证明
Lemma 14 ([Local convex convergence]). Let O⊂^m be open and convex. A finite…Lemma 15 ([One-dimensional conjugacy]). Let u:→∪+∞ be proper, lower…观点
We give the details of the canonical limit first. Two finite-volume bounds control the density adjustments in its construction. A trial bound on bounded density intervals. Fix D_0<∞ and choose a grid spacing q∈(0,1/2) with q^ -3 >D_0+1 . For large L , at least D_0L^3 grid centers lie in Λ_L at distance q/8 from its boundary. For n≤D_0L^3 , put one particle in each of n distinct balls of radius q/8 around such centers. Every pair has distance at least 3q/4 , and each particle has at most Cq^ -3 neighbors at distance less than one. The core is bounded on [3q/4,1] , so the positive energy of these configurations is at most E_q n for a finite constant E_q . We need not count the negative energy. The n! assignments of the labeled particles to the balls have disjoint integration regions and cancel the canonical factorial. If v_q>0 is the volume of a ball, this gives Z_ L,n (β)≥v_q^n e^ -βE_q n . Consequently a_L(β,n)≤C_ D_0,β (0≤n≤D_0L^3, L sufficiently large). The constant is uniform over the physical class, and can be chosen uniformly for β in a compact subset of (0,∞) . A lower bound. There are at most (L/b+2)^3 cropped cells in Λ_L . Cauchy--Schwarz in and the inequality n!≥(n/e)^n therefore give, with d_L=n/L^3 , a_L(β,n) ≥(βκ)/(t(1+2b/L)^3)d_L^2 -(βB_0)/(t)d_L +(d_L d_L-d_L)/(t). Dropping the positive quadratic term also gives the same inequality with only its last two terms. In particular, Z_ L,n is positive and finite, and a_L is bounded above and below when the density stays in a fixed bounded interval and L is large. Convergence at a positive density. Fix β>0 and d>0 . The increasing infima I_j=_L(β,n): L≥j, |n/L^3-d|≤1/j converge to a finite number I . Select ℓ_j≥j and n_j in this set such that a_ ℓ_j (β,n_j)≤I_j+1/j . Then n_j/ℓ_j^3→d and a_ ℓ_j (β,n_j)→I . Every sequence with L→∞ and n/L^3→d has liminf at least I_j for each j , and hence at least I . To prove the reverse inequality, tile part of a large cube by k=(L)/(ℓ_j+2)^3 cubes of side ℓ_j , leaving gaps of width two. Every interaction between distinct cubes is nonpositive. For any prescribed counts m_1,…,m_k , restricting the integral to these cubes gives Z_ L,∑_i m_i (β) ≥∏_ i=1 ^k Z_ ℓ_j,m_i (β). Indeed there are (∑_i m_i)!/∏_i m_i! assignments of labels to the cubes, and these give exactly the factorials on the right. Start with n_j particles in every small cube. Define a higher count n_ hi =(4d+4)ℓ_j^3 and the diluted densities d_b=(n_j)/((ℓ_j+2)^3), d_ hi = n_ hi (ℓ_j+2)^3 . For large j , n_j>0 , n_ hi >n_j , and d_b→d , d_ hi →4d+4 . If a target count n exceeds kn_j , change some cubes to n_ hi ; if it is smaller, empty some cubes. One additional intermediate count realizes n exactly. For fixed sufficiently large j , this is possible for all large L with n/L^3→d . The limiting upper fraction of changed cubes is at most c_j= (d-d_b)_+ d_ hi -d_b , ((d_b-d)_+)/(d_b) , c_j⟶0. All new counts are bounded by (4d+4)ℓ_j^3 , so their values of a_ ℓ_j have a common upper bound by. The baseline values are bounded in absolute value. Applying , taking the large-volume limit with j fixed, and then letting j→∞ therefore gives _ L→∞; n/L^3→d a_L(β,n) ≤_ j→∞ ((ℓ_j)/(ℓ_j+2))^3 (a_ ℓ_j (β,n_j)+C c_j)=I. The single intermediate cube contributes zero in the first limit. This proves the all-sequence convergence for d>0 . Zero density and convexity. When n/L^3→0 , the linear lower bound derived above has limit zero. For an upper bound, fix a sufficiently large side ℓ and use separated cubes as in, with empty baseline. Fill some cubes with ℓ^3 particles and at most one with an intermediate count. Their number is o(L^3) , so gives a limiting upper bound zero. Thus H_φ(β,0)=0 . For convexity, take 0≤d_1<d_2 and a density strictly between them. For a large fixed ℓ , use counts d_1ℓ^3 and d_2ℓ^3 in the separated cubes. The desired density lies strictly between their diluted densities for all sufficiently large ℓ . Choose their proportions to attain the target count, with at most one intermediate cube. Inequality, followed first by L→∞ and then by ℓ→∞ , gives the convexity inequality for H_φ(β,·) . The endpoint cases of that inequality are immediate. The function is finite and convex on [0,∞) , hence continuous on (0,∞) . The linear lower bound gives lower semicontinuity at zero; since its value there is zero, extension by +∞ to negative densities is a closed convex function. For each fixed density, H\"older's inequality makes - Z_ L,n (β) concave in β . Passage to the limit gives finite concavity of H_φ(·,d) on (0,∞) . Finally gives . We have established the canonical limit with enough control over particle number to pass to the grand-canonical ensemble. In particular, the density variable in the next maximization will not require a separate approximation of the canonical free energy. Pressure and conjugacy. On a compact subset of Ω , implies that, for sufficiently large D and all large L , e^ thn Z_ L,n (β)≤e^ -c n^2/L^3 (n>DL^3) with c>0 independent of the potential in the class. The sum of these terms is exponentially small in L^3 , whereas the term n=0 is one. Among the remaining at most DL^3+1 terms, choose one of maximal size. Every subsequence has a further subsequence on which its density converges to some d∈[0,D] . The all-sequence canonical limit identifies the limiting exponent as hd-H_φ(β,d) . Conversely, the term n= dL^3 supplies this exponent as a lower bound for every d≥0 . This proves. The finite-volume logarithms are finite and jointly convex in (β,h) , by H\"older's inequality applied to their defining integrals and sums. The limit is finite by the density bound, and it is nonnegative because the empty configuration is allowed. Lemma 14 ([Local convex convergence]), the finite-dimensional convex convergence principle of , upgrades pointwise convergence to uniform convergence on compact subsets of the open convex set Ω . Finally, the closed convexity in density proved above permits the one-dimensional Fenchel biconjugacy identity, proved in Lemma 15 ([One-dimensional conjugacy]) and treated in , yielding .