The Boltzmann distribution

Our story so far...

  • We have a formula (complicated) for the thermodynamic probability $w_B(N_1,N_2,N_3,...)/\Omega$ of a quantum system (particles in a box).
  • The values of the occupation numbers $\{N_j\}$ which result in the maximum value for $w_B$ (subject to some constraints) should be the equilibrium state of the system.

Our next mission: To find that equilibrium state!

In classical thermodynamics the equilibrium state is a function of things like volume and temperature. You will not see $T$ and $V$ directly in what follows...

But

  • $U$ (one of our constraints) is a function of temperature.
  • The allowed energy levels depend on $V$.

Finding the maximum value of $w_B$

We were hoping for *some* way to maximize the function which counts the number of microstates: $$w_B(N_1, N_2, N_3...N_n) = N! \prod_{i=1}^n \frac{g_i^{N_i}}{N_i!}.$$ Gotta try varying all $n$ of those occupation number, $N_1, N_2,...$ until we find the combination that maximizes $w_B$...!

It's generally easier to maximize $\ln(w)$, which can be written as $$\ln(w_B) = \ln N! + \sum_i N_i \ln g_i - \sum_i \ln N_i!.[*]$$ [*] It's OK to maximize $\ln(w_B)$ because $\ln(x)$ is a monotonically increasing function of $x$.

That Scottish mathematician (who feared only "assassination on account of having discovered a trade secret of the glassmakers of Venice") swoops in to profer his approximation-- $\ln(n!) \approx n\ln(n) - n$, and so... $$\ln(w_B) = {\rm const} + \sum_i N_i \ln g_i - \sum_i N_i\ln N_i + \sum_i N_i .$$

Does $g_i$ depend on $N_i$?

We've got two constraints to worry about as we search for the maximum value of $\ln w_B$: $$\phi=\sum_i N_i = N; \ \ \eta=\sum_i N_i E_i=U.$$

But Giuseppe has roused himself from his melancholic reflections to remind us to solve a bunch of equations like $\frac{\partial f}{\partial x_i} +\alpha\frac{\partial \phi}{\partial x_i} +\beta\frac{\partial \eta}{\partial x_i} = 0$. Translating into our context, we have to solve $n$ equations, like $$\frac{\partial \ln w_B}{\partial N_j} +\alpha\frac{\partial \phi}{\partial N_j}+\beta\frac{\partial \eta}{\partial N_j} = 0.$$

Substituting in the actual form of our constraint equations... $$\begin{align}\frac{\partial}{\partial N_j}\[ \sum_i N_i\ln g_i - \sum_i N_i\ln N_i + \sum_i N_i \]& \\
+\alpha\frac{\partial }{\partial N_j}\sum_i N_i +\beta\frac{\partial }{\partial N_j} \sum_i N_i E_i & =0.\end{align}$$

That partial derivative fishes out just one term from each sum, the one where $i=j$ $$\begin{align} \ln g_j - \ln N_j - \frac{N_j}{N_j} + 1 & \\ +\alpha 1 +\beta E_j & =0.\end{align}$$

Simplifying a bit: $$ \alpha +\beta E_j = \ln\frac{N_j}{g_j}.$$

Exponentiating both sides...

$$e^{\alpha}e^{\beta E_j} = \frac{N_j}{g_j}\equiv f_j(E_j).$$

This $f_j$ is the number of particles per quantum state at equilibrium in the energy level $j$.

We _still_ don't know what $\alpha$ and $\beta$ are.

Arnold Sommerfeld

German physicist, Arnold Sommerfeld, 1868-1951, was nominated for the Nobel prize 81 times, but never won it.

Sommerfeld (l) with Annie Schrödinger, Peter Debye, 1926

...said

"Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you anymore."

Entropy

Our expression for entropy was: $$\begin{align}\frac{S}{k_B} & =\sigma = \ln(w_B) \\ & = \ln N! + \sum_i N_i \ln g_i - \sum_i N_i\ln N_i + \sum_i N_i .\end{align}$$ Now we will find an expression for the middle two terms in the entropy (above):

The 'fished out' equation--*one* of the $n$ Lagrange equations--was $$\ln g_j - \ln N_j +\alpha +\beta E_j = 0.$$

  • Multiply by $N_j$:
    $$N_j\ln g_j - N_j\ln N_j + \alpha N_j + \beta N_j E_j = 0.$$
  • One expression for each $j$. *All* of them are equal to 0. So, adding a bunch of them still gives 0. $$\sum_j N_j\ln g_j - \sum_j N_j\ln N_j +\alpha \sum_j N_j + \beta \sum_j N_j\ E_j = 0.$$
  • Two of the sums can be simplified:
    $$\sum_j N_j\ln g_j - \sum_j N_j\ln N_j + \alpha N + \beta U = 0.$$ Re-arranging: $$\sum_j N_j\ln g_j - \sum_j N_j\ln N_j = - \alpha N - \beta U$$

Substituting this last expression into the entropy, we get $$\sigma = \ln(w_B) = \ln N! -\alpha N -\beta U + N .$$

Classically, we had: $$T\,dS = dU + P\,dV \Rightarrow dS = \frac{1}{T}dU +\frac{P}{T}dV.$$

If we take $k_B \sigma= k_B \ln w_B = S=S(U,V)$, then this implies $$\begin{align} \frac{1}{T} & = \( \frac{\partial S}{\partial U} \)_V = \( \frac{\partial k_B\ln w_B}{\partial U} \)_V \\ & = - k_B \beta .\end{align}$$

Where does $V$ come into things? $N$ is not a function of $V$. That leaves $g_j=g_j(E_j)$ and $E_j$ itself. In the case of the particle in the box, $$E_j = \frac{\pi^2\hbar^2}{2mV^{2/3}}n_j^2.$$

So $E_j$ is a function of $V$, and for partial derivatives, that means holding $V$ constant is the same as holding the energy levels $\{ E_j \}$ constant when taking partial derivatives. So, all of the $U$ and $V$ dependence of the entropy is concentrated in $U(V,N)$.

 

With $\beta = -1/(kT)$, from the energy level occupancy expression: $$N_j = g_j e^{\alpha}e^{-E_j/(kT)}.$$

The sum over all $j$... $$N = \sum_j^n N_j = e^{\alpha} \sum_j g_j e^{-E_j/(kT)}.$$

So, $e^{\alpha}$ can be written as: $$e^{\alpha} = \frac{N}{\sum_j g_j e^{-E_j/(kT)}}.$$

Now, at last, we can write down that expression for the probability of occupation of a single state with one Lagrange multiplier (the temperature) in it as:

$$f_j(E_j) = \frac{N_j}{g_j}= e^{\alpha}e^{\beta E_j} =\frac{N e^{-E_j/kT}}{\sum_j g_j e^{-E_j/kT}}.$$

This is the Boltzmann distribution for distinguishable particles: The expected number of particles per quantum state for energy level $j$.

Carter describes it as a "probability" of occupation for any quantum state in level $j$, which is a bit jarring at first because this number can be more than 1. But it's possible to have more than one particle in any given quantum state.

Solving this for $N_j = f_j g_j$, we find: $$N_j= \frac{N}{Z}g_je^{-E_j/kT}.$$

The partition function

So, our equilibrium state is characterized by: $$f_j(E_j) = \frac{N e^{-E_j/kT}}{\sum_j g_j e^{-E_j/kT}} \equiv \frac{N e^{-E_j/kT}}{Z} $$ where the denominator, $Z$, is the partition function ("Zustandssumme" = sum of states): $$Z \equiv \sum_j g_j e^{-E_j/kT}.$$

It turns out that *all* of the thermodynamic properties of a dilute gas can be expressed in terms of the logarithm of the partition function and its derivatives.

For example, the Gibbs free energy turns out to be: $$G=\mu N = NkT\ln\frac{N}{Z}.$$

Let's verify just one: The expression for the internal energy $U$ which is: $$U=NkT^2\(\frac{\partial \ln Z}{\partial T}\)_V = \frac{N}{Z}kT^2 \(\frac{\partial Z}{\partial T}\)_V.$$

Since the energies $E_j=E_j(V)$, keeping V constant amounts to treating the energies as constant when taking derivatives. The partition function was: $$Z= \sum_j g_je^{-E_j/kT}.$$

The derivative of the partition function is... $$\begin{align}\(\frac{\partial Z}{\partial T}\)_V &=-\sum_j g_j E_j e^{-E_j/kT} \frac{d}{dT}\(\frac{1}{kT}\) = \\
& = \frac{1}{kT^2}\sum_j g_jE_j e^{-E_j/kT}.\end{align}$$

Subbing into the expression for $U$: $$U = \frac{N}{Z}\sum_j g_jE_j e^{-E_j/kT}.$$

Since $N_j = f_j g_j =\frac{N}{Z}g_je^{-E_j/kT}$,

$$U = \sum_j N_j E_j.$$ Which should be recognizable as the internal energy!

And....

This can also be considered to be an equation of constraint in the sense of Lagrange multipliers.

Fermions / Bosons / Dilute gases

Important modifications for our scheme: quantum particles are empirically indistinguishable, so we need to modify our counting rules.

Fermions - particles with spin $s=1/2 + n$.

Most prominent example: electrons (spin 1/2).

The Pauli exclusion principle says, that taking into account spin,

No two fermions can be in the same state. $$\Rightarrow f(E)=\frac{1}{e^{(E-\mu)/kT}+1}.$$

Fermi-Dirac distribution.

Bosons - particles with spin $s=n$.

Any number of bosons can be in a particular quantum state. $$\Rightarrow f(E)=\frac{1}{e^{(E-\mu)/kT}-1}.$$

Bose-Enstein distribution.

Dilute gas - distinguishable particles, but assuming occupation numbers are typically small, $N_j << g_j$. In this limit, it shouldn't matter whether a particle is a fermion, or a boson, or a classical, distinguishable particle. $$\Rightarrow f_j=\frac{N_j}{g_j}=\frac{Ne^{E_j/kT}}{Z}.$$

The Maxwell-Boltzmann distribution.

Image credits

Wales Museum, Finkelstein, et. al