## PHY452H1S Basic Statistical Mechanics. Lecture 11: Statistical and thermodynamic connection. Taught by Prof.\ Arun Paramekanti

Posted by peeterjoot on February 27, 2013

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# Disclaimer

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).

# Ideal gas

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

we have

so that

For and , the exponential can be approximated by

so that

or

This is the Maxwell distribution.

# Non-ideal gas. General classical system

Breaking the system into a subsystem and the reservoir so that with

we have

and for the subsystem

# Canonical ensemble

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

we have

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