9 Statistical physics

9.1 Foundations: From Microstates to Thermodynamics

Statistical physics is the art of predicting macroscopic laws from microscopic uncertainty. We never know the exact $10^{23}$-body state; instead we organize ignorance into ensembles whose averages reproduce temperature, pressure, magnetization, and friends. This section builds the core: microstates vs macrostates, the fundamental postulate, the three standard ensembles, and the thermodynamic limit. We also show how the information-theoretic view turns “maximum entropy” into the unique engine behind the familiar distributions.

9.1.1 Microstates, macrostates, and phase space

A microstate is a complete specification of the system’s degrees of freedom. For $N$ classical point particles, it is a point in $6N$-dimensional phase space with coordinates

A macrostate is defined by a few bulk variables, e.g., $(E,V,N)$ for an isolated gas. Many microstates map to the same macrostate. Counting how many leads to entropy.

Liouville’s theorem states that Hamiltonian flow is incompressible in phase space, so volume elements are preserved along trajectories. This makes uniform distributions over energy shells dynamically consistent.

9.1.2 Fundamental postulate and Boltzmann’s principle

Fundamental postulate. For an isolated system with fixed $(E,V,N)$ at equilibrium, all microstates compatible with these constraints are equally likely.

Let $\Omega(E,V,N)$ be the number of microstates with energies in $[E,E+\delta E]$ for a small but macroscopic $\delta E$. Then the entropy is

Thermodynamic temperature follows from

and pressure and chemical potential are

For weakly interacting subsystems $A$ and $B$, additivity of $S$ at fixed total $E$ yields the equilibrium condition $T_{A}=T_{B}$ by maximizing $S_{A}(E_{A})+S_{B}(E_{B})$ subject to $E_{A}+E_{B}=E$.

9.1.3 Thermodynamic limit, extensivity, and concavity

Define the thermodynamic limit as $N\to\infty$, $V\to\infty$ with $n=N/V$ fixed. In this limit:

• Entropy becomes extensive: $S(\lambda E,\lambda V,\lambda N)\approx \lambda S(E,V,N)$
• $S$ is concave in its natural variables, ensuring stability and positive heat capacities where appropriate
• Fluctuations of intensive variables shrink like $1/\sqrt{N}$, making thermodynamics sharp

Short-range interactions guarantee ensemble equivalence; long-range cases (self-gravity, unscreened Coulomb) can violate additivity and need care.

9.1.4 Microcanonical ensemble

The microcanonical density is uniform on the constant-energy hypersurface:

Expectations are

Maximizing $S=k_{B}\ln\Omega$ subject to constraints reproduces thermodynamic relations. For many systems this is the conceptual anchor; in practice the canonical ensemble is often easier to compute with and becomes equivalent at large $N$.

9.1.5 Canonical ensemble via a heat bath

Place system $S$ weakly coupled to a huge reservoir $R$ with fixed $E_{\text{tot}}$. The probability that $S$ is found in microstate $i$ with energy $E_{i}$ is proportional to the number of reservoir states at energy $E_{\text{tot}}-E_{i}$:

with inverse temperature $\beta \equiv 1/(k_{B}T)$. Normalize to get

$Z$ is the partition function. Thermodynamic potentials follow:

Energy fluctuations satisfy

so relative fluctuations scale as $1/\sqrt{N}$ for normal systems.

9.1.6 Grand canonical ensemble: exchanging particles

If both energy and particles exchange with a reservoir, the probability of microstate $i$ with $(E_{i},N_{i})$ is

with grand partition function

Define the grand potential

Then

Fluctuations:

and energy–number covariances come from mixed derivatives of $\ln \Xi$.

9.1.7 Legendre transforms and thermodynamic potentials

Thermodynamics is organized by Legendre transforms that swap natural variables:

• $S=S(E,V,N)$
• $F=U-TS$ with natural variables $(T,V,N)$
• $G=F+pV$ with $(T,p,N)$
• $H=U+pV$ with $(S,p,N)$
• $\Phi_{G}=F-\mu N$ with $(T,V,\mu)$

Maxwell relations follow from equality of mixed partial derivatives, e.g.,

9.1.8 Information-theoretic foundation (Jaynes view)

Let $\{p_{i}\}$ be probabilities over microstates. Maximize the Shannon entropy

subject to known constraints. With only normalization and mean energy fixed, Lagrange multipliers give

With energy and particle number constraints, one obtains the grand canonical form $p_{i}\propto e^{-\beta(E_{i}-\mu N_{i})}$. Thus ensembles are the least biased distributions compatible with macroscopic data.

Relative entropy $D(p\Vert q)=\sum p_{i}\ln(p_{i}/q_{i})\ge 0$ measures distance to a prior $q$. Many fluctuation theorems and linear-response results are compact in this language.

9.1.9 Large deviations, concentration, and ensemble equivalence

For $N\gg 1$, probabilities concentrate exponentially near thermodynamic values. If $A$ is an additive observable,

with a rate function $I(a)\ge 0$ that vanishes at the equilibrium value. Under standard convexity conditions:

• Microcanonical, canonical, and grand canonical ensembles are equivalent in the thermodynamic limit
• Nonconcavities signal phase coexistence; Maxwell constructions repair them at the level of thermodynamic potentials

9.1.10 Worked mini-examples

(a) Two-level system
Energy levels $0$ and $\epsilon$ for $N$ noninteracting sites. Canonical partition function per site

Total $Z=z^{N}$. Average energy $U=N\,\epsilon/(e^{\beta\epsilon}+1)$ and heat capacity shows a Schottky peak near $k_{B}T\sim \epsilon$.

(b) Classical harmonic oscillator
Single oscillator with $E_{n}=(n+\tfrac12)\hbar\omega$ gives

Average energy $U=\frac{\hbar\omega}{2}+\frac{\hbar\omega}{e^{\beta\hbar\omega}-1}$ tends to $k_{B}T$ at high $T$ (equipartition) and to $\hbar\omega/2$ at $T\to 0$.

(c) Coin toss and Stirling
For $N$ independent fair coins, the multiplicity of $k$ heads is $\binom{N}{k}\approx \exp\!\left[N\,H(k/N)\right]/\sqrt{2\pi N x(1-x)}$ with $x=k/N$ and binary entropy $H(x)=-x\ln x-(1-x)\ln(1-x)$. This is the large-deviation archetype.

(d) Ideal gas microcanonical temperature
For a classical ideal gas, counting states yields

so $S=k_{B}\ln\Omega$ gives $U=\tfrac{3}{2}Nk_{B}T$ and $pV=Nk_{B}T$ after Stirling.

(e) Canonical fluctuations
Show $\langle (\Delta E)^{2}\rangle = k_{B}T^{2}C_{V}$ by taking $\partial^{2}_{\beta}\ln Z$ and relate the heat capacity to curvature of $\ln Z$.

9.1.11 Minimal problem kit

• Starting from $S=k_{B}\ln\Omega$, derive $1/T=(\partial S/\partial E)_{V,N}$ and $p/T=(\partial S/\partial V)_{E,N}$
• Derive the canonical ensemble by maximizing $\mathcal S[p]$ with constraints $\sum p_{i}=1$ and $\sum p_{i}E_{i}=U$
• Show that $F=-k_{B}T\ln Z$ implies $S=-\partial F/\partial T$ and $p=-\partial F/\partial V$
• For a classical ideal gas, compute $\Omega(E,V,N)$ up to a constant using phase-space volume and obtain the Sackur–Tetrode entropy
• Prove $\langle (\Delta N)^{2}\rangle = k_{B}T\,(\partial N/\partial\mu)_{T,V}$ in the grand canonical ensemble
• Discuss when ensemble equivalence fails and give a physical example with long-range forces

9.2 Partition Functions & Thermodynamic Potentials

Partition functions are the generators of everything thermodynamic. Once you have $Z$ or $\Xi$ or the $N\!pT$ analogue, derivatives give you $U,p,S,\mu$ and all the response coefficients. This section builds the canonical and grand-canonical machinery, adds the isothermal–isobaric ensemble, and shows how Legendre transforms and saddle points tie them back to microcanonical entropy. Worked examples anchor the formulas.

9.2.1 Canonical partition function and cumulants

For fixed $(T,V,N)$ the canonical weights are $P(i)\propto e^{-\beta E_{i}}$ with

Helmholtz free energy

Standard derivatives

Energy cumulants follow from derivatives of $\ln Z$ with respect to $\beta$. For instance

Generalized forces for a control parameter $\lambda$ that enters $H(\lambda)$ obey

Examples: $M = -\left(\partial F/\partial B\right)_{T,V,N}$ for a magnetic field $B$ and $p = -\left(\partial F/\partial V\right)_{T,N}$ for volume.

9.2.2 Grand partition function, fugacity, and pressure

When particle number fluctuates at fixed $(T,V,\mu)$

Define the grand potential

Thermodynamics in this ensemble is compact

It is common to use the fugacity $z\equiv e^{\beta \mu}$. Number fluctuations and compressibility are linked by

For a uniform ideal gas one finds the handy identity $pV = k_{B}T \ln \Xi$.

9.2.3 Isothermal–isobaric ensemble and Gibbs free energy

Many experiments hold $(T,p,N)$ fixed. The isothermal–isobaric partition function is the Laplace transform of $Z$ over $V$

Its thermodynamic potential is the Gibbs free energy

Then

Fluctuations of $V$ at fixed $(T,p,N)$ are generated by derivatives of $\ln \Delta$.

9.2.4 Legendre structure and microcanonical link

Canonical and grand-canonical ensembles are Laplace transforms of the microcanonical state count

For large systems the integral is dominated by a saddle point $E^{\ast}$ with $\partial_{E}\ln\Omega|_{E^{\ast}}=\beta$, i.e., $1/T = \partial S/\partial E$. Evaluating at the saddle yields the Legendre relation

Similarly, the grand-canonical transform over both $E$ and $N$ yields

These are the reasons thermodynamic potentials are Legendre transforms of $S$.

9.2.5 Ideal classical gas in the canonical ensemble

Take $N$ indistinguishable particles of mass $m$ with Hamiltonian $H=\sum p_{i}^{2}/2m$ and ignore interactions. The canonical partition function factorizes into a translational part

with thermal wavelength

Then

Equation of state follows immediately

Entropy is the Sackur–Tetrode form

and the internal energy is $U=\tfrac{3}{2}Nk_{B}T$ with $C_{V}=\tfrac{3}{2}Nk_{B}$. The $1/N!$ cures the Gibbs paradox by enforcing indistinguishability.

9.2.6 Quantum ideal gases via the cluster expansion

For noninteracting bosons or fermions with one-particle energies $\{\epsilon_{\alpha}\}$

upper sign for bosons, lower for fermions. Occupations are

In the continuum limit one replaces $\sum_{\alpha}\to \int d\epsilon\,g(\epsilon)$ to obtain thermodynamics of photon, phonon, or electron gases. Interactions can be included perturbatively via virial or cluster expansions that correct $\ln \Xi$ by coefficients $B_{2},B_{3},\dots$.

9.2.7 Generalized forces and response matrices

If a Hamiltonian depends on external controls $\boldsymbol{\lambda}=(\lambda_{1},\dots)$, the matrix of second derivatives of $F$ gives linear response

where $X_{a}\equiv -\partial H/\partial \lambda_{a}$ is the conjugate observable. Examples include magnetic susceptibility, compressibility, and heat capacities. Mixed derivatives encode Maxwell relations, e.g., $\left(\partial S/\partial V\right)_{T,N}=\left(\partial p/\partial T\right)_{V,N}$.

9.2.8 Ising-like lattice systems and counting practice

For a finite lattice with spins $s_{i}=\pm 1$ and Hamiltonian $H=-J\sum_{\langle ij\rangle}s_{i}s_{j}-h\sum_{i}s_{i}$ the canonical partition function is

While exact evaluation is hard beyond 1D, derivatives of $\ln Z$ still define magnetization $M=\left(\partial \ln Z/\partial (\beta h)\right)$ and susceptibility $\chi=\partial M/\partial h$. Mean-field or transfer-matrix approaches approximate $Z$ systematically.

9.2.9 Gibbs–Duhem and extensivity checks

For an extensive system with $U(S,V,N)$ homogeneous of degree one, Euler’s relation holds

Differentiating yields the Gibbs–Duhem equation

This is a consistency check for any equation of state you derive from a partition function.

9.2.10 Laplace tricks, saddle points, and when ensembles match

Because $Z$ and $\Delta$ are Laplace transforms, large-$N$ thermodynamics follows from their leading saddle. Concavity of $S(E,V,N)$ guarantees equivalence of ensembles. At first-order transitions $Z$ develops multiple competing saddles; Legendre transforms pick the convex envelope, and fluctuations become non-Gaussian in finite systems.

9.2.11 Worked mini-examples

(a) $N\!pT$ ideal gas

Compute $\Delta$ for the ideal gas using $Z=(V/\lambda_{T}^{3})^{N}/N!$ to show

and verify $G=Nk_{B}T\ln\!\left(p\,\lambda_{T}^{3}/k_{B}T\right) + Nk_{B}T$ and $V = \left(\partial G/\partial p\right)_{T,N} = Nk_{B}T/p$.

(b) Pressure from $Z$ via the virial

For a classical interacting gas with pair potential $u(r)$, show that $p = Nk_{B}T/V - \tfrac{2\pi n^{2}}{3}\int_{0}^{\infty} dr\, r^{3} u'(r) g(r)$ using $p=-(\partial F/\partial V)_{T,N}$ and a uniform scaling argument, where $n=N/V$ and $g(r)$ is the pair distribution.

(c) Number fluctuations and compressibility

In the grand ensemble, verify $\langle(\Delta N)^{2}\rangle = k_{B}T\,\left(\partial N/\partial \mu\right)_{T,V}$ and relate to the isothermal compressibility $\kappa_{T}=\frac{1}{n^{2}}\left(\partial n/\partial \mu\right)_{T}$.

(d) Microcanonical ↔ canonical

Starting from $Z=\int dE\,\exp[S(E)/k_{B}-\beta E]$, perform a quadratic expansion of $S(E)$ about $E^{\ast}$ to obtain Gaussian energy fluctuations with variance $k_{B}T^{2}C_{V}$.

(e) Two-level system free energy

For $N$ independent two-level sites with energies $0$ and $\epsilon$, compute $Z=(1+e^{-\beta\epsilon})^{N}$, then $F$, $S$, and $C_{V}$ and locate the Schottky peak.

9.2.12 Minimal problem kit

• Derive all canonical relations for $U,S,p$ starting from $F=-k_{B}T\ln Z$ and verify the Maxwell relation $\left(\partial S/\partial V\right)_{T,N}=\left(\partial p/\partial T\right)_{V,N}$
• Show that in the grand ensemble, $pV=k_{B}T\ln\Xi$ for an ideal gas and obtain $N$ and $S$ from derivatives of $\ln\Xi$
• Compute the isothermal–isobaric partition function for a classical harmonic solid treated as $3N$ independent oscillators and extract $G(T,p,N)$ in the limit of weak compressibility
• Using $\ln \Xi = \pm\sum_{\alpha}\ln(1\mp z e^{-\beta\epsilon_{\alpha}})$, recover the Bose–Einstein and Fermi–Dirac distributions and the classical limit when $z\ll 1$
• From extensivity, prove Euler’s relation $U=TS-pV+\mu N$ and then the Gibbs–Duhem equation, and test them on the ideal gas results above

9.3 Classical Ideal Gas & Equipartition

The classical ideal gas is the training montage of statistical physics: you learn to count states, take derivatives of $\ln$ partition functions, and watch macroscopic laws fall out like it’s nothing. We derive the Maxwell–Boltzmann distribution, prove equipartition, recover $pV=Nk_{B}T$, and build intuition for adiabats, sound speed, and when the classical picture breaks (hello, $\lambda_{T}$).

9.3.1 Canonical counting recap and the ideal-gas Z

For $N$ indistinguishable point particles of mass $m$ in volume $V$ with Hamiltonian $H=\sum_{i=1}^{N}\boldsymbol p_{i}^{2}/2m$, the canonical partition function factorizes:

with thermal wavelength

Thermodynamics then follows immediately:

Heat capacities:

Entropy is the Sackur–Tetrode form:

The $1/N!$ kills the Gibbs paradox by encoding indistinguishability.

9.3.2 Maxwell–Boltzmann distribution (velocities, speeds, energies)

Each momentum component is Gaussian:

Equivalently, each velocity component is Gaussian with variance $\langle v_{x}^{2}\rangle = k_{B}T/m$. The speed distribution (radial in velocity space) is

Useful moments:

The kinetic energy distribution is $\chi^{2}$-like with $3$ degrees of freedom:

9.3.3 Equipartition theorem

Statement. Every quadratic degree of freedom in the Hamiltonian contributes $\tfrac{1}{2}k_{B}T$ to the mean energy.

Sketch of proof (canonical, integration by parts). If $H$ contains a term $\alpha x^{2}$, then

For $H=\sum p_{i}^{2}/2m + \cdots$, each quadratic $p_{i}^{2}/2m$ gives $\langle p_{i}^{2}/2m\rangle = \tfrac{1}{2}k_{B}T$. A monatomic gas has $3N$ translational momenta → $U=\tfrac{3}{2}Nk_{B}T$.

When it fails (and why that’s good). If a degree of freedom isn’t effectively quadratic (e.g., quantized rotations/vibrations at $k_{B}T$ below level spacings), equipartition “freezes out.” This explains the drop of specific heats at low $T$ and the success/failure of Dulong–Petit.

9.3.4 Kinetic-theory

From elastic impacts on a wall, the pressure can be written

Combine with $U=\tfrac{3}{2}Nk_{B}T$ to get $pV=Nk_{B}T$. More generally, for pairwise interactions $u(r)$ the virial theorem yields

The ideal gas has $\boldsymbol F_{ij}=0$, so the second term vanishes.

9.3.5 Adiabats, sound speed, and quick thermodynamic moves

For a reversible adiabatic change ($dQ=0$) in a monatomic ideal gas,

with $\gamma=5/3$. The speed of sound (small adiabatic density waves) is

For mixtures, replace $m$ by the mean molecular mass and $\gamma$ by the appropriate ratio of heat capacities.

9.3.6 External fields: barometric formula and separability

In a uniform gravitational field $U(z)=mgz$, the one-body Boltzmann weight factorizes:

The momentum and position parts separate, so kinetic-energy results (equipartition, MB speeds) survive untouched.

9.3.7 Transport in one screen: mean free path and friends

For hard spheres of diameter $d$ at number density $n$, the mean free path is

Using typical speed $\bar v \sim \langle v\rangle$,

These kinetic-theory scalings are remarkably good for dilute gases; real cross sections and mixtures refine the prefactors.

9.3.8 Validity of the classical picture

Classical, nondegenerate behavior requires the diluteness parameter to be small:

If $n\lambda_{T}^{3}\gtrsim 1$ (low $T$ or high $n$), quantum statistics kicks in: bosons want Bose–Einstein, fermions go Pauli, and the MB distribution is no longer the move.

9.3.9 Beyond monatomic: what equipartition counts

For a rigid linear diatomic at intermediate $T$:

• $3$ translational + $2$ rotational quadratic modes → $f=5$ effective d.o.f., so $C_{V} \approx \tfrac{5}{2}Nk_{B}$, $\gamma \approx 7/5$
• Vibrations add $2$ more quadratic modes but only when $k_{B}T$ exceeds the vibrational quantum spacing; otherwise they’re frozen

Moral: count only the modes that are active at your $T$.

9.3.10 Worked mini-examples

(a) RMS speed for air at room $T$

For $\mathrm{N}_{2}$ with $m\approx 28\,\mathrm{u}$ at $T=300\ \mathrm{K}$,

Plug numbers to get a few hundred $\mathrm{m/s}$; compare to $v_{\text{mp}}$ and $\langle v\rangle$.

(b) Adiabatic compression

A monatomic gas is compressed from $(p_{1},V_{1})$ to $(p_{2},V_{2})$ adiabatically. Use $pV^{\gamma}=\text{const}$ to find $T_{2}/T_{1}=(V_{1}/V_{2})^{\gamma-1}$.

(c) Barometric scale height of helium

For He at $T=300\ \mathrm{K}$, compute $H=k_{B}T/(m g)$ and compare to $\mathrm{N}_{2}$ to see the mass dependence pop.

(d) Mean free path at STP

Take $d\approx 0.3\ \mathrm{nm}$, $n\approx 2.5\times 10^{25}\ \mathrm{m}^{-3}$; estimate $\ell$ and comment on why smells diffuse fast.

(e) Equipartition sanity check

Show directly from $\int dp\;p^{2} e^{-\beta p^{2}/2m}$ that $\langle p^{2}/2m\rangle = \tfrac{1}{2}k_{B}T$ and extend to three components.

9.3.11 Minimal problem kit

• Starting from $f(v)$, derive $v_{\text{mp}}$, $\langle v\rangle$, and $v_{\text{rms}}$
• Prove the equipartition identity $\langle x\,\partial H/\partial x\rangle=k_{B}T$ for a quadratic coordinate and apply it to $p^{2}/2m$
• Derive $p=\tfrac{2}{3}U/V$ via wall-collision counting and recover $pV=Nk_{B}T$
• For a linear diatomic with rotational constant $B$, estimate the temperature where rotations “turn on” by comparing $k_{B}T$ to $\hbar^{2}/(2I)$
• Check the classicality criterion for a cold atomic gas of density $n$ and temperature $T$: compute $n\lambda_{T}^{3}$ and discuss when quantum statistics is required

9.4 Fluctuations & the Fluctuation–Response Principle

Macroscopic laws look crisp because relative fluctuations scale like $1/\sqrt{N}$—until they don’t. This section organizes what fluctuates, how much, and why responses equal fluctuations. We connect the canonical, grand-canonical, and isothermal–isobaric ensembles; introduce static fluctuation–dissipation relations; link pair correlations to compressibility; and preview what breaks near critical points.

9.4.1 Energy and number: the two flagship variances

Canonical (fixed $T,V,N$):

Relative size shrinks as $\langle(\Delta E)^{2}\rangle^{1/2}/U \sim 1/\sqrt{N}$ for normal systems.

Grand canonical (fixed $T,V,\mu$):

Equivalently, in terms of density $n=N/V$ and isothermal compressibility $\kappa_{T} \equiv -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T,N}$,

for a single-component fluid where $dn = \chi_{\mu}\,d\mu$ and standard thermodynamic identities have been used.

9.4.2 Volume fluctuations at fixed pressure

Holding $(T,p,N)$ fixed, the isothermal–isobaric partition function $\Delta$ generates

Again, fluctuation = response: compressibility measures how much $V$ wiggles when $p$ nudges; that same coefficient sets the equilibrium variance.

9.4.3 Static fluctuation–response in one line

Let a control $\lambda$ couple to an observable $X$ via $H \to H - \lambda X$ (so $X=-\partial H/\partial \lambda$). In the canonical ensemble,

Special cases:

• $X=E$, $\lambda$ shifts $\beta$: $\chi_{EE}$ reduces to $C_{V}/T$ and reproduces $\langle(\Delta E)^{2}\rangle=k_{B}T^{2}C_{V}$
• $X=N$ in the grand ensemble gives $\left(\partial N/\partial \mu\right)_{T,V}=\beta\,\langle(\Delta N)^{2}\rangle$

Positivity of variances $\Rightarrow$ stability: $C_{V}\ge 0$, $\kappa_{T}\ge 0$ in ensembles with fixed $T$.

9.4.4 Correlations, structure factor, and the compressibility sum rule

Define the microscopic density $\hat n(\boldsymbol r)=\sum_{i}\delta(\boldsymbol r-\boldsymbol r_{i})$, the pair correlation $g(\boldsymbol r)$, and the static structure factor

In a uniform fluid,

Taking $\boldsymbol k\to 0$ connects microscopic correlations to macroscopic response:

This is the compressibility sum rule. Near a critical point, $\kappa_{T}\to \infty$ and $S(0)$ spikes—aka critical opalescence.

9.4.5 Saddle points, curvature, and large deviations

Microcanonical entropy $S(E)$ controls fluctuations through its curvature. Expand around the most probable energy $E^{\ast}$:

Performing the Laplace transform to the canonical ensemble gives a Gaussian for $E$ with variance $( -k_{B}/S'' )$, equivalent to $k_{B}T^{2}C_{V}$. More generally, additive observables obey a large-deviation principle:

where $I(a)\ge 0$ is the rate function (Legendre transform of the cumulant generating function). At first-order transitions the relevant thermodynamic potential develops multiple competing saddles → non-Gaussian fluctuations and phase coexistence.

9.4.6 Critical fluctuations and correlation length

Let $\phi$ be an order parameter (e.g., magnetization density). Define the connected correlator

Away from criticality,

with finite correlation length $\xi$. The static susceptibility $\chi \equiv \int d^{d}r\, G(\boldsymbol r)$ scales as

Therefore, in a volume $V$, the variance of the spatially averaged order parameter scales like $\mathrm{Var}(\bar\phi) \sim \chi/V$. When $\xi$ is finite, relative fluctuations $\sim 1/\sqrt{V}$. As $T\to T_{c}$, $\xi\to\infty$ and the $1/\sqrt{N}$ law breaks down.

9.4.7 Finite-size scaling: when N is large but not infinite

• For noncritical systems: $\mathrm{Var}(A) \propto N$ for additive $A$ → relative size $\propto N^{-1/2}$.
• Near criticality: with $\xi$ comparable to system size $L$, replace $\xi$ by $L$ in scaling formulas; peaks in specific heat or susceptibility round and shift with $L$ in predictable ways (used to extract critical exponents numerically).

9.4.8 Example: noninteracting paramagnet and Curie law

$N$ spins $s=\tfrac12$ in field $B$ with Hamiltonian $H=-\mu B\sum_{i}\sigma_{i}$, $\sigma_{i}=\pm 1$. The single-spin partition function is $z=2\cosh(\beta\mu B)$, magnetization per spin

Variance of $M$ is binomial:

Static susceptibility $\chi \equiv \left(\partial \langle M\rangle/\partial B\right)_{T}$ matches fluctuation–response:

In the weak-field limit, Curie law:

9.4.9 Example: ideal gas density fluctuations and S(0)

In a classical ideal gas, particles are uncorrelated: $g(\boldsymbol r)=1$. Then $S(\boldsymbol k)=1$ for all $\boldsymbol k$ and

so $\kappa_{T}=1/(n k_{B}T)$, the ideal-gas result. Number variance in a subvolume $V'$ is Poissonian: $\langle(\Delta N')^{2}\rangle=\langle N'\rangle$.

9.4.10 Common pitfalls

• Mixing ensembles. Don’t use canonical $C_{V}$ with grand-canonical number fluctuations unless you’ve matched the ensemble to the measurement.
• Forcing Gaussian fits near transitions. At first order, distributions are bimodal (two phases); at criticality, power-law tails win.
• Ignoring constraints. Conserved totals suppress variances (e.g., fixed total $N$ reduces subvolume fluctuations).
• Calling negative $C_{V}$ “instability.” In microcanonical small systems with long-range forces, $C_{V}<0$ can appear; it signals ensemble inequivalence, not algebra failure.

9.4.11 Worked mini-examples

(a) $N\!pT$ volume variance
From $\Delta=\int dV\,e^{-\beta pV}Z$, show $\langle(\Delta V)^{2}\rangle = k_{B}T\,V\,\kappa_{T}$ and verify it for an ideal gas.

(b) Compressibility sum rule
Starting with $S(\boldsymbol k)=1+n\int d^{3}r\,e^{-i\boldsymbol k\cdot\boldsymbol r}[g(r)-1]$, take $\boldsymbol k\to 0$ and use thermodynamics to derive $S(0)=n k_{B}T\kappa_{T}$.

(c) Energy large deviations
From $Z=\int dE\,e^{S(E)/k_{B}-\beta E}$, do the quadratic expansion and read off $\langle(\Delta E)^{2}\rangle$; identify when the Gaussian approximation fails.

(d) Paramagnet variance → Curie
Compute $\langle M\rangle$ and $\langle M^{2}\rangle$ from $z$ and show $\chi=\beta\langle(\Delta M)^{2}\rangle$; take $B\to 0$ to get $\chi\propto 1/T$.

(e) Subvolume fluctuations
For an ideal gas, partition a box into two volumes $V'$ and $V-V'$. With fixed total $N$, show $\mathrm{Var}(N')=N(V'/V)(1-V'/V)$ (binomial), contrasting with grand-canonical Poisson variance.

9.4.12 Minimal problem kit

• Prove the general static relation $\chi_{XX}=\beta\langle(\Delta X)^{2}\rangle$ from derivatives of $\ln Z$ with a source term
• Derive $\langle(\Delta N)^{2}\rangle = k_{B}T\left(\partial N/\partial\mu\right)_{T,V}$ and connect it to $\kappa_{T}$ for a single-component fluid
• Show that $S(0)=n k_{B}T\kappa_{T}$ implies $S(0)\to\infty$ as a liquid approaches its gas–liquid critical point and explain “critical opalescence” qualitatively
• Using a Landau $\phi^{4}$ free energy $f=a(T-T_{c})\phi^{2}+b\phi^{4}$, compute Gaussian fluctuations above $T_{c}$ and extract $\chi\propto 1/(T-T_{c})$ at mean-field level
• For a lattice model with conserved order parameter, discuss how conservation modifies long-time relaxation (but not equal-time static variances) at fixed $T$