**1. INTRODUCTION**

* Statistical Mechanics* originated from the intellectual need of understanding the connection between the molecular interactions in a substance and its macroscopic behaviour. The problem is the following: Knowing the force of attraction/repulsion between *two* molecules of a certain substance as a function of the distance (in general, attractive at long distances, repulsive at short distances), can we describe the full physical behaviour of a large population of them? Can we determine, for example, whether they are a gas, a liquid or a solid at a given temperature and pressure?

The pioneers of the field, * Clausius*, *Maxwell* and * Boltzmann*, concentrated their efforts in understanding the gas phase. This was a magnificent task, considering that the existence of atoms was not yet fully accepted by the scientific community. Their work showed that gases can be nicely explained as low-density ensembles of molecules in constant random motion that hardly interact with each other, except for occasional collisions. From this picture, two important facts were clear:

**Temperature**is a measure of the kinetic energy of the molecules or, in other words, a measure of the (average)*speed*of the molecules.**Pressure**exerted on the walls of any gas container arises from the continuous collisions of the gas molecules with them.

Shortly after, *van der Waals* was able to explain the gas and liquid phases consistently in a unique theoretical frame for the first time. Liquids are described as high-density ensembles of molecules in constant random motion that, unlike gases, interact with each other through collisions and, most importantly, their molecular interaction. However, the distinction with a gas was somehow blurred and, in fact, another important fact was highlighted:

- Gases can be transformed into liquids, or vice versa, in a
*continuous*manner (without any evaporation/condensation) if the physical conditions of temperature and pressure are changed appropriately. By doing so, our system goes through physical states that are neither a gas nor a liquid, as commonly known.

Gases and liquids, grouped under the common name of ** Fluids**, have been studied intensively ever since, notably by means of

*Integral Equation Theory*.

Solids, on the other hand, resisted for longer to be included in a unified theory with the gas and liquid phases. It wasn’t till the arrival of * Density Functional Theory* (DFT) in the 1970’s that the three phases started being studied together. In solids, molecules are ordered following a regular pattern. This happens in two general situations:

- When the temperature is so low that the movement of the molecules is not enough to overcome the intermolecular forces between them.
- When the density is so high that the molecules need to order themselves in order to
*fit*more or less loosely in the volume they occupy.

In other words, the solid phase occurs at low temperatures, and at high pressures.

**2. PERTURBATION THEORIES**

Calculating the physical properties of a system of molecules is far from being an easy task. For real systems, the complexity of the mathematical problem is generally intractable. There are just a few cases in which an exact mathematical solution can be obtained, and, in fact, all of these cases are quite idealized systems. However, these exactly (or quasi-exactly) solvable models are of great help since we can use them to describe the properties of more realistic models by means of a trick called * Perturbation Theory* (PT). This mathematical trick consists in approximating the energy of a “realistic” (unsolvable) model as the energy of an idealized (solvable) model (called the *reference*) plus a “small” contribution (called the *perturbation*). The idea is that the perturbation should account for a particularly important feature of the “realistic” system lacking in the idealized model. Although the results of PT vary enormously depending on how similar both systems are (and range from an excellent quasi-exact solution to the realistic problem to a dreadful catastrophe that bears no resemblance to the system we want to describe), it must be said that, more often than not, the results of PT are physically meaningful. PT has been widely applied in many fields of physics, in particular in the description of fluids and solids.

The key ingredient in PT is a function of special relevance called * Radial Distribution Function * that appears both in Integral equation theory and in Density functional theory, and is usually denoted \(\,g(r)\). This function is important for the following reasons:

- It contains all the (macroscopic) thermodynamic information of the system and also quite a lot of (microscopic) structural information.
- Importantly, \(\,g(r)\,\) is proportional to the (averaged) number of molecules one would find at a certain distance \(\,r\,\) if, instead of staying in a fixed system of reference, was travelling
*on one of the molecules*. - Knowing this function for an “idealized” system, one can obtain an approximation for all the properties of a “realistic” system.

The simplest model for gases is the * Ideal Gas *, where molecules are supposed to be independent point-like particles that move randomly without interacting with each other. This model can be solved exactly and provides an excellent approximation for diluted gases, but using it as system of reference in PT is not a good idea: adding any attractive interaction would make all particles collapse, since there is no repulsion between them. In particular, it is completely unsuitable for solids. The reason for that is simple geometry: molecules in a solid order themselves following a regular pattern because they cannot penetrate each other (they repel each other at very close distances). However, the molecules of an ideal gas do not “see” each other and are, therefore, incapable of any order.

An alternative to the ideal gas is given by the * Hard Sphere * model (HS), where molecules are modelled as spheres that do not interact with each other apart from being impenetrable (like balls of billiards, say). This deceptively simple model is not exactly solvable. However, a lot of effort has been put into understanding it. The reason for this lays in a remarkable feature found in computer simulations of the model in 1957 (see * here *): at high densities, the hard spheres go through a phase transition and become an ordered solid. At the time, this was quite unexpected since the solid was thought to originate from the attractive interactions between the molecules, and the interaction in this model is purely repulsive!

Hard Spheres is one of the favourite systems of reference in perturbation theory. Here we show an schematic plot of the Radial Distribution Function for: an ideal gas, a HS gas, a HS liquid and a HS solid:

Recall that the radial distribution function is proportional to the (averaged) number of molecules one would find at a distance \(\,r\,\) if one was travelling *on one of the molecules*. Note that:

- For the ideal gas, this function is constant. This happens because ideal particles do not “see” each other and can be at any point of the volume with the same probability.
- For the HS model, the function \(\,g(r)\,\) is identically zero from \(\,r=0\,\) to the HS diameter \(\,r=\sigma\,\). This is a consequence of the impenetrability of the spheres: the minimum distance between (the centres of) two HS is their diameter \(\,\sigma\).
- Also, notice that the higher the density (GAS$\to$LIQUID$\to$SOLID), the higher the local order, manifested in the “peaks” of the function \(\,g(r)\).

These functions are used in perturbation theory to describe more realistic models, like molecules interacting with a * Lennard-Jones potential *, for example.

The study of fluids and solids is important for different purposes. One is to predict and describe the occurrence of different phases of matter that are unknown to us (for example, some of the 15 solid types of ice were predicted theoretically before they were found in experiments), or phases that can’t be created in a laboratory (some minerals under the extreme pressures or temperatures of a planet inner core, say). Another purpose is the description of colloidal solutions which consist in colloidal particles (or * colloids*) in solution. These particles can be likened to a gas, a liquid or a solid (where the colloids play the role of the atoms, and the solvent the role of the vacuum). Colloids are everywhere in the food and cosmetic industry nowadays: milk, mayonnaise, ink, paint…