PHY452H1S Basic Statistical Mechanics. Lecture 11: Statistical and thermodynamic connection. Taught by Prof.\ Arun Paramekanti
Posted by peeterjoot on February 27, 2013
Peeter’s lecture notes from class. May not be entirely coherent.
Connections between statistical and thermodynamic views
- “Heat”. Disorganized energy.
- . This is the thermodynamic entropy introduced by Boltzmann (microscopic).
Let’s isolate the contribution of the Hamiltonian from a single particle and all the rest
so that the number of states in the phase space volume in the phase space region associated with the energy is
With entropy defined by
For and , the exponential can be approximated by
This is the Maxwell distribution.
Non-ideal gas. General classical system
Breaking the system into a subsystem and the reservoir so that with
and for the subsystem
Can we use results for this subvolume, can we use this to infer results for the entire system? Suppose we break the system into a number of smaller subsystems as in fig. 1.2.
We’d have to understand how large the differences between the energy fluctuations of the different subsystems are. We’ve already assumed that we have minimal long range interactions since we’ve treated the subsystem above in isolation. With the average energy is
We define the partition function
Observe that the derivative of is
allowing us to express the average energy compactly in terms of the partition function
Taking second derivatives we find the variance of the energy
We also have
Recalling that the heat capacity was defined by