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A CONTINUUM TEMPERATURE SINGULARITY FOR A RADIAL PAIR POTENTIAL
version 2 -- rewritten to be read by a student who knows calculus
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Contents
0. What is proved, and why it is not obvious
1. The potentials we allow, and the theorem
2. The core, the packing density, and stability
3. Thermodynamic limits: canonical free energy and pressure
4. The quadratic density transform
5. From finite range to mean field: the Kac limit
6. Three competing branches and the initial triangle of slopes
7. Keeping a corner: the persistence lemma
8. One potential, one ball of slopes, one interval of densities
A. Appendix: the convex analysis that is used
Notation used everywhere. |x| is the Euclidean length of x in R^3.
conv S is the convex hull of S. B(g,r) is the closed ball of radius r
about g. A function F of two variables is convex if
F(t u + (1-t) v) <= t F(u) + (1-t) F(v) for every 0 <= t <= 1, and
concave if the inequality is reversed.
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0. WHAT IS PROVED, AND WHY IT IS NOT OBVIOUS
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Put N identical particles in a cube Lambda = [-L/2,L/2]^3. Each pair of
particles interacts through a function Phi of the distance between them,
so the energy of a configuration (x_1,...,x_N) is a sum over pairs,
(0.1) U_N(x_1,...,x_N) = sum_{i<j} Phi(x_i - x_j).
At inverse temperature beta > 0 a configuration occurs with probability
proportional to e^{-beta U_N}. Everything thermodynamic is therefore
computed from the canonical partition function
(0.2) Z(L,N,beta) = (1/N!) * integral over Lambda^N of
e^{-beta U_N} dx_1 ... dx_N,
and the canonical free energy density at density rho > 0 is
(0.3) f(beta,rho) = - lim_{L -> oo}
log Z(L, floor(rho L^3), beta) / (beta L^3),
where floor(rho L^3) is the particle number that makes the density as
close to rho as it can be. Where f is differentiable in beta, its
derivative is minus the energy per particle. A corner of f is therefore
a jump of the energy at a fixed temperature and a fixed density, which
is the hallmark of a first order phase transition. This document proves
that one single radial pair potential can have such a corner at one
fixed temperature, and that the same temperature works throughout a
whole interval of densities.
Two words recur, so here they are before anything else.
Supporting vector. Let F be a convex function of the two variables
(beta,h), the first playing inverse temperature and the second a scaled
chemical potential. A vector v supports F at the point lambda if
(0.4) F(lambda') >= F(lambda) + v.(lambda' - lambda)
for every lambda' in the domain.
The set of all such v is written dF(lambda) and is called the
subdifferential of F at lambda. Geometrically, v is the slope of a
plane that touches the graph of F at lambda from below. If the graph is
smooth there, exactly one such plane exists, and v is the gradient. If
the graph has a corner, there are many, and the set dF(lambda) records
the corner completely. The whole proof is a device for keeping a
two dimensional ball of such vectors, all of them sitting at points with
the same density, but with different first coordinates.
Pressure and its conjugate. Instead of fixing the number of particles,
fix a parameter h and sum over all particle numbers with weight
e^{t h N}, for a fixed positive scale factor t. Dividing the logarithm
of that sum by t L^3 and letting L grow gives a convex function
Q(beta,h). Its Legendre transform in the h variable alone,
(0.5) H(beta,d) = sup over h of { h d - Q(beta,h) },
is the free energy as a function of the density d, up to the harmless
factor t. A supporting vector (v_beta,d) of Q at (beta,h) encodes both
a density d and a slope v_beta in the temperature direction. Keeping a
ball of such vectors with a fixed density coordinate d and a whole
range of first coordinates is exactly what produces the corner of f
as a function of beta at that density.
Why the result is not obvious. Three things fight each other.
(a) Attraction lowers the energy of a dense cluster, so it wants to
create a jump in density.
(b) A potential with a hard core can hold only finitely many particles
inside a fixed small region, so the energy cannot be pushed to
minus infinity, and stability holds. Stability is what makes the
free energy finite and the thermodynamic limit controlled.
(c) The mean field limit that produces a clean corner is the limit in
which the attraction becomes infinitely weak and infinitely long
ranged. A single finite range potential is never of that form,
and passing from the mean field expression to a genuine finite
range potential can smooth the corner away.
Point (c) is where the work is. The construction answers it by choosing
the strength of each attraction before the range that realises it, with
each new strength smaller than the accuracy already achieved. Then the
corner survives infinitely many approximations. Section 7 is the
technical heart: it shows that adding a small attraction, together with
an error small compared with the square of that attraction, moves the
corner only slightly and keeps all three groups of slopes apart.
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1. THE POTENTIALS WE ALLOW, AND THE THEOREM
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Definition 1.1 (admissible potential). Write Phi(x) = phi(|x|). We call
Phi admissible when phi is Borel measurable, phi(0) = +oo, phi is finite
and locally bounded on (0,oo), and
(i) phi(r) -> +oo as r -> 0;
(ii) there are 0 < r_1 < r_2 and a > 0 with phi(r) <= -a for every
r in [r_1,r_2];
(iii) integral from R to oo of r^2 |phi(r)| dr < oo for some finite R;
(iv) stability: there is B < oo with U_N(x_1,...,x_N) >= -B N for
every N and every configuration.
Conditions (i)-(iii) say that the potential repels violently at zero
distance, is honestly attractive somewhere, and dies fast enough at
infinity for the integral of the interaction to be finite. Condition
(iv) is the physical requirement that N particles cannot gain more than
a fixed amount of energy per particle. Potentials are functions of the
difference of positions only: no external field and no many-body terms.
Note that (iii) collects |phi|, not phi, so it allows an attractive tail
decaying like r^{-3-eps} for any eps > 0, and also slower tails such as
r^{-3} (log r)^{-2}. The theorem below produces a tail of size
o(r^{-3}) without committing to any power law.
Theorem 1.2 (main result). There exist an admissible radial pair
potential Phi(x) = phi(|x|) on R^3, a nonempty open interval I inside
(0,oo), and one beta_c with 1/2 < beta_c < 3/2, with all of the
following properties.
(a) phi(r) = o(r^{-3}) as r -> oo.
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