The Partition Function
Any good undergrad statistical mechanics class analyzes the partition function. However, Russ Hobbie and I don’t introduce the partition function in Intermediate Physics for Medicine and Biology. Why not? It
Russ and I do discuss the Boltzmann factor. Suppose you have a system that is in thermal equilibrium with a reservoir at absolute temperature T. Furthermore, suppose your system has discrete energy levels with energy Ei, where i is just an integer labeling the levels. The probability Pi of the system being in level i, is proportional to the Boltzmann factor, exp(–Ei/kBT),
Pi = C exp(–Ei/kBT),
where exp is the exponential function, kB is the Boltzmann constant and C is a constant of proportionality. How do you find C? Any probability must be normalized: the sum of the probabilities must equal one,
Σ Pi = 1 ,
where Σ indicates a summation over all values of i. This means
Σ C exp(–Ei/kBT) = 1,
or
C = 1/[Σ exp(–Ei/kBT)] .
The sum in the denominator is the partition function, and is usually given the symbol Z,
Z = Σ exp(–Ei/kBT) .
In terms of the partition function, the probability of being in state Pi is simply
Pi = (1/Z) exp(–Ei/kBT) .
An Introduction to Thermal Physics, by Daniel Schroeder. |
Here is what Daniel Schroeder writes in his excellent textbook An Introduction to Thermal Physics,
The quantity Z is called the partition function, and turns out to be far more useful than I would have suspected. It is a “constant” in that it does not depend on any particular state s [he uses “s” rather than “i” to count states], but it does depend on temperature. To interpret it further, suppose once again that the ground state has energy zero. Then the Boltzmann factor for the ground state is 1, and the rest of the Boltzmann factors are less than 1, by a little or a lot, in proportion to the probabilities of the associated states. Thus, the partition function essentially counts how many states are accessible to the system, weighting each one in proportion to its probability.
To see why it’s so useful, let’s define β as 1/kBT. The Boltzmann factor is then
exp(–βEi)
and the partition function is
Z = Σ exp(–βEi) .
The average energy, E>
The denominator is just Z. The numerator can be written as the negative of the derivative of Z with respect to
I won’t go on, but there are other quantities that are similarly related to the partition function. It
Is the partition function hidden in IPMB? You might recognize it in Eq. 3.37, which determines the average kinetic energy of a particle at temperature T (the equipartition of energy theorem) It looks a little different, because there
Why didn
Source: http://hobbieroth.blogspot.com/2023/06/the-partition-function.html
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