it is necessary to implement a trick to minimize surface effects. The most common trick involves the implementation of periodic boundary conditions (p.b.c.).

The implementation of p.b.c. for a one-dimensional "box" of length L converts the segment L into a circle of radius r, where $2 \pi r = L$. The distance between any two particles is measured along the arc of such circle and hence the maximum separation between any two particles is L/2. One simple way of implementing p.b.c. for particles in a 1-dimensional box of length L is to add(subtract) L to the coordinate of any particle that crosses the left(right) boundary of the box after being slightly displaced to a new position. This can be implemented with an if-statement as follows:

$$\begin{array}{l}\text{if} (\text{x} > \text{L}) \\\hspace{......text{x} = \text{x} + \text{L} \\\text{end if} \\\end{array}$$ (386)

To compute the minimum distance dx between particles p1 and p2 at x1 and x2, respectively, we can write

$$\begin{array}{l}\text{dx} = \text{x1} - \text{x2} \\\text{......xt{dx} = \text{dx} + \text{L} \\\text{end if} \\\end{array}$$ (387)

Note that when programing Eqs. (386) and (387) you will need to implement the logic characters and the punctuation marks required by the specific programing language of choice. The generalization of Eqs. (386) and (387) to the implementation of p.b.c. in two and three dimensions is straightforward.