Energy Fluctuations in the Canonical Ensemble

Canonical Ensemble: First, a Quick Review The Probability that the system is in quantum state r at temperature T is: Ur  εr = energy of state r.  “Partition Function”

Entropy in The Canonical Ensemble This general Definition of Entropy, in combination with The Canonical Distribution allows the calculation of all of the system thermodynamic properties:

Helmholtz Free Energy Internal Energy: Note: This means that Z can be written: Z  exp[-F/(kT)] Internal Energy: Ū  Ē  Average Energy of the system Helmholtz Free Energy, F. F = Ū - TS  Average Helmholtz Free Energy The Partition Function Z acts as a “Bridge” linking microscopic physics (quantum states) to the energy & so to all macroscopic properties of a system.

How large are the fluctuations? Mean Internal Energy Ū  Thermal Average of the system Internal Energy. The actual internal energy fluctuates due to the system interacting with the heat bath. How large are the fluctuations? Are they important?

Fluctuations in Internal Energy A measure of the departure from the mean is the standard deviation, as in any statistical theory. Some detailed manipulation shows that

The Variance The relative fluctuation in energy (U/Ū) gives the most useful information. More manipulation shows that:

N  Number of Particles in the system. The relative fluctuation in energy (U/Ū): Ū & CV are extensive properties proportional to the size of the system. So, they are both proportional to N  Number of Particles in the system. This implies that

For Macroscopic Systems with ~1024 particles, the relative fluctuations are So, the fluctuations about Ū are very tiny, which means that U & Ū can be considered identical for practical purposes.

So, U & Ū can be considered identical for practical purposes. Based on this, it is clear that Macroscopic Systems interacting with a heat bath effectively have their energy determined by that interaction. Similar relationships also hold for relative fluctuations of other macroscopic properties.