Thermodynamic Limit and Extensivity

A tablespoon of water and a swimming pool of water have the same temperature, the same density, the same boiling point. Double the amount and you double the energy, the entropy, the volume — but the temperature and pressure do not budge. This split of thermodynamic quantities into two clean families — extensive ones that scale with system size and intensive ones that do not — is so familiar it feels like a definition. It is not. It is a theorem, and it holds only in a special mathematical limit and only for a special class of forces. The thermodynamic limit is $N\to\infty$ , $V\to\infty$ at fixed density $n=N/V$ (and fixed energy density). In this limit the free energy per particle $f=F/N$ converges to a finite, size-independent function of the intensive variables alone, and the awkward boundary and surface contributions — which scale like $N^{2/3}$ , one power of $N$ smaller than the bulk — wash out. Extensivity is the statement that $F$ grows linearly with $N$ , so that per-particle quantities settle to well-defined bulk values. Only then can one meaningfully say "the entropy density of water" without specifying how much water. The catch is that this requires short-range interactions. If forces are long-range — gravity being the canonical villain — the energy of a system grows faster than $N$ (each particle feels every other, so $U\sim N^2$ ), the per-particle energy diverges, and extensivity fails. Self-gravitating systems are non-extensive: they have no proper thermodynamic limit, exhibit negative heat capacity, and their microcanonical and canonical descriptions disagree. This article makes the thermodynamic limit precise, proves that extensivity is equivalent to short-range forces plus proper indistinguishability counting, and shows where — spectacularly — it breaks.

Key takeaways

The thermodynamic limit takes $N,V\to\infty$ at fixed density $n=N/V$ ; in it, the free energy per particle $f=F/N$ converges to a finite intensive function.
Extensive quantities ( $U,S,V,N,F$ ) scale linearly with size; intensive ones ( $T,p,\mu,n$ ) are size-independent. This is a theorem, not a definition.
Corrections to $f/N$ are surface terms of order $N^{-1/3}$ (boundary $\sim N^{2/3}$ vs bulk $\sim N$ ) that vanish in the limit.
Extensivity requires short-range interactions and the $1/N!$ indistinguishability factor; either failure breaks it.
Long-range forces break it: self-gravity gives $U\sim N^2$ (non-extensive), no thermodynamic limit, negative heat capacity, and ensemble inequivalence.
Key idea: extensivity — the linear scaling of thermodynamic potentials with system size — is guaranteed only in the thermodynamic limit and only for short-range forces; it is the condition that makes intensive bulk quantities well-defined.

The core insight

In the thermodynamic limit ( $N,V\to\infty$ , $n=N/V$ fixed), the free energy per particle converges to a finite intensive function with surface corrections of order $N^{-1/3}$ . Extensivity — linear scaling of $F,U,S$ with $N$ — is a theorem that holds for short-range interactions with proper indistinguishability counting, and fails for long-range forces like gravity.

Where this lives today

The thermodynamic limit is the silent assumption under nearly every equilibrium calculation, and its failures are where much current research lives. Nanoscience is the physics of systems too small for the limit: nanoparticles melt at lower temperatures than bulk, catalysts change activity with size, and quantum dots have size-tunable properties — all consequences of the $N^{-1/3}$ surface term that bulk thermodynamics discards. Engineers designing nanomaterials must keep exactly the corrections the thermodynamic limit throws away. At the opposite extreme, the non-extensivity of long-range systems is an active frontier. Self-gravitating systems (star clusters, dark-matter halos, the early universe) have negative specific heat and inequivalent ensembles, forcing a rethink of statistical mechanics for gravity; plasmas, two-dimensional turbulence, and some spin models with long-range couplings show the same anomalies. This has spurred both rigorous work on when a thermodynamic limit exists and speculative generalizations (Tsallis's non-extensive statistics) for systems where it does not. And in the theory of phase transitions, the thermodynamic limit is indispensable in a precise sense: a genuine phase transition — a true non-analyticity in the free energy — can only occur as $N\to\infty$ . Any finite system has a perfectly smooth partition function; the sharp singularities of boiling, freezing, and criticality are strictly infinite- $N$ phenomena, which is why the thermodynamic limit is not a convenience but a necessity for the physics of transitions.

The physical picture

Break a large system into blocks. If the blocks barely interact across their shared faces — because the forces are short-ranged, felt only by near neighbours — then the total energy is (bulk energy of each block) plus a small correction living on the interfaces. The interface area scales as $N^{2/3}$ while the bulk scales as $N$ , so as the system grows the interfaces become negligible: energy, and every extensive quantity, becomes additive. This additivity is exactly what lets us write $F=Nf(n,T)$ with $f$ a function of intensive variables only. Extensivity is additivity in the large- $N$ limit.

Figure

Bulk versus surface: the interior scales as N (volume), the boundary as N^{2/3} (area). The surface fraction ~N^{-1/3} vanishes as the system grows.

The intensive quantities are the slopes of this linear scaling. Temperature is $\partial U/\partial S$ , pressure is $-\partial U/\partial V$ , chemical potential is $\partial U/\partial N$ — all ratios of extensive changes, so they are size-independent. In the thermodynamic limit these slopes settle to fixed bulk values, and the fundamental relation becomes $U=TS-pV+\mu N$ (the Euler relation), the mathematical face of extensivity: energy is a first-order homogeneous function of its extensive arguments. Now the dramatic failure. Gravity is unscreened and long-range: every particle attracts every other, so the potential energy of $N$ particles in a fixed region grows like the number of pairs, $\sim N^2$ , not $\sim N$ . The energy per particle then grows without bound as the system enlarges, and there is no finite $f=F/N$ — no thermodynamic limit at all. The punchline plot below contrasts the two scalings: short-range energy climbs linearly (extensive), while gravitational energy curves upward as $N^2$ (non-extensive), which is why a star cluster can heat up when it loses energy and defies ordinary thermodynamics.

Figure

Energy scaling: short-range forces give U ∝ N (extensive, straight line); long-range gravity gives U ∝ N² (non-extensive, curving up) — no thermodynamic limit.

The core equations

The thermodynamic limit is defined by letting the size grow with the intensive ratios held fixed:

N\to\infty,\quad V\to\infty,\quad \frac{N}{V}=n=\text{const},\quad \frac{U}{N}=u=\text{const} \qquad (1)

where: $n$ — number density; $u$ — energy per particle. The claim is that in this limit the free energy density $f(n,T)=\lim_{N\to\infty}F/N$ exists and is finite — a nontrivial theorem, proved rigorously by van Hove, Ruelle, Fisher, and Lieb for short-range stable potentials. Extensivity is the linear scaling of the potentials, equivalently first-order homogeneity:

S(\lambda U,\lambda V,\lambda N)=\lambda\,S(U,V,N)\quad\Longleftrightarrow\quad U=TS-pV+\mu N \qquad (2)

where: $\lambda>0$ — a scaling factor; the right side is the Euler relation, obtained by differentiating the homogeneity condition with respect to $\lambda$ and setting $\lambda=1$ . Homogeneity of degree one is extensivity. The corrections to the bulk value are surface terms, one power of $N$ down:

F(N) = N f_\infty + a\,N^{2/3} + \cdots \quad\Rightarrow\quad \frac{F}{N} = f_\infty + a\,N^{-1/3}+\cdots \xrightarrow{N\to\infty} f_\infty \qquad (3)

where: $f_\infty$ — the bulk free-energy density; $a$ — a surface coefficient (surface tension–like); the correction scales as $N^{-1/3}$ because the boundary area is $\sim N^{2/3}$ while the bulk is $\sim N$ . This is why a small droplet has measurable surface effects and an ocean does not. Failure of extensivity is diagnosed by the energy scaling. For a long-range potential $\sim 1/r^s$ in $d$ dimensions, extensivity requires $s>d$ ; gravity ( $s=1$ , $d=3$ ) violates it:

U_{\text{grav}} \sim -\frac{Gm^2 N(N-1)}{2\langle r\rangle} \propto N^2 \qquad (4)

where: the $N(N-1)/2$ counts pairs; because every particle interacts with every other, $U\propto N^2$ and $U/N\propto N\to\infty$ — no finite energy density, no thermodynamic limit. Short-range forces, by contrast, give each particle a bounded number of neighbours, so $U\propto N$ .

Derivation AExtensivity ⇔ short-range forces, with the surface correction of order N^{−1/3}

The goal is to show that a system with short-range interactions has a free energy that scales linearly with size up to a surface term of order $N^{2/3}$ , so that the free-energy density converges in the thermodynamic limit — and to see explicitly where long-range forces break this. The strategy is to partition the system into equal cells, add up their free energies, and estimate the interaction across the cell boundaries.

Step 1. Divide a large system of $N$ particles into $M$ identical cubic cells, each with $N/M$ particles. Because the interaction range $\xi$ is finite (short-range), a particle interacts only with others within a shell of thickness $\xi$ around it. The total free energy is the sum of the cell free energies plus interactions confined to the cell boundaries. Why: short range means the coupling between two cells is limited to the particles within $\xi$ of the shared face — a surface effect, not a bulk one.

Step 2. Count the two contributions. The bulk term scales as the number of cells times the per-cell free energy: $\propto N$ . The boundary term scales as the total interface area times $\xi$ : the area of a system of $N$ particles at fixed density is $\propto N^{2/3}$ , so the boundary free energy is $\propto N^{2/3}$ . Why: volume-like quantities scale as $N$ and area-like quantities as $N^{2/3}$ in three dimensions — the fundamental surface-to-volume ratio.

Step 3. Assemble: $F(N)=N f_\infty + a N^{2/3}+\cdots$ , so $F/N = f_\infty + a N^{-1/3}+\cdots$ . As $N\to\infty$ the surface term vanishes and the density converges to the intensive bulk value $f_\infty(n,T)$ . Extensivity ( $F\propto N$ ) holds up to sub-leading corrections. Why: dividing the $N^{2/3}$ surface term by $N$ gives $N^{-1/3}\to0$ ; the bulk dominates and per-particle quantities settle down.

Step 4. Break it with long-range forces. If the potential falls off slower than $1/r^d$ , a particle interacts with a number of others that grows with the system size, and the "boundary" is no longer a surface: the interaction energy scales as $N^2$ (all pairs), so $F/N\propto N\to\infty$ . No finite density exists. Why: unscreened long-range forces make every particle feel the whole system, converting the surface-only coupling into a bulk-defeating $N^2$ term — extensivity fails. *Verified: the Sackur–Tetrode entropy is shown homogeneous of degree one, $S(\lambda E,\lambda V,\lambda N)=\lambda S$ ; a model $F(N)=Nf_\infty+aN^{2/3}$ has $F/N\to f_\infty$ with relative surface correction $\sim N^{-1/3}$ ; and the gravitational $U\propto N^2$ gives $U(\lambda N)/U(N)\to\lambda^2$ (non-extensive), all in verify_aq-sm-thermodynamic-limit-extensivity.py.*

Go deeper · Derivation B · 1 minThe free-energy density converges with a computable rate — the owned result

Derivation BThe free-energy density converges with a computable rate — the owned result

This is the article's owned structural result: not only does the free energy per particle converge in the thermodynamic limit, but its approach to the bulk value is controlled — a $1/N^{1/3}$ surface correction — and the limiting density is the intensive Sackur–Tetrode free energy, a fully explicit function of $n$ and $T$ . The goal is to exhibit the convergence quantitatively for the ideal gas and confirm the limit is intensive. The strategy is to compute $f=F/N$ from the extensive entropy and show it depends only on the density and temperature.

Step 1. From the Sackur–Tetrode free energy $F=-Nk_BT[\ln(V/N\lambda_T^3)+1]$ , form the density $f=F/N=-k_BT[\ln(1/(n\lambda_T^3))+1]$ with $n=N/V$ . This depends only on $n$ and $T$ — not on $N$ or $V$ separately. Why: the $1/N!$ correction turned $V$ into $V/N=1/n$ , so the free-energy density is manifestly a function of intensive variables alone, the hallmark of a proper thermodynamic limit.

Step 2. Add a boundary. A finite droplet has a surface free energy $\sim \gamma A \sim \gamma N^{2/3}$ , so $f(N)=f_\infty + \gamma' N^{-1/3}$ . The correction is positive (surface tension costs free energy) and shrinks as the cube root of the size. Why: the surface term is one power of $N$ below the bulk, so its per-particle contribution falls as $N^{-1/3}$ — slow, but convergent.

Step 3. Estimate the crossover. Surface effects matter when $\gamma N^{2/3}\sim N f_\infty$ , i.e. $N\lesssim(\gamma'/f_\infty)^3$ . For everyday samples ( $N\sim10^{23}$ ) the correction is $\sim10^{-8}$ ; for a nanodroplet ( $N\sim10^3$ ) it is $\sim10\%$ . Why: this is why bulk thermodynamics is superb for macroscopic matter and why nanoscience must keep surface terms — the $N^{-1/3}$ law quantifies exactly when.

Step 4. Confirm intensivity of the limit. The limiting $f_\infty(n,T)$ yields intensive $p=n^2\partial f/\partial n$ , $s=-\partial f/\partial T$ , etc., all size-independent — the bulk thermodynamics. Why: once the density converges, every thermodynamic derivative is a derivative of an intensive function, so all intensive quantities are well-defined bulk properties. *Verified: $f(n,T)$ from the extensive free energy is intensive (no $N,V$ separately); the surface-corrected model converges to $f_\infty$ with relative error $\sim N^{-1/3}$ , monotonically, in the verify script.*

Limiting cases

The thermodynamic limit organizes several regimes:

Case	Result	Meaning
Macroscopic sample	$N^{-1/3}\sim10^{-8}$	Surface corrections utterly negligible; bulk thermodynamics is exact.
Nanoscale	$N^{-1/3}\sim0.1$	Surface terms matter; melting points, phase behaviour become size-dependent.
Long-range (gravity)	$U\propto N^2$	No thermodynamic limit; non-extensive; negative heat capacity possible.
Finite system	fluctuations $\sim N^{-1/2}$	Relative fluctuations vanish as $N\to\infty$ , justifying sharp thermodynamics.
Phase transition	sharp only as $N\to\infty$	Non-analyticities (true singularities) exist strictly in the thermodynamic limit.

Common trap“Extensivity is just the definition of extensive quantities — there is nothing to prove”

This confuses a definition with a theorem. Yes, we call quantities that scale with size "extensive" — but whether the free energy actually does scale linearly with $N$ , so that a well-defined bulk density exists, is a genuine mathematical result that can fail. It was proved (van Hove 1949, Ruelle, Fisher, Lieb) only under specific conditions: the interaction must be stable (energy bounded below by $-BN$ ) and tempered (short-range enough, faster than $1/r^d$ ). Drop either and extensivity collapses: unstable potentials give energy $\propto N^{5/3}$ (implosion), long-range gravity gives $\propto N^2$ . Self-gravitating systems are the standard example of matter with no thermodynamic limit — they have negative specific heat, inequivalent ensembles, and no meaningful "temperature of a star cluster" in the ordinary sense. The mundane extensivity of a glass of water is a nontrivial fact about the short range of intermolecular forces.

Worked examples

1 · How big must a droplet be for surface effects to disappear? (a real crossover)

A water droplet has bulk free-energy density $f_\infty$ and a surface free energy $\gamma A$ with $\gamma\approx0.072\ \mathrm{J/m^2}$ . Estimate the fraction of the free energy that is "surface" for (a) a $10\,\mathrm{nm}$ droplet and (b) a $1\,\mathrm{mm}$ droplet, using the number of molecules and the $N^{-1/3}$ scaling.

Step 1. Molecules in a $10\,\mathrm{nm}$ droplet. Volume $=\tfrac43\pi(5\times10^{-9})^3=5.2\times10^{-25}\,\mathrm{m^3}$ ; water has $\sim3.3\times10^{28}$ molecules/m³, so $N\approx5.2\times10^{-25}\times3.3\times10^{28}=\mathbf{1.7\times10^4}$ molecules.

Step 2. Surface fraction scales as $N^{-1/3}$ . For $N=1.7\times10^4$ : $N^{-1/3}=(1.7\times10^4)^{-1/3}=1/25.7=\mathbf{0.039}$ , i.e. roughly $4\%$ of the free energy is surface — non-negligible. A $10\,\mathrm{nm}$ droplet has measurably depressed melting and boiling points.

Step 3. A $1\,\mathrm{mm}$ droplet: volume $=\tfrac43\pi(5\times10^{-4})^3=5.2\times10^{-10}\,\mathrm{m^3}$ , so $N\approx1.7\times10^{19}$ . Then $N^{-1/3}=(1.7\times10^{19})^{-1/3}=1/2.6\times10^{6}=\mathbf{3.8\times10^{-7}}$ — surface effects are one part in a few million, utterly negligible.

Step 4. The crossover. Between the two, surface effects fall by a factor $\sim10^5$ — a spectacular illustration of the $N^{-1/3}$ law. This is exactly why bulk thermodynamics works flawlessly for beakers and why nanoscience must always keep the surface term.

2 · Why a self-gravitating gas has no thermodynamic limit (exam level)

Consider $N$ particles of mass $m$ in a fixed region of size $R$ , interacting gravitationally. (a) Show the potential energy scales as $N^2$ . (b) Explain why the energy per particle diverges as $N\to\infty$ at fixed $R$ . (c) State one thermodynamic anomaly this produces.

Step 1. Pair-counting: each of the $N(N-1)/2$ pairs contributes $\sim -Gm^2/R$ , so $U_{\text{grav}}\sim -\frac{Gm^2 N(N-1)}{2R}\propto \mathbf{N^2}$ at fixed $R$ . Every particle interacts with every other — no screening.

Step 2. Energy per particle: $U/N \propto N^2/N = N \to\infty$ . There is no finite energy density, so $f=F/N$ does not converge — the thermodynamic limit does not exist for gravity at fixed size.

Step 3. (Even letting $R$ grow, the standard limit $N,V\to\infty$ at fixed $n$ fails because the gravitational energy still outruns $N$ .) The scaling test: $U(\lambda N)/U(N)\to\lambda^2\ne\lambda$ , so $U$ is homogeneous of degree two, not one — non-extensive.

Step 4. Anomaly: self-gravitating systems have negative heat capacity — add energy and they expand and cool (a star that gains energy puffs up and its surface temperature drops). This is impossible for an extensive system in canonical equilibrium, and it signals that gravity's microcanonical and canonical ensembles are inequivalent. Gravity is the textbook case of thermodynamics without a thermodynamic limit.

Origins · 1 minDavid Ruelle and how this was discovered

David Ruelle

1935– · Bures-sur-Yvette, France

Ruelle, a Belgian-French mathematical physicist at the IHÉS, is one of the architects of the rigorous theory of the thermodynamic limit. Building on van Hove's 1949 result, Ruelle (with Fisher, and later Lieb and Lebowitz) proved under precise conditions — that the interaction be stable and tempered (short-range) — that the free energy per particle converges to a well-defined, convex limit as the system grows, putting extensivity on a firm mathematical footing rather than a physical assumption. His 1969 monograph Statistical Mechanics: Rigorous Results is the standard reference for these existence theorems, which delineate exactly which interactions admit a thermodynamic limit and which (like gravity) do not. Ruelle later became a founder of the modern theory of chaos and dynamical systems, coining (with Takens) the notion of a "strange attractor." His work made explicit the conditions under which the everyday extensivity of matter is guaranteed.

History

1949

— van Hove proves the existence of the thermodynamic limit for a class of short-range interactions.

1963

— Fisher and Ruelle establish general existence theorems for stable, tempered potentials.

1969

— Ruelle's Statistical Mechanics: Rigorous Results codifies the theory of the thermodynamic limit.

1972

— Lieb and Lebowitz prove the thermodynamic limit exists even for charged systems (with overall neutrality and screening).

1968

— Lynden-Bell analyzes the negative heat capacity of self-gravitating systems, a canonical non-extensive example.

Going further

References. Ruelle, Statistical Mechanics: Rigorous Results (1969) — the existence theorems · Pathria & Beale, Statistical Mechanics, 3rd ed., §1.1, §3.1 · Kardar, Statistical Physics of Particles, §4.1 · Sethna, Statistical Mechanics, Ch. 6 · Landau & Lifshitz, Statistical Physics Part 1, §2, §24 · Lynden-Bell, "Negative specific heat in astronomy" (1999) for the gravitational case.

Provenance. The definition and the existence of the thermodynamic limit are standard (Ruelle 1969; Pathria §1.1). A verifier should reproduce: first-order homogeneity of the Sackur–Tetrode entropy $S(\lambda E,\lambda V,\lambda N)=\lambda S$ ; convergence $F/N\to f_\infty$ with $N^{-1/3}$ surface correction; intensivity of the ideal-gas free-energy density $f(n,T)$ ; and the gravitational $U\propto N^2$ non-extensivity ( $U(\lambda N)/U(N)\to\lambda^2$ ). Numerical instances ( $10\,\mathrm{nm}$ droplet $N\approx1.7\times10^4$ , surface fraction $\approx4\%$ ; $1\,\mathrm{mm}$ droplet surface fraction $\approx4\times10^{-7}$ ) are recomputed in verify_aq-sm-thermodynamic-limit-extensivity.py.

Builds on: Thermodynamic Potentials and the Fundamental Relation · Second Law of Thermodynamics and Entropy · First Law of Thermodynamics and Internal Energy. Leads to: Euler Relation and Gibbs–Duhem Equation · Legendre Transforms · Helmholtz Free Energy, Enthalpy, and Gibbs Free Energy · Equivalence of Ensembles.

Why isn't extensivity automatic? Doesn't more stuff just mean proportionally more of everything?

It feels automatic because we live among short-range forces, where it happens to be true. But "more stuff = proportionally more energy" requires that adding a particle costs a bounded amount of energy regardless of how big the system already is — which holds only if each particle interacts with a bounded number of neighbours. Short-range forces guarantee that (a molecule feels only its immediate surroundings), so energy, entropy, and free energy all scale linearly with $N$ . If forces are long-range, adding a particle makes it interact with all the existing ones, so the cost grows with system size and the total energy climbs faster than $N$ . Then doubling the amount more than doubles the energy, and there is no well-defined "energy per particle." Extensivity is a consequence of the short range of the forces, not a logical necessity.

What exactly goes wrong for self-gravitating systems?

Several linked things. First, the energy is non-extensive: $U\propto N^2$ because gravity is unscreened and every pair contributes, so there is no finite energy density and no ordinary thermodynamic limit. Second, self-gravitating systems have negative heat capacity — remove energy from a star and it contracts and gets hotter (the gravothermal effect that powers stellar evolution). Third, because of this, the microcanonical (fixed energy) and canonical (fixed temperature) ensembles give different answers — ensemble equivalence, which relies on extensivity, breaks down; a self-gravitating system can even have no stable canonical equilibrium at all (the "gravothermal catastrophe"). None of this violates physics — it is correct statistical mechanics for long-range forces — but it means the comfortable machinery built on extensivity (well-defined intensive variables, ensemble equivalence, positive specific heat) simply does not apply to gravity.

If real systems are always finite, why do we take N to infinity at all?

Because the infinite limit is where thermodynamics becomes clean and universal, and real macroscopic systems are so close to it that the difference is invisible. For $N\sim10^{23}$ , the surface corrections are $\sim N^{-1/3}\sim10^{-8}$ and the relative fluctuations are $\sim N^{-1/2}\sim10^{-12}$ — far below any measurement. So the thermodynamic limit is not a fiction; it is an extraordinarily accurate idealization for laboratory-scale matter. More deeply, certain phenomena only exist in the limit: a true phase transition is a non-analyticity of the free energy, and any finite system has a perfectly smooth (analytic) partition function, so sharp boiling points and critical singularities are strictly infinite- $N$ features. Taking $N\to\infty$ is how we get the sharp, universal laws; keeping $N$ finite is what you do when the system is genuinely small (nanoscience) or when you care about the fluctuations themselves.

How is extensivity related to the Euler relation U = TS − pV + μN?

They are the same statement. Extensivity says the internal energy $U(S,V,N)$ is a homogeneous function of degree one in its extensive arguments: scale $S,V,N$ all by $\lambda$ and $U$ scales by $\lambda$ too. Euler's theorem on homogeneous functions then says that such a function equals the sum of each variable times its partial derivative: $U = S(\partial U/\partial S) + V(\partial U/\partial V) + N(\partial U/\partial N) = TS - pV + \mu N$ , since $\partial U/\partial S=T$ , $\partial U/\partial V=-p$ , $\partial U/\partial N=\mu$ . So the Euler relation is the mathematical fingerprint of extensivity, and differentiating it gives the Gibbs–Duhem equation, which constrains how the intensive variables can vary together. If extensivity fails (long-range forces), the Euler relation fails too, and these tidy identities among intensive variables no longer hold.

Do surface effects always make things less stable, or can they matter in surprising ways?

Surface free energy is generally positive (creating an interface costs energy — that is what surface tension is), so small systems tend to be less stable in ways that lower melting and boiling points and drive coalescence (small droplets merge to reduce total surface). But the $N^{-1/3}$ surface term shows up in many guises, not all destabilizing. It underlies the Kelvin equation (vapour pressure depends on droplet curvature), the Gibbs–Thomson effect (melting point depends on size), nucleation barriers (a small nucleus is mostly surface, which is why supercooling and superheating are possible), and the enhanced catalytic activity of nanoparticles (a large surface-to-volume ratio exposes more reactive sites). In each case the physics is the competition between a bulk term $\propto N$ and a surface term $\propto N^{2/3}$ , exactly the crossover the thermodynamic limit erases. Nanoscience is, in a real sense, the science of keeping the term that thermodynamics discards.

Does the thermodynamic limit require the interactions to be attractive, repulsive, or neither?

Neither specifically — what it requires is that the interaction be stable and tempered. Stability means the total energy is bounded below by $-BN$ for some constant $B$ (the system cannot collapse to $-\infty$ energy); a purely attractive $1/r$ potential fails this because particles could pile up and release unlimited energy, whereas a short-range repulsive core plus short-range attraction (like real molecules) is stable. Tempered means the interaction falls off faster than $1/r^d$ at large distance (short-range), so distant parts of the system barely interact. Under these two conditions — proved by Ruelle, Fisher, Lieb — the free-energy density converges and is convex, regardless of the detailed sign of the force. Ordinary matter (Lennard-Jones molecules, screened Coulomb interactions in neutral plasmas) satisfies both; gravity fails temperedness, and a naked attractive $1/r$ fails stability.