Statistical Mechanics: the Partition Function as a Sum Over States

For the reader from (mathematics).

Read $\beta$ as a formal variable and $Z(\beta)=\sum_{i}\mathrm{e}^{-\beta E_{i}}$ as a generating function: a Laplace/Dirichlet-type transform of the energy spectrum. Every thermodynamic quantity is a logarithmic derivative of it. The only “physics” input is the principle that selects the weights $\mathrm{e}^{-\beta E_{i}}$ , which we derive rather than postulate.

The partition function is born as a normalizing constant. What makes it the central object of the theory is that, once you have it, you have everything: the entire thermodynamics of a system is encoded in the single function $\log Z$ .

1 Where the Boltzmann weight comes from

Fix a system with microstates $i$ of energy $E_{i}$ . We do not know which microstate the system occupies; we know only its mean energy. Among all probability distributions $\{p_{i}\}$ with $\sum_{i}p_{i}=1$ and $\sum_{i}p_{i}E_{i}=\langle E\rangle$ fixed, which should we choose? The principle of maximum entropy says: the least-committal one, maximizing the Gibbs–Shannon entropy $S=-\sum_{i}p_{i}\log p_{i}$ . Lagrange multipliers $\alpha,\beta$ enforce the two constraints; we extremize

-\sum_{i}p_{i}\log p_{i}-\alpha\Big{(}\sum_{i}p_{i}-1\Big{)}-\beta\Big{(}\sum_% {i}p_{i}E_{i}-\langle E\rangle\Big{)}

(1)

over each $p_{i}$ in turn. This gives one first-order condition per state $i$ -an ordinary unconstrained calculus problem, one equation per unknown-namely $\log p_{i}=-1-\alpha-\beta E_{i}$ , that is

\boxed{\,p_{i}=\frac{\mathrm{e}^{-\beta E_{i}}}{Z},\qquad Z=\sum_{i}\mathrm{e}% ^{-\beta E_{i}}.\,}

(2)

The multiplier $\beta$ enforcing the energy constraint is the inverse temperature, $\beta=1/(k_{B}T)$ , as we will see shortly. The normalizer $Z$ is the partition function: it is what partitions unit probability among the states. Nothing was postulated about exponentials; the exponential is forced by maximizing entropy at fixed mean energy. We return to this point, hard, in the Interlude.

2 Everything is a derivative of $\log Z$

Because $Z$ packages the whole spectrum, its logarithm generates the thermodynamics:

	$\displaystyle\langle E\rangle$	$\displaystyle=\frac{\sum_{i}E_{i}\mathrm{e}^{-\beta E_{i}}}{Z}=-\frac{\partial% \log Z}{\partial\beta},$		(3)
	$\displaystyle\operatorname{Var}(E)$	$\displaystyle=\langle E^{2}\rangle-\langle E\rangle^{2}=\frac{\partial^{2}\log Z% }{\partial\beta^{2}},$		(4)

so $\log Z$ is literally the cumulant generating function of the energy.

Where does the identification $\beta=1/(k_{B}T)$ from §1.1 actually come from? Evaluate the informational entropy on the maximizing distribution itself: since $\log p_{i}=-\beta E_{i}-\log Z$ ,

S=-\sum_{i}p_{i}\log p_{i}=\beta\langle E\rangle+\log Z.

(5)

Multiply by $k_{B}$ to give $S$ thermodynamic units and differentiate with respect to $\langle E\rangle$ at fixed volume and particle number, using $\partial\log Z/\partial\beta=-\langle E\rangle$ from above-the $\partial\beta/\partial\langle E\rangle$ terms cancel exactly, leaving

k_{B}\frac{\partial S}{\partial\langle E\rangle}=k_{B}\beta.

(6)

Classical thermodynamics defines temperature by $\partial S/\partial E=1/T$ ; comparing the two fixes $\beta=1/(k_{B}T)$ exactly. The Lagrange multiplier of §1.1 and the thermodynamic inverse temperature are the same object because both are, at bottom, $\partial S/\partial E$ .

The free energy is $F=-\beta^{-1}\log Z$ , the entropy $S=-\partial F/\partial T$ , the heat capacity $C=\partial\langle E\rangle/\partial T$ , and so on. One function, differentiated, yields the lot.

For the reader from (physics).

If you already know all this, the only new emphasis is structural: keep your eye on (2) as a map from a spectrum to a number, and on the multiplicativity of §4. Those two features-and not the thermodynamics-are what survive into quantum statistics, field theory, and representation theory.

3 Two elementary systems: the boson and the fermion appear

Example 1.1 (Two-level system).

A single degree of freedom with energies $\{0,\epsilon\}$ has $Z=1+\mathrm{e}^{-\beta\epsilon}$ . Its heat capacity peaks at $k_{B}T\sim\epsilon$ (the Schottky anomaly)¹¹ 1 Explicitly, $C(T)=k_{B}\left(\frac{\epsilon}{k_{B}T}\right)^{2}\frac{\mathrm{e}^{-\epsilon/% k_{B}T}}{(1+\mathrm{e}^{-\epsilon/k_{B}T})^{2}}$ -a bump vanishing both as $T\to 0$ (frozen out) and $T\to\infty$ (equal population, no distinguishing power), unlike the monotonic heat capacity of a classical system. Named for Walter Schottky (1922); such bumps have since been used to diagnose gapped low-lying levels-impurity spins, nuclear hyperfine levels, tunneling two-level systems in glasses-directly from bulk specific-heat data.: the system can only “notice” temperature once $k_{B}T$ is comparable to its gap.

Example 1.2 (A single harmonic oscillator - the boson).

With $E_{n}=\hbar\omega(n+\tfrac{1}{2})$ , $n=0,1,2,\dots$ , the geometric series sums to

Z_{\mathrm{b}}=\sum_{n\geq 0}\mathrm{e}^{-\beta\hbar\omega(n+1/2)}=\frac{% \mathrm{e}^{-\beta\hbar\omega/2}}{1-\mathrm{e}^{-\beta\hbar\omega}}=\frac{1}{2% \sinh(\beta\hbar\omega/2)}.

(7)

A bosonic mode admits any occupation number $n\geq 0$ ; that is why we get a full geometric series.²² 2 Readers who know the Feynman path integral will recognize (7) lurking inside the harmonic-oscillator propagator, $K(x_{b},t;x_{a},0)\propto(\sin\omega t)^{-1/2}\exp(\mathrm{i}\cdots)$ : continue to imaginary time, $t\to-\mathrm{i}\beta\hbar$ (so $\sin\omega t\to-\mathrm{i}\sinh\beta\hbar\omega$ ), set $x_{b}=x_{a}$ , and integrate over $x_{a}$ -that is, trace over closed paths-and the Gaussian integral yields exactly $1/(2\sinh(\beta\hbar\omega/2))$ . No accident: $\mathrm{e}^{-\beta H}$ is the time-evolution operator at imaginary time, so the thermal $Z$ is the Feynman amplitude to return, summed over starting points. Lecture 3 is this footnote, industrialized.

Thread (The $-c/24$ thread starts here).

The factor $\mathrm{e}^{-\beta\hbar\omega/2}$ in (7) is the zero-point energy $\tfrac{1}{2}\hbar\omega$ of the oscillator. It looks like a harmless prefactor now. Summed over infinitely many modes it will diverge and get regularized in L3 via $\sum n\overset{\zeta}{=}-\tfrac{1}{12}$ (the energy itself carrying the extra $\tfrac{1}{2}$ : $-\tfrac{1}{24}$ per unit of mode spacing), become the $q^{1/24}$ of the Dedekind eta in L4, and end as the $-c/24$ grading shift in L5.

Example 1.3 (A single fermionic mode - the fermion).

With occupation $n\in\{0,1\}$ (Pauli exclusion) and energy $E=\epsilon n$ ,

Z_{\mathrm{f}}=1+\mathrm{e}^{-\beta\epsilon}.

(8)

The contrast with (7) is the whole of quantum statistics in miniature: bosons sum a geometric series over all $n$ , fermions truncate at two terms.

4 Independence becomes multiplication

If a system splits into non-interacting parts with additive energy $E_{(i,j)}=E^{A}_{i}+E^{B}_{j}$ , then

Z=\sum_{i,j}\mathrm{e}^{-\beta(E^{A}_{i}+E^{B}_{j})}=\Big{(}\sum_{i}\mathrm{e}% ^{-\beta E^{A}_{i}}\Big{)}\Big{(}\sum_{j}\mathrm{e}^{-\beta E^{B}_{j}}\Big{)}=% Z_{A}\,Z_{B},

(9)

and more generally $Z=\prod_{k}Z_{k}$ , $\log Z=\sum_{k}\log Z_{k}$ . This innocuous identity is the seed of every product formula in these notes.

Thread (The Boltzmann factor as a homomorphism).

Look closely at (9). The single fact that powers it is

\mathrm{e}^{-\beta(E_{1}+E_{2})}=\mathrm{e}^{-\beta E_{1}}\,\mathrm{e}^{-\beta E% _{2}}.

(10)

The map $E\mapsto\mathrm{e}^{-\beta E}$ sends addition of energies to multiplication of weights. That is the defining property of a one-dimensional character of the additive group of energies-and it is why independent systems multiply, why $Z$ factorizes over modes, and (eventually) why $\eta$ and $\theta$ are infinite products. We will also soon want to weight states by a second additive quantity, a conserved charge $N$ , via $\mathrm{e}^{-\beta(E-\mu N)}$ : the new knob $\mu$ is the chemical potential, developed in Lecture 2.

For the curious (the symplectic route to $\beta=1/(k_{B}T)$ ).

The identification $\beta=1/(k_{B}T)$ above leaned on classical thermodynamics as an external input. There is a purely geometric route that never invokes it. Phase space carries a Liouville volume, preserved by Hamiltonian flow; let $\Omega(E)=\operatorname{Vol}\{H\leq E\}$ be the enclosed volume below energy $E$ -a symplectic quantity, nothing thermal about it yet. Define the microcanonical entropy $S(E)=\log\Omega(E)$ and define $\beta:=dS/dE$ : the logarithmic growth rate of a volume, pure calculus so far. This $\beta$ agrees with the Lagrange multiplier of §1.1: for large systems $Z(\beta)=\int dE\,\Omega^{\prime}(E)\,\mathrm{e}^{-\beta E}$ is dominated by the energy $E^{\ast}$ where $\log\Omega^{\prime}(E)-\beta E$ is stationary, i.e. $\beta=d\log\Omega^{\prime}/dE|_{E^{\ast}}$ ; and since the derivatives of $\log\Omega$ and $\log\Omega^{\prime}$ agree up to subextensive corrections, Laplace’s method makes the two definitions coincide in the thermodynamic limit. The one physical, not purely geometric, ingredient left is the postulate that equilibrium maximizes accessible phase volume: two weakly coupled systems distribute their energy so as to maximize $\Omega^{\prime}_{A}(E_{A})\,\Omega^{\prime}_{B}(E_{\mathrm{tot}}-E_{A})$ , which forces equal $\beta$ at the maximum-the statement that underwrites the zeroth law, with lighter machinery than the full Clausius–Kelvin apparatus. The same Liouville volume returns as the path-integral measure in Lecture 3.

Rosetta notes

•

$\beta\leftrightarrow$ inverse temperature, soon to be reinterpreted as an imaginary time and then as $\operatorname{Im}\tau$ .
•

$\log Z\leftrightarrow$ the free energy, the generating function of energy cumulants.
•

$Z=\prod_{k}Z_{k}\leftrightarrow$ the product formulas $\prod(1-q^{n})^{-1}$ and $\prod(1+q^{n})$ that become $\eta$ and $\theta$ .
•

The sum $\sum_{i}$ over a labelled basis is about to become a basis-independent trace.