Correctness Report: “The mean-field Potts model”
Source: pott.qmd Date: 2026-02-25
Summary
All formulas and qualitative claims in the note are correct. Five computations were verified independently; no errors found. A few minor expositional remarks are collected at the end.
Claim-by-claim verification
1. Energy rewriting: \(E(\sigma) \to E(\rho)\)
Claim. \(E(\sigma)=-\frac{1}{2N}\sum_{i,j=1}^N\mathbf{1}\{\sigma_i=\sigma_j\}\) rewrites as \(E(\rho)=-\frac{N}{2}\sum_a\rho_a^2\) up to an additive constant.
Verification. Let \(N_a=|\{i:\sigma_i=a\}|=N\rho_a\). The double sum decomposes by color:
\[ \sum_{i,j=1}^N\mathbf{1}\{\sigma_i=\sigma_j\} = \sum_{a=1}^q N_a^2 = N^2\sum_a\rho_a^2 \]
Dividing by \(-2N\):
\[ E(\sigma) = -\frac{N}{2}\sum_a\rho_a^2 \]
The sum includes diagonal terms \(i=j\) (contributing \(N\) to the total), so the exact identity is \(E=-\frac{N}{2}\sum_a\rho_a^2\), which equals the off-diagonal version plus \(-1/2\). The document correctly flags “up to an additive constant irrelevant for Gibbs weights.”
PASS.
2. Free energy functional \(\Phi_\beta\)
Claim. Typical samples from \(\mu_\beta\) concentrate near minimizers of
\[ \Phi_\beta(\rho)=\sum_a\rho_a\log\rho_a - \frac{\beta}{2}\sum_a\rho_a^2 \]
Verification. The number of configurations with empirical proportions \(\rho\) is the multinomial coefficient
\[ \binom{N}{N\rho_1,\ldots,N\rho_q} = \frac{N!}{\prod_a(N\rho_a)!} \sim \exp\{NH(\rho)\} \]
by Stirling, where \(H(\rho)=-\sum_a\rho_a\log\rho_a\). The Boltzmann weight is \(e^{-\beta E(\rho)}=\exp\{\frac{\beta N}{2}\sum_a\rho_a^2\}\). So the probability of macrostate \(\rho\) scales as
\[ \mu_\beta(\rho)\propto\exp\left\{N\!\left(-\sum_a\rho_a\log\rho_a+\frac{\beta}{2}\sum_a\rho_a^2\right)\right\} \]
Typical \(\rho\) maximizes the exponent, equivalently minimizes its negation \(\Phi_\beta\).
PASS.
3. Lagrange condition and critical point types
Claim (a). Stationary points on the simplex satisfy \(\log\rho_a+1-\beta\rho_a=\lambda\) for all \(a\).
Verification. \(\frac{\partial\Phi_\beta}{\partial\rho_a}=\log\rho_a+1-\beta\rho_a\). The simplex constraint \(\sum_a\rho_a=1\) yields a single Lagrange multiplier \(\lambda\). Correct.
Claim (b). All local minima are disordered (\(\rho_a=1/q\)) or ordered with one dominant color (\(k=1\)).
Verification. The scalar function \(f(x)=\log x+1-\beta x\) is strictly concave (\(f''=-1/x^2<0\)), so \(f(x)=\lambda\) has at most 2 solutions. Each \(\rho_a\) takes one of these solutions. This gives stationary points where \(k\) coordinates equal \(r\) and \(q-k\) equal \(s=(1-kr)/(q-k)\). Checking which are local minima requires the Hessian on the simplex tangent space.
The Hessian of \(\Phi_\beta\) is diagonal: \(H_{ab}=(1/\rho_a-\beta)\,\delta_{ab}\). For the \(k\)-ordered point:
- Within-\(r\)-block eigenvalue: \(1/r-\beta\), multiplicity \(k-1\)
- Within-\(s\)-block eigenvalue: \(1/s-\beta\), multiplicity \(q-k-1\)
- Cross eigenvalue: \(1/r+1/s-2\beta\)
For \(k\geq 2\): these ordered branches emerge from the uniform state at \(\beta=q\) with \(r\approx 1/q\), giving \(1/r-\beta\approx q-q=0\). Along the branch, \(r\) increases while \(\beta\) increases, so \(1/r-\beta<0\): the within-\(r\)-block direction is a saddle direction. Partially ordered states (\(k\geq 2\)) are therefore never local minima.
Only the disordered point and \(k=1\) ordered points can be local minima.
PASS.
4. Coexistence formula \(\beta_c=\frac{2(q-1)\log(q-1)}{q-2}\)
Claim. At \(\beta_c\), the disordered and ordered free energies are equal.
Verification. Parametrize the order parameter as \(m\in[0,1]\) with \(r=\frac{1+(q-1)m}{q}\), \(s=\frac{1-m}{q}\).
Stationarity condition: \(\log\frac{1+(q-1)m}{1-m}=\beta m\), giving \(\beta=\frac{1}{m}\log\frac{1+(q-1)m}{1-m}\).
Equal free energy condition (\(\Phi(m)=\Phi(0)\)): after substituting \(\beta\) and simplifying,
\[ \left(1+\tfrac{(q-1)m}{2}\right)\log(1+(q-1)m) +\left((q-1)-\tfrac{(q-1)m}{2}\right)\log(1-m)=0 \]
Solving. Try \(m^*=\frac{q-2}{q-1}\). Then:
- \(1+(q-1)m^*=q-1\)
- \(1-m^*=\frac{1}{q-1}\)
- Both prefactors equal \(\frac{q}{2}\)
LHS \(= \frac{q}{2}\log(q-1)+\frac{q}{2}\log\frac{1}{q-1}=0\). Confirmed.
Substituting \(m^*\) into the stationarity condition:
\[ \beta_c=\frac{q-1}{q-2}\cdot\log\frac{q-1}{1/(q-1)} =\frac{q-1}{q-2}\cdot 2\log(q-1) =\frac{2(q-1)\log(q-1)}{q-2} \]
Numerical checks:
| \(q\) | \(\beta_c\) (formula) | Decimal |
|---|---|---|
| 3 | \(4\log 2\) | 2.773 |
| 4 | \(3\log 3\) | 3.296 |
| 5 | \(\frac{8\log 4}{3}\) | 3.697 |
Consistency with \(q=2\): The formula is \(0/0\) at \(q=2\), but \(\lim_{q\to 2}\frac{2(q-1)\log(q-1)}{q-2}=2\) by L’Hopital, recovering the mean-field Ising critical point \(\beta_c=2\). The document correctly restricts to \(q\geq 3\).
PASS.
5. Phase diagram and qualitative claims
Claim (a): \(\beta_s<\beta_c\) for \(q\geq 3\). The ordered local minima appear (via saddle-node bifurcation) before they become global minimizers. Standard result, consistent with the metastable window. CORRECT.
Claim (b): First-order transition for \(q\geq 3\). At \(\beta_c\), the order parameter jumps from \(m=0\) to \(m^*=(q-2)/(q-1)>0\). This is a discontinuous change in the minimizer. CORRECT.
Claim (c): Second-order transition for \(q=2\). The mean-field Ising model (\(q=2\)) has a continuous pitchfork bifurcation at \(\beta=2\). The ordered state emerges smoothly from the disordered one. CORRECT.
Claim (d): “Second spinodal on the low-temperature side.” The disordered point loses local stability when the Hessian becomes indefinite. The Hessian of \(\Phi_\beta\) at \(\rho^{\rm dis}\) restricted to the simplex tangent space is \((q-\beta)I\). So the disordered point loses stability at \(\beta=q\). For \(q=3\): \(\beta=3>\beta_c\approx 2.773\), confirming that the disordered local minimum persists above \(\beta_c\) and vanishes at a second spinodal. CORRECT.
Claim (e): Three-regime summary.
| Regime | Condition | Global min | Ordered local min? |
|---|---|---|---|
| High temp | \(\beta<\beta_s\) | disordered | no |
| Metastable | \(\beta_s<\beta<\beta_c\) | disordered | yes |
| Low temp | \(\beta>\beta_c\) | ordered | yes (disordered is metastable) |
All descriptions match. CORRECT.
PASS.
6. Spinodal threshold \(\beta_s\)
The note defines \(\beta_s\) qualitatively as the inverse temperature where ordered local minima first appear. No closed form exists, but there is a clean implicit characterization.
Derivation. Using the order parameter \(m\), the stationarity condition for ordered points is \(\beta=g(m)\) where
\[ g(m)=\frac{1}{m}\log\frac{1+(q-1)m}{1-m} \]
The spinodal \(\beta_s\) is the minimum of \(g\) over \(m\in(0,1)\): it is the smallest \(\beta\) for which the stationarity equation has a non-trivial solution.
Setting \(g'(m_s)=0\) and simplifying gives the tangency condition
\[ \beta_s = \frac{q-1}{1+(q-1)m_s}+\frac{1}{1-m_s} \]
where \(m_s\in(0,1)\) simultaneously satisfies
\[ m_s\!\left(\frac{q-1}{1+(q-1)m_s}+\frac{1}{1-m_s}\right)=\log\frac{1+(q-1)m_s}{1-m_s} \]
In the document’s notation (\(r_s = \frac{1+(q-1)m_s}{q}\), \(s_s = \frac{1-m_s}{q}\)), this reads:
\[ \beta_s = \frac{q-1}{q}\cdot\frac{1+(q-2)r_s}{r_s(1-r_s)} \]
This is a transcendental equation (no closed form), but it pins down \(\beta_s\) uniquely.
Numerical values:
| \(q\) | \(\beta_s\) | \(\beta_c\) | \(q\) (disordered spinodal) |
|---|---|---|---|
| 3 | 2.746 | 2.773 | 3 |
| 4 | 3.219 | 3.296 | 4 |
| 5 | 3.565 | 3.697 | 5 |
| 10 | 4.560 | 4.944 | 10 |
The ordering \(\beta_s<\beta_c<q\) holds in all cases, confirming the metastable window.
PASS.
Expositional remarks
These are not errors. They are minor points of precision that a reader might notice.
“All local minima are necessarily of the following two types” (line 48). Strictly, the Lagrange analysis identifies all stationary points, which include saddle points. The subsequent paragraph (“The remaining work is algebra…”) correctly acknowledges that determining minima vs. saddles requires further analysis. The claim is true as stated (local minima are a subset of stationary points, and the only stationary points that can be minima are the two listed types), but the phrasing could be read as saying the Lagrange condition alone classifies minima.
“The global minimizer changes discontinuously” at \(\beta_c\) (line 95). At \(\beta=\beta_c\) exactly, the disordered and ordered states are both global minimizers (equal free energy). The discontinuity is in the sense that for \(\beta<\beta_c\) the global minimizer is disordered and for \(\beta>\beta_c\) it is ordered, with a jump in the order parameter. This is the standard usage, but a pedantic reader might note that “changes discontinuously” glosses over the coexistence at \(\beta_c\) itself.
No explicit formula for \(\beta_s\) in the note. The implicit characterization (section 6 above) is clean enough to include. Adding it would complete the quantitative picture alongside \(\beta_c\).
Conclusion
All five mathematical claims (energy rewriting, free energy functional, Lagrange condition, coexistence formula, phase diagram) are correct. The coexistence formula \(\beta_c=\frac{2(q-1)\log(q-1)}{q-2}\) was verified both analytically (via the ansatz \(m^*=(q-2)/(q-1)\)) and numerically for \(q=3,4,5\). The \(q\to 2\) limit correctly recovers the mean-field Ising critical point. No mathematical errors were found.