We start by splitting the ways we categorize a system into two categories: extensive and intensive.

Before going into detail about this, however, a little explanation of the notation about studying Thermal properties.

Typically, we're going to deal with the following variables (although theoretically, the structure of Thermal Physics allows other variables to be considered, like the Magnetic Field (B), for example): Internal Energy (U), Entropy (S), Volume (V), Number of Particles (N), Temperature (T), Pressure (P), and Chemical Potential (μ). Now, as according to our goal, given a certain set of initial conditions, there should be an equilibrium that tells us all the information of the system. Mathematically, that means we will have a set of functions to explain our system, most generally and typically this is:

U = U(S,V,N)

Or we can invert to obtain:

V = V(U,S,N)

Or we might be able to find through experiment:

T = T(S,P,N)

Maybe we have multiple particles:

U = U(S,V,N_{1},N_{2}

And so on...

Now, as mentioned before it is useful to categorize these variables.

An *extensive* parameter, like Internal Energy (U), Entropy (S), Volume (V), or Number of Particles (N), are parameters that 'scale with the system'. In other words U(aS,aV,aN)=aU(S,V,N).

An *intensive* parameter, like temperature (T), pressure (P), or chemical potential (μ), are parameters that DON'T 'scale with the system'. In other words, T(aS,aV,aN)=T(S,V,N).

The other reason it's useful to define extensive/intensive parameters is because it tells us what the system is dependent on. Typically, any parameter will only at most be a function of all intensive parameters, or all other extensive parameters.