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I have recently started studying Statistical Mechanics, and through my study of Classical Statistical Mechanics, I have studied how do we write distribution functions for equilibrium systems which can be isolated(micro canonical ensemble) or exchange only energy (canonical ensemble) or both matter and energy (grand canonical ensemble) with the surrounding. However, we never deal with processes like we did in thermodynamics i.e. isobaric, isochoric, isothermal, adiabatic etc, in which we move our system from one equilibrium to another equilibrium through a series of quasi-static equilibriums and calculate change in thermodynamic variables like $P$, $T$, $V$, $S$ etc.

How do we deal with this in the light of statistical mechanics? I mean how does the distribution function change with time so that we can calculate the changes in thermodynamic variables at the end of the processes?

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    $\begingroup$ In thermodynamics we only could only extract information about the initial and final states of a process, but in stat mech we have a microscopic model that allows us to calculate the properties of the final state from first principles, so why bother with the abstract thermodynamic process? (and if we really do want to calculate the final state from the initial state we can always use thermodynamics) $\endgroup$ – By Symmetry Apr 20 at 9:14
  • $\begingroup$ @BySymmetry So what is the exact procedure that lets us calculate the properties of the final state from first principles? $\endgroup$ – Naman Agarwal Apr 20 at 9:36
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Like the comment says, once you have calculated the partition function of the system, you essentially have all the information of the system. What remains is to extract the information you need.

For example, for an ideal gas, the partition function (in the canonical ensemble) is: $ z = \frac{V^N (2\pi mkT)^\frac{3N}{2}}{h^{3N}N!}$. You can then use the Helmholtz free energy, $A = -kTlnz$, to calculate the thermodynamic quantities like pressure and entropy.

$P = -\frac{\partial A}{\partial V} \implies PV = NkT$

So, if you know what the volume and temperature in the final state, you can calculate what the pressure will be. You can calculate similar expressions for all thermodynamics quantities from the partition function.

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