Statistical Mechanics is the discipline that, with just the knowledge of the basic interaction between (usually) two molecules, is capable of describing the macroscopic behaviour of an enormous number of molecules interacting with each other. In the general theory, one assumes that the interacting molecules are all identical or that there is a simple mixture of molecules (2 or 3 types, maximum). Of course, this is a perfectly valid assumption for many molecular systems, since molecules are indistinguishable.
However, there are some systems where molecules are far from being identical. Probably, the most common example is polymers , the molecules that make up rubber, cellulose and most of plastics. Polymers are long chains of smaller “links” called monomers , whose length varies considerably: from a few monomers to hundreds of thousands.
Another system of study with unequal particles are colloidal particles or colloids : large molecular aggregates of nanoscopic size (1 to 1000 nanometers). When colloids are dispersed in a solvent, they interact with each other (very much like gas or liquid molecules interact with each other) and adopt configurations which 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). However, colloids are all slightly different from each other, and standard techniques of statistical mechanics that consider identical particles fail to explain the rich behaviour of this systems.
Systems with non-identical interacting particles, like polymers and colloids, are called polydisperse. Those with identical particles, like molecular gases and fluids, are called monodisperse. Polydispersity originates because particles have different size, different shape, different electric charge, different chemical composition or a combination of them.
There are statistical mechanics tools to study polydisperse systems, but the theory is considerably more complex. To start, theses systems need more extra information to be characterized: the number of particles of each type (size, charge… ). This is given by a function called the distribution function.
Notice that, if a polydisperse system undergoes a phase transition, the emerging phase does not have, in general, the same distribution function as the original system. For example, if a polydisperse liquid evaporates, the gas will in general has a distribution function different from that of the liquid. Suppose, for example that smaller particles are more likely to evaporate than bigger ones. In turn, as gas evaporates, the distribution function of the liquid also changes. All this phenomenology is absent in the monodisperse systems.
We studied theoretically the properties of a polydisperse system close to its Critical Point (the point where the gas-liquid coexistence line ends). Comparison was made with the monodisperse case. Among other differences, we pointed out that, while the monodisperse case needs only 2 parameters to characterize the properties of the system close to the critical point, the polydisperse case needs none less than 7.