Distribution functions in classical monatomic fluids

Consider a system of N particles in a volume V and at temperature T. The probability that each of the N particles is in its volume element dr_N (i.e. 1 in dr₁, 2 in dr₂ etc.), is given by

where Z_N is the configuration integral. The r represent a three dimensional volume element, either dxdxydz or df sinq dq r ²dr. The above probability is valid for all of the particles in their respective volume elements simultaneously. If particle 1 is in dr₁ etc. through n in dr_n while the remaining n+1 through N particles can be anywhere, the probability is

The probability that any particle is in dr₁, …, dr_n irrespective of the configuration of the remaining particles is given by

The distribution function r ⁽¹⁾ indicates the probability that any of the N particles will be found in dr₁. For a solution this probability is a constant since the N particles will be found anywhere if we wait a sufficiently long time. In a fluid, therefore

We can see that this so by substituting into the above equations

We can define a correlation function such that the probabilities for particles being found in n volume elements can be expressed in terms of the probability that a particle is found in a single volume element.

where g⁽ⁿ⁾ is a correlation function. If the interactions between the particles were zero, then g⁽ⁿ⁾ would be equal to one. The pair correlation function g⁽²⁾(r₁,r₂) is particularly important since it can related to the experimental structure factor. In a liquid of spherically symmetric molecules g⁽²⁾(r₁,r₂) depends only on the relative distance, r₁₂ between the molecules. We can express g⁽²⁾(r₁,r₂) as g⁽²⁾(r₁₂) or simply g(r). These definitions are useful since we can use them to give the probability of observing a molecule at r given that we have a molecule at the origin. In other words we can define a coordination number as a function of distance. The function r g(r)dr gives the probability of observing a particle at r given that there is a molecule at the origin. Using the spherical volume element (and the spherical symmetry of the molecule that we have assumed) we can calculate the probability as follows

The integral states that there are N-1 particles outside the central particle we are observing. What is interesting is the form of g(r). The function g(r) can be thought of as a factor that multiplies the bulk density r to give a local density about our central molecule of interest. The function g(r) is called the radial distribution function of the liquid. This function is useful because it can be related to liquid structure and thermodynamic properties.

Relating g(r) to Energy

From the classical definition of the partition function we can write Q_N = Z_N/N!L ^3N .

The energy is

where

The first terms is the mean kinetic energy and the second term is the mean potential energy. If we assume that U can be represented by the pair potential then we have

The total energy is

Relating g(r) to Pressure

The pressure P is given by

where

Before differentiating with Z_N with respect to V, we change the variables of integration in such a way that the limits become constants and U becomes an explicit function of V. The new variables are x₁', y₁', z₁' etc. where

X_K = V^1/3x_K' etc.

Then

U is a pair potential in this integral as well.

The derivative with respect to volume is

where

Having carried out the differentiation with respect to V, we transform back to the original variables.

which, when substituted into the equation above gives

This equation is known as the pressure equation.