Engineering a temperature transition with radial pair interactions

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Family 228 · Probability and statistical mechanics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

Can identical particles moving freely in three-dimensional space undergo a mathematically demonstrable temperature transition when their energy is just a sum of radial pair interactions?

This resembles the everyday question of why matter changes phase, but the mathematical task is demanding even as an existence problem. The particles are not attached to lattice sites, their number grows with the volume, and an attractive interaction must avoid catastrophic collapse. One also needs a genuine singularity for one fixed interaction, rather than a corner that appears only in an additional mean-field limit.

Family 228 contains two constructions. Both claim a stable radial pair potential whose canonical free energy has a strict derivative jump at one common positive inverse temperature for every density in an open interval. The potentials differ: one has a divergent repulsive core and an integrable tail; the other has a bounded continuous core and an explicit power-law tail bound.

These are engineered examples. Neither manuscript identifies the phases as ordinary liquid, vapor, or solid, and neither asserts the result for the standard Lennard–Jones potential.

2. Background and essential definitions

Energy, stability, and the thermodynamic limit

For particle positions $x_1,\ldots,x_N\in\mathbb R^3$, a radial pair potential $\varphi$ gives energy

$$U_N(x_1,\ldots,x_N)=\sum_{1\leq i<j\leq N}\varphi(|x_i-x_j|).$$

Only the distance between each pair matters. There is no fixed spatial grid in this physical interaction and no separate three-body or four-body term.

A potential is stable if some finite constant $B$ satisfies

$$U_N\geq-BN$$

for every configuration and every particle number. The lower bound grows only linearly with $N$, preventing attraction from producing energy of order $-N^2$ through collapse. Stability is a statement about total energy; it does not require the pair potential itself to be nonnegative.

Let $\Lambda_L$ be a cube of volume $L^3$. At inverse temperature $\beta>0$, the canonical partition function is

$$Z_{L,N}(\beta)=\frac1{N!}\int_{\Lambda_L^N}e^{-\beta U_N}\,dx_1\cdots dx_N.$$

The factor $N!$ accounts for identical particles. The source uses position integration with a fixed, temperature-independent measure. No kinetic or temperature-dependent normalization is included.

At density $\rho>0$, take $N=\lfloor\rho L^3\rfloor$. The free-energy density is

$$f(\beta,\rho)=-\lim_{L\to\infty}\frac1{\beta L^3}\log Z_{L,\lfloor\rho L^3\rfloor}(\beta).$$

The volume limit is what permits a singularity. For a large finite system, a sharp but smooth crossover can resemble a transition without giving the claimed nondifferentiability.

What derivative jumps mean

The manuscripts claim finite one-sided derivatives satisfying

$$\partial_\beta^-f(\beta_c,\rho)> \partial_\beta^+f(\beta_c,\rho).$$

This is a downward jump as inverse temperature increases. The free energy is continuous at the transition, but its slope changes abruptly.

Where differentiability is available, $\partial_\beta(\beta f)$ is the configurational energy per volume. Thus a corner in the stated normalization carries an energy discontinuity. The claim is made at fixed density, not merely by varying a chemical potential and observing two densities.

3. Development history and previous partial results

Continuum phase transitions form part of the program highlighted in Simon’s problems in mathematical physics. The precise interaction class matters: a theorem for several species, many-body forces, or an altered spatial interaction may address an important model without settling every radial single-species formulation.

Ruelle’s 1971 theorem established a transition in the two-species Widom–Rowlinson system. Israel’s convex constructions also produced coexistence for hard-core continuum particles with radial pair perturbations. A hard core excludes a whole positive interval of pair distances, unlike M1’s potential, which is finite at every positive separation.

Lebowitz, Mazel, and Presutti proved continuum liquid–vapor transitions for a model with attractive pair interactions and repulsive four-body interactions. The many-body repulsion is a substantive feature of that model.

A recent theorem of He, Jauslin, Lebowitz, and Peled treats a finite-range modified Kac pair interaction and distinct-density coexistence. Its interaction uses a fixed partition into boxes. The current manuscripts emphasize that their final physical potentials depend only on particle separation, even though boxes are used to estimate them.

The main long-range tool has a classical origin. Lebowitz and Penrose related weak, spread-out attraction to a mean-field free energy followed by convexification. Extending this idea to one fixed potential requires care: a sequence of smooth free energies can converge to a nonsmooth one even when no member has a transition.

Likewise, two states of different density at a fixed chemical potential do not automatically yield a temperature corner at one prescribed density. Family 228 seeks the stronger supporting-slope geometry needed for that canonical conclusion.

4. What the new manuscripts claim

A divergent core with an integrable tail

M1 claims a radial Borel potential $\varphi$, finite and locally bounded at every positive distance, with $\varphi(0)=+\infty$ and

$$\varphi(r)\longrightarrow+\infty\quad(r\downarrow0).$$

It has a genuine attractive interval: for some $0<r_1<r_2$ and $a>0$, one has $\varphi(r)\leq-a$ throughout $[r_1,r_2]$. It is stable, and its tail satisfies

$$\int_R^\infty r^2|\varphi(r)|\,dr<\infty, \qquad \varphi(r)=o(r^{-3}).$$

The first condition is absolute tail integrability in three dimensions. The second says that $r^3\varphi(r)\to0$. Neither statement supplies a fixed positive power margin beyond $r^{-3}$.

For this potential, the free-cube canonical limit is claimed to exist and be finite for every $\beta>0$ and every $\rho>0$. There is a nonempty open density interval $I$ and one $\beta_c\in(1/2,3/2)$ at which the strict downward derivative jump occurs for all $\rho\in I$.

A bounded continuous core with a power-law margin

M2 constructs a different stable radial potential, bounded and continuous even at zero, satisfying

$$|\varphi(r)|\leq Cr^{-3-1/32}\qquad(r\geq1).$$

For an open interval of positive densities, its canonical free energy is finite at every positive inverse temperature and has the same kind of derivative jump at a common $\beta_c\in[7/8,9/8]$.

The interval is centered at $5p/3$, where $p$ is the limiting maximal number of mutually distance-at-least-one points per unit volume. The numerical value of this packing constant is unnecessary for the construction. The temperature bounds use the energy units selected by the construction, rather than naming a temperature for a particular material.

These two results must not be merged into a single potential having every listed property. M1 retains divergence at the origin but proves only its integrable-tail conclusion. M2 proves a fixed algebraic decay margin with a bounded core. Each has its own density interval and transition point.

5. Why the advance matters

Importance: 4/5. The claimed constructions would establish a genuine canonical temperature singularity for single-species continuum particles with ordinary radial pair interactions. That is a major existence result in a setting where obtaining mathematically controlled phase transitions has required careful model design.

The common transition temperature across an entire density interval is stronger than separately finding a singularity at unrelated densities and temperatures. It arises from a two-dimensional collection of competing pressure slopes.

The bounded-core result also gives an explicit decay exponent beyond the integrability threshold. That extra margin restricts how the infinitely many attraction scales may be assembled and requires quantitative estimates.

The rating reflects the conceptual advance while preserving its engineered character. The claims do not classify the phases of a standard molecular potential or explain a particular observed fluid transition.

6. The main idea

Small clusters prevent uncontrolled collapse

The short-range reference interaction is strongly repulsive except for a very narrow low-energy shell at a tiny positive distance. Four particles can sit at the vertices of a tetrahedron with every pair distance in that shell. Five points in three-dimensional space cannot all have nearly the same pair distance within the selected tolerance.

This geometric restriction yields a quadratic penalty for large particle populations in small cubes. It is a concrete form of superstability: beyond a linear correction, the energy controls the sum of squared local particle counts. The bound survives the attractive terms added later and supplies thermodynamic limits and occupation estimates.

At a suitable construction scale, three configurations compete: near-vacuum, separated particles, and four-particle clusters. They generate three groups of slopes of a pressure depending on inverse temperature and a field conjugate to density.

Add attraction without losing the corner

A spread-out attraction has the form

$$-aR^{-3}w(x/R),$$

where $w$ is a nonnegative radial kernel of integral one. Its total mass is proportional to $a$, while increasing $R$ makes it weaker and longer-ranged. A Kac comparison relates its pressure to an ideal transformation containing a quadratic density reward.

The manuscripts add a summable sequence of such attractions at successively larger ranges. The difficult point is to preserve actual supporting slopes at each finite stage. Uniform approximation alone could round a corner. The construction therefore keeps three separated slope groups and arranges errors to be small compared with the next attraction mass.

M2 gives a quantitative pressure error of order $\sqrt{R_0/R}$, where $R_0$ is the range of the previously constructed reference. A sufficiently rapid range schedule and suitably shrinking masses make the losses summable while ensuring the required power-law tail bound.

Recover the fixed-density temperature jump

For a convex pressure $Q(\beta,h)$, a supporting vector $(e,\rho)$ describes an affine plane lying below the pressure and touching it at the chosen parameter point. The second coordinate is density; the first tracks the energy-related slope.

The construction preserves an entire two-dimensional ball of supporting vectors at one limiting pressure point. A horizontal slice of that ball fixes $\rho$ but leaves two different first coordinates. Partial convex duality then transfers those two supports into distinct one-sided canonical slopes. The ball supplies a whole interval of such densities, all at the same $\beta_c$.

The final interaction is fixed before taking the thermodynamic limit. Its many scales are ingredients of one potential, not an instruction to send the attraction range to infinity together with the volume.

7. Scope, limitations, and open questions

The potentials are infinite-range constructions with tailored short-distance shells and summable attractions. The theorem is not a finite-range radial-potential result and is not a statement for the ordinary 12–6 Lennard–Jones interaction.

M1 explicitly does not establish $|\varphi(r)|\leq Cr^{-3-\epsilon}$ for any fixed $\epsilon>0$. M2 supplies such a margin but changes the core. Historical problem formulations requiring both kinds of hypotheses cannot be declared solved by combining conclusions about different potentials.

The claimed singularity does not identify conventional liquid, vapor, or crystalline phases. Nor does it classify all Gibbs states or supply a phase diagram at every density. Those are separate physical and mathematical questions.

Natural extensions would impose simpler interaction shapes, faster or finite-range decay, or stronger geometric descriptions of the coexisting states. The present contribution is the stated existence and slope-jump construction, reported without independent proof verification.

8. Sources and further reading

Simon’s problem collection gives the historical motivation. Ruelle, Lebowitz–Penrose, Lebowitz–Mazel–Presutti, and He–Jauslin–Lebowitz–Peled provide the primary continuum precedents linked above.

M1 and M2 below should be read as complementary constructions with different core and tail assumptions. Their introductions state those distinctions explicitly and explain the supporting-slope route from pressure coexistence to a canonical temperature singularity.

Source manuscripts

• M1. OpenAI, A continuum temperature singularity for a radial pair potential (September 24, 2026).
• M2. OpenAI, A radial continuum phase transition with algebraic decay (September 24, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.