# How is macroscopic "volume" of a classical system related to microscopic variables?

In my book it says:

In classical ensemble theory, every macroscopic observable of a system is directly connected to a microscopic function of the coordinates and momenta of the system.

I can see that given $$\mathbf q_i$$ and $$\mathbf p_i$$ (coordinates and momenta), we have functions $$P(\mathbf q_i)$$, $$T(\mathbf p_i), E(\mathbf q_i,\mathbf p_i)$$ etc. But what about volume? How do you relate the microscopic set of $$\mathbf q_i$$ and $$\mathbf p_i$$ to macroscopic volume? (For the general case of interacting particles)

In other words, how can we define volume as a function of trajectory?

P.S: ensemble average of these functions should give the macroscopic parameters.

Generally, for an ideal gas the volume of the system is $$\int \Pi_i d^3q_i$$ over the entire phase space.

• So does that mean we can't define volume for a particular point in phase space like we can do for pressure? Also what about the general case of interacting particles? (I edited the question)
– Ali
Apr 18, 2020 at 6:05
• I'm not sure what you mean by defining a volume for a point! Usually, you use the partition function to calculate quantities like the pressure, etc. The partition function for an interacting system also turns out to be a function of the ideal partition function which contains the volume of the gas.
– user171881
Apr 18, 2020 at 18:14

The basic ideia is that in statistical mechanics you will need to establish a link between the micro and the macro, in the microcanonical ensemble the number of microstates acessible is related to entropy by $$S(V,N,U)=K_b\ln\Omega(V,N,U)$$. But to obtain $$\Omega$$ we must integrate over all the states acessible in the phase, and that is the key point for your question, to do this the variables $$(q,p)$$ must respect some constrains, like conservation of energy, and in the case of the volume you must think that the system is occupying some volume, like the harmonic oscilator that oscilate between $$-A,A$$. In general you must think that your system is in some laboratory so it's confined in some volume.

The volume is the most dificult variable to identify in the micro point of view.