We've stressed that our analysis of systems rests upon knowing just a few variables, instead of trying to find out variables affecting individual particles. To this end, we will talk about 6 variables in particular that can be used to determine the energy of a system.
We have already been introduced to the entropy σ and the temperature τ as variables. There are two more variables that are so common in everyday usage that they don't warrant a close look, namely the number N of particles in a system and the volume V of a system. That leaves two more variables to understand before we can dive into the study of systems.
The Chemical Potential
Suppose that we have two systems, each consisting of the same single chemical species, which come into thermal and diffusive contact (meaning that particles can move between them). Note that thermal contact alone prohibits such an exchange. Imagine what happens when you touch a radiator - there is certainly a thermal contact, as you feel the heat of the radiator. However, there isn't much of a diffusive contact, as your hand doesn't suddenly melt into the radiator and become replaced in part by metal!
Now, our chemical intuition tells us that the particles will flow from the denser system to that which is less dense. We will formalize this notion by introducing the chemical potential μ, which governs how particles will flow between two systems. For now, we can think of the chemical potential as follows:
The chemical potential can be defined in different manners as well, and we will address this shortly.
Nevertheless, we can say now that particles will flow from a system with a higher chemical potential to a system with a lower chemical potential if the two are in diffusive and thermal contact.