Continuum phase transitions for radial pair potentials

Source and verification

This article explains family 228 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

A finite gas has a smooth partition function, but its thermodynamic limit can develop a temperature singularity. Family 228 constructs radial pair potentials in three dimensions for which a first-order transition occurs at one common temperature throughout a nonempty interval of densities.

🏷️ Preliminaries: stability and canonical free energy

For particles $x_1,\dots ,x_N$ in a cube $\Lambda \subset {\mathbb{R}}^3$, a radial pair potential gives energy

$$E_{\varphi }(x_1,\dots ,x_N)=\sum _{i<j}\varphi (|x_i-x_j|).$$

Stability means $E_{\varphi }\ge -BN$ for a constant independent of the configuration and particle number. It prevents the energy from decreasing quadratically in $N$ through collapse of many particles.

With free boundaries, the configurational canonical partition function is

$$Z_{\Lambda ,N}(\beta )=\frac{1}{N!}{\int }_{{\Lambda }^N}{e}^{-\beta E_{\varphi }(x)}dx.$$

At density $\rho$, the thermodynamic free energy per volume is the limit of $-(\beta |\Lambda |{)}^{-1}\log Z_{\Lambda ,N}$ as $N/|\Lambda |\to \rho$. Stability and tail control establish its existence in the models under discussion. A phase transition here means a strict finite jump between the one-sided derivatives of this limiting function with respect to inverse temperature.

🏷️ Two constructions

The first companion constructs a stable radial potential with a divergent repulsive core, an attractive interval, and an integrable tail satisfying $\varphi (r)=o(r^{-3})$. Its free energy is finite at every positive density and inverse temperature. On one open density interval it has a strict downward derivative jump at the same finite positive ${\beta }_c$. (OpenAI, 2026)

The second companion obtains a bounded continuous stable radial potential with quantitative decay

$$|\varphi (r)|\le C{r}^{-3-1/32},r\ge 1,$$

and again one common transition inverse temperature throughout an open interval of positive densities. The potential is fixed before the thermodynamic limit and does not depend on the density chosen in that interval. (OpenAI, 2026b)

These are constructed interactions, not claims about every attractive radial potential or a particular experimentally fitted fluid. Nor is the first potential’s singular core retained in the bounded construction.

🏷️ Convexity and the direction of the jump

It is useful to remove the prefactor $1/\beta$. Write $H(\beta ,\rho )=\beta f(\beta ,\rho )$. At finite volume,

$$\begin{array}{rl}& \frac{d^2}{d{\beta }^2}\left(-\frac{1}{|\Lambda |}\log Z_{\Lambda ,N}\right)\\ & =-\frac{1}{|\Lambda |}{\mathrm{Var}}_{\beta }(E_{\varphi })\le 0.\end{array}$$

Thus $H$ is concave in $\beta$, and its thermodynamic limit can have a downward derivative jump. If $H$ is continuous at ${\beta }_c$, then

$$\begin{array}{c}{f}_{-}^{\prime }({\beta }_c)-f_{+}^{\prime }({\beta }_c)\\ =\frac{H_{-}^{\prime }({\beta }_c)-H_{+}^{\prime }({\beta }_c)}{{\beta }_c}.\end{array}$$

The jump has the same sign. This calculation fixes the convention: a derivative in inverse temperature is not the same observable as a derivative in temperature.

The grand-canonical pressure is the partial convex dual of $H$ in density. A supporting slope records an energy coordinate and a density coordinate. Two supporting slopes with the same density but different energy coordinates give the canonical corner at that density.

🏷️ Structure of the construction

The microscopic core favors a narrow shell of very small separations. Four particles can occupy the vertices of a tetrahedron with all six pair distances in that shell; five cannot satisfy the corresponding narrow-shell constraint. This geometric restriction supplies a quadratic local occupancy bound and hence stability.

At a large fixed parameter, the reference pressure has competing branches from vacuum, separated particles, and four-particle clusters. Spread-out radial attractions add a quadratic density contribution. The proof arranges three separated groups of supporting slopes whose convex hull contains a two-dimensional ball. Horizontal slices of that ball retain multiple energy slopes at every nearby density.

The key difficulty is preserving these slopes while replacing an ideal long-range interaction by a genuine finite-range approximation. Uniform convergence of convex functions is insufficient: smooth functions can converge uniformly to a function with a corner. The manuscript controls the approximation error relative to each newly added attractive mass, so the separated slope groups persist at every stage.

For the bounded potential, an attraction of range $R$ approximates its ideal pressure transform with error of order $\sqrt{R_0/R}$, where $R_0$ is the earlier interaction range. Mesoscopic boxes of side comparable to $\sqrt{R_{0}R}$ balance boundary losses against variation of the new kernel.

An iterated schedule with rapidly growing ranges and summable masses preserves a common ball of supports and converges to one interaction. The same schedule ensures the power-law tail bound. The proof therefore has two linked quantitative obligations: preserving the thermodynamic corner and keeping the final potential stable with the promised decay.

🏷️ Key steps and proof outlines

The construction has two separate quantitative tasks: a local occupancy penalty must control every later attraction, and the pressure approximation must preserve a two-dimensional set of supporting slopes. The latter is stronger than merely approximating a pressure with a corner.

A narrow-shell packing obstruction gives quadratic stability

In the divergent-core construction, the low-cost separations lie in $[e^{-t},(1+\delta )e^{-t}]$, with $\delta =10^{-3}$. Five points cannot have all pair distances in this shell. To see this, rescale the shell radius to one and take the Gram matrix of the four displacement vectors from one point. It differs in operator norm by at most $4(2\delta +{\delta }^2)<1/2$ from the matrix with diagonal one and off-diagonal $1/2$. That matrix has smallest eigenvalue $1/2$, so the Gram matrix would have rank four, impossible in ${\mathbb{R}}^3$.

The shell graph therefore has no five-vertex clique. Its edge count on $n$ vertices is at most $3n^2/8$: maximize ${\sum }_{ij\text{ edge}}{a}_i{a}_j$ over nonnegative weights summing to one, and choose a maximizer with minimal support. Two nonadjacent positive-weight vertices can merge their weights without decreasing the objective. The minimal support is thus a clique of size at most four, on which the objective is at most $(1-1/4)/2$.

Partition space into small cells of diameter below one. If $n_i$ particles occupy cell $i$, every nonshell pair pays at least $t^2$. Consequently the core energy is bounded below by

$$\frac{t^2}{8}\sum _i{n}_i^2-\frac{t^2}{2}N.$$

For the total attraction, let $J_{ij}$ bound its absolute value between cells and assume ${\sup }_i{\sum }_j{J}_{ij}\le M=O(t)$. Symmetry and $2n_i{n}_j\le n_i^2+n_j^2$ bound the attraction by $(M/2){\sum }_i{n}_i^2$. Fixing $t$ sufficiently large leaves a positive quadratic coefficient. This proves stability uniformly over all the later attraction ranges (OpenAI, 2026a).

Occupation moments make interaction deletion quantitative

The bounded construction retains a corresponding cell penalty $D_0{\sum }_i{n}_i^2-B_{0}N$ and a sufficiently small attraction row bound relative to $D_0$. Let $F_i=\mathbb{E}\exp (\beta D_0{n}_i^2/4)$. First finite-volume stability proves each $F_i$ is finite. Conditional on the particles outside cell $i$, its attraction to them is at most $n_i{G}_i$, where $G_i={\sum }_{j\ne i}{J}_{ij}{n}_j$.

Young’s inequality gives $n_i{G}_i\le D_0{n}_i^2/4+G_i^2/D_0$. Summing the remaining quadratic-decay series in $n_i$ yields

$$\mathbb{E}[\exp (\beta D_0{n}_i^2/4)\mid \text{outside}]\le C\exp (\beta G_i^2/D_0).$$

Weighted Cauchy–Schwarz bounds $G_i^2\le M{\sum }_j{J}_{ij}{n}_j^2$. Convexity of the exponential and concavity of a small power then give

$$F_{\max }\le C{F}_{\max }^{\eta },\eta =4M^2/D_0^2\le 1/16.$$

Hence $F_{\max }\le C^{1/(1-\eta )}$, independently of volume and all attraction ranges. In particular $\mathbb{E}(n_i{n}_j)$ is uniformly bounded. Interpolate between two interactions, differentiate log partition functions, and bound the absolute pairwise difference by a cell matrix. These moments control the derivative. This works when deleting positive and negative interactions between large boxes, because the full core inside each small cell remains intact.

The ideal attraction has an explicit active-density formula

Use the source’s normalized pressure $Q$ and canonical function $H_{\ast }$, for which $f=t{H}_{\ast }/\beta$. Adding ideal attraction mass $a$ acts by

$$T_{a}Q(\beta ,h)=\underset{u}{\sup }\left\{Q(\beta ,h+\beta au)-\frac{\beta a{u}^2}{2}\right\}.$$

Insert density duality and complete the square. The expression becomes a supremum in density $d\ge 0$ of $hd-H_{\ast }(\beta ,d)+\beta a{d}^2/2$, minus $\beta a(u-d{)}^2/2$. Thus active shifts are exactly maximizing densities. The quadratic occupancy bound keeps them uniformly bounded on compact parameter sets.

To approximate this operation by an actual range-$R$ kernel when the reference range is $R_0$, partition into boxes of side $s$. Deleting reference interactions across boxes costs $O(R_0/s)$. Replacing the new kernel by boxwise upper or lower values costs $O(s/R)$. The occupation bounds make both estimates uniform even when $a$ is very small. Choosing $s\asymp \sqrt{R_{0}R}$ gives pressure error $O(\sqrt{R_0/R})$ (OpenAI, 2026b).

Secants preserve separated slope groups

Suppose an ideal pressure $P$ has subgradients generated near three separated density-energy slopes, and its physical approximation $q$ satisfies $\Vert P-q\Vert \le \delta$. Add ideal attraction mass $a$ to $q$. At an active shifted point, the maximizing property gives a parabolic upper bound in the field direction. Testing increments of size proportional to $a$ confines nearby density slopes to one narrow band, selecting a single one of the three groups.

Inside a ball of radius proportional to $a$, test the supporting inequality in every direction. Passing from $P$ to $q$ costs at most $2\delta$ in each secant, so the slope error is $O(\delta /a)$. The attraction itself shifts an energy slope by $a{u}^2/2=O(a)$. Thus one step enlarges the groups by

$$O(\delta /a+a)$$

and shrinks the available parameter domain by $O(a)$. This division by $a$ explains why uniform pressure convergence without a rate relative to the next mass is insufficient.

For the singular-core paper, choose the next mass first, then the current range so large that its approximation error is at most the square of that next mass. The reference is fixed at each such choice; no uniform rate is needed. The resulting summable masses imply $r^3|\varphi (r)|\to 0$.

For the bounded paper use the explicit schedule

$$R_n=R_1^{2^{n-1}},a_1=A,a_n=R_n^{-1/16}(n\ge 2).$$

The range estimate gives ${\delta }_n=O(R_n^{-1/4})$, whereas $a_{n+1}=R_n^{-1/8}$. Therefore both $a_{n+1}$ and ${\delta }_n/a_{n+1}$ are summable and their sums become small as $R_1$ increases. Simultaneously ${\sum }_n{a}_n{R}_n^{1/32}<\infty$, which yields the stated $r^{-3-1/32}$ tail bound for the same fixed potential.

One pressure support ball gives a whole density interval

Minimize the ideal pressure minus the linear functional corresponding to the barycenter of the three initial slopes. The minimum stays inside the parameter domain. The barycenter lies outside every two-group convex hull, so all three groups must supply actual supports at that point. Their triangle contains a fixed ball. The iteration preserves that ball, not just three approximate function values.

Uniform convergence to the pressure of the final potential passes every supporting inequality to a common limiting parameter $({\beta }_c,h_c)$. For each density $\rho$ in a short interval, a horizontal slice of the ball contains two supports $(s_1,\rho )$ and $(s_2,\rho )$ with $s_1<s_2$. Partial convex duality gives

$$H_{\ast ,+}^{\prime }({\beta }_c,\rho )\le -s_2<-s_1\le H_{\ast ,-}^{\prime }({\beta }_c,\rho ).$$

Finiteness and concavity make these one-sided derivatives finite. Multiplication by $t/\beta$ preserves the strict downward jump: the derivative of the prefactor contributes the same term from both sides. Because all slices came from one support ball at one pressure parameter, the inverse temperature is common to the entire density interval.

See Also

• Family 215 treats phase behavior in lattice spin systems with a different microscopic state space.
• Family 234 uses thermodynamic pressure and variational limits for random quadratic interactions.

🏷️ Manuscripts and verification scope

• A continuum temperature singularity for a radial pair potential — pinned manuscript and companion source.
• A radial continuum phase transition with algebraic decay — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page describes both the divergent-core and bounded-continuous constructions, with a common inverse-temperature derivative jump on an open density interval. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A continuum temperature singularity for a radial pair potential.

🐻  OpenAI 2026b. A radial continuum phase transition with algebraic decay.