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Boltzmann Distribution
Adrian Down
September 16, 2005
1 Chapter 2 summary
1.1 Concepts
- The Fundamental assumption is that a system is equally likely to be in any accessible quantum state.
- The ergotic hypothesis says that the time average of a quantity is equiv- alent to the ensemble average.
- The ensemble average is given by
〈x〉 =
s
x(s)p(s)
- The multiplicity function of a combined system is given by
g(N, U ) =
U <U 1
g 1 (U 1 )g 2 (U − U 1 )
This sum is dominated by a single term, the most probable configura- tion.
σ = ln g
- Temperature is defined as 1 τ
∂σ ∂U
|〈U 〉
- The Laws of Thermodynamics show the connection between our sta- tistical formulation and classical thermodynamics.
Zeroth If two systems are in thermal equilibrium with a third system, then the two are in equilibrium with each other.
τ 1 = τ 3 and τ 2 = τ 3 ⇒ τ 1 = τ 2
first Heat (not covered yet) second Entropy can never decrease in a physical process, σf ≥ σi third The entropy of a system approaches a constant value as temper- ature of the system approaches 0
1.2 Main ideas
- Formulate the problem in terms of probability to find the multiplicity function
- Calculate the entropy, which usually involves logarithms of factorials. Use the Sterling approximation for large factorials.
- Use the entropy to determine other properties of the system.
2 Chapter overview
Reservoirs The reservoir is a very large system. It provides a temperature, but its other properties are unimportant.
Boltzmann factor The Boltzmann factor relates the relative probability for the system to be in states of different energy.
Partition function The Partition function is the sum of the Boltzmann factors for all states of the system. It is one of the most important ideas in statistical mechanics and relates macroscopic quantities to our microscopic formulation. We will do an example of a paramagnetic system.
Definition. The exponential of the negative of the energy divided by the temperature is called the Boltzmann factor.
The Boltzmann factor is important for finding the relative probability of a system to be in two different states. As the energy of the microstate increases, the multiplicity function of the reservoir decreases, meaning that we are less likely to find the small system in a state of higher temperature.
4 Partition function
We know that
∑
l
Pl = 1 =
l
Ae−^
l τ
⇒ A =
l e
− τl
Definition.
Z =
l
e−^
l τ
is called the partition function.
Z tells us how the probability is partitioned among all the possible mi- crostates of our system depending in their individual energies.
4.1 Importance of Z
We can use our microscopic model to get the microstate energies for our sys- tem. These energies can be used to calculate the partition function. Once we have this function, we will see in this chapter how to go from Z to ther- modynamic properties. Z is the bridge between thee microscopic model and the real world.
4.2 Degenerate energy levels
Suppose you specify energy with n instead of l, where n is the energy value instead of l, which specifies the state with a particular set of quantum num-
bers. Then
Z =
n
gn(n)e−^
nτ
Note. This is a discreet version of a LaPlace transform.
4.3 Z and thermodynamic properties
4.3.1 Average energy
Average energy is often our first quantity of interest when dealing with a system.
〈〉 = U
〈〉 =
l
lPl
We know the probability of being in the lth^ microstate,
〈〉 =
Z
l
le−^
l τ
We introduce a quantity to simplify the calculations,
β =
τ
then
〈〉 =
Z
l
le−βl
Here we use a trick. Note that ∂ ∂β
e−βl^ = −le−βl
We use this to get
〈〉 = −
Z
∂β
l
e−βl^ = −
Z
∂Z
∂β ∂Z ∂τ
∂Z
∂β
∂β ∂τ
τ 2
∂Z
∂β
〈〉 =
τ 2 Z
∂Z
∂τ
