An Experimental Mathematics Derivation of Onsager's famous Explicit formula for the Free Energy of the Ising Model By Manuel Kauers' Computer and Doron Zeilberger's Computer [Shalosh B. EKhad] Abstract: One of the most celebrated results in Statistical Physics is Lars Onsager's tour-de-force explicit formula for the free energy of the 2-dimensional Ising model with zero magnetic field. (Physical Reviews v.65 (1944), p. 117ff). His proof, as well as all the subsequence proofs, are extremely complicated and ad hoc. In this note we will rederive his famous formula AB INITIO. While we use an experimental mathematics approach, based on guessing, that is non-rigorous, it is nevertheless absolutely certain, and should satisfy physicists who do not share the mathematicians' dogmatic insistence on rigor. Recall that many people got Physics Nobel prizes for physical theories based on non-rigorous mathematics. Let -psi/(k*T), the Onsager free energy, be denoted by Ons(v), where , v=J/(k*T), T is the absolute temperature , and J is the coupling constant. By the so-called (elementary!) combinatorial approach (due to B.L. van der Waerden, 1941), that is described, inter alia, in Colin Thompson's classic monograph "Mathematical Statistical Physics", MacMillan, 1972, [Note that this method predates Onsager's 1944 proof] /infinity \ | ----- | | \ {1} r| Ons(v) = ln(2) + 2 ln(cosh(v)) + | ) a[r] w | | / | | ----- | \ r = 1 / where w=tanh(v). See Eq. (1.18) in Chapter 6 of Thompson's book (p. 150) . {1} Here for each, r, a[r] , is the coefficient of N in the polynomial expression , n[r](N), for the number of so-called lattice polygons with r edges that can be placed in a rectangular torus with N lattice points whose sides are larger than r. Lattice polygons are subgraphs of the rectangular lattice where every vertex has an even number of neighbors (i.e. 0,2, or 4) . By using the (finite!) transfer matrices for width <=7, one can guess explicit expressions for the first, 10, polynomials n[2] = 0 n[4] = N n[6] = 2 N N (9 + N) n[8] = --------- 2 n[10] = 2 N (6 + N) N (7 + N) (32 + N) n[12] = ------------------ 6 2 n[14] = N (N + 21 N + 130) 3 2 N (N + 102 N + 1715 N + 11766) n[16] = -------------------------------- 24 3 2 N (N + 49 N + 776 N + 5876) n[18] = ----------------------------- 3 4 3 2 N (N + 210 N + 7415 N + 118830 N + 980904) n[20] = --------------------------------------------- 120 of course n[odd] are all zero. {1} Taking the coefficients of N, we get that the first , 10, terms of , a[2 r] , are {1} {1} {1} {1} {1} {1} {1} {1} {1} {1} a[2] = 0, a[4] = 1, a[6] = 2, a[8] = 9/2, a[10] = 12, a[12] = 112/3, a[14] = 130, a[16] = 1961/4, a[18] = 5876/3, a[20] = 40871/5 By the Kramres-Wannier duality argument, given a combinatorial explanation by van der Waerden in 1941, Ons(v) satisfies the FUNCTIONAL EQUATION Ons(v*)=Ons(v)-log(sinh(2v)) where v*=arctanh(exp(-2*v)) . This is equivalent to Ons(v*)-log(2cosh(2v*))=Ons(v)-log(2cosh(2v)) . Since sinh(2 v) z = ---------- 2 cosh(2 v) is invariant under v<->v*, it is natural to consider the function Ons(v) - ln(2 cosh(2 v)) and later express it in terms of z, that is the most natural variable. This function equals : /infinity \ | ----- | | \ {1} r| -ln(2 cosh(2 v)) + ln(2) + 2 ln(cosh(v)) + | ) a[r] w | | / | | ----- | \ r = 1 / where w=tanh(v) Since 2 -ln(2 cosh(2 v)) + ln(2) + 2 ln(cosh(v)) = -ln(w + 1) this function, entirely in terms of w, is /infinity \ | ----- | 2 | \ {1} (2 r)| -ln(w + 1) + | ) a[2 r] w | | / | | ----- | \ r = 1 / 20 whose beginning, up to, and including, the power, w , is 4 6 8 10 12 14 16 18 20 2 3 w 5 w 19 w 59 w 75 w 909 w 3923 w 17627 w 81743 w 22 -w + ---- + ---- + ----- + ------ + ------ + ------- + -------- + --------- + --------- + O(w ) 2 3 4 5 2 7 8 9 10 changing to the natural z variable, that in terms of w is 2 w (1 - w) (w + 1) z = ------------------- 2 2 (w + 1) we get that the beginning Taylor series is 2 4 25 6 1225 8 3969 10 17787 12 184041 14 41409225 16 147744025 18 2133423721 20 22 - 1/4 z - 9/32 z - -- z - ---- z - ---- z - ----- z - ------ z - -------- z - --------- z - ---------- z + O(z ) 48 1024 1280 2048 7168 524288 589824 2621440 The first, 10, coefficients are -9 -25 -1225 -3969 -17787 -184041 -41409225 -147744025 -2133423721 [-1/4, --, ---, -----, -----, ------, -------, ---------, ----------, -----------] 32 48 1024 1280 2048 7168 524288 589824 2621440 factoring suggests that there is a closed form 2 2 2 2 4 2 2 2 2 2 2 4 2 2 2 2 2 2 2 1 (3) (5) (5) (7) (3) (7) (3) (7) (11) (3) (11) (13) (3) (5) (11) (13) (5) (11) (13) (17) [- -----, - -----, - ----------, - -----------, - -----------, - -----------------, - -------------------, - -------------------------, - --------------------------, 2 5 4 10 8 11 10 19 16 2 (2) (2) (2) (3) (2) (2) (5) (2) (2) (7) (2) (2) (3) 2 2 2 2 (11) (13) (17) (19) - ---------------------------] 19 (2) (5) The sequence of ratios is 50 147 324 605 1014 1575 2312 3249 [9/8, --, ---, ---, ---, ----, ----, ----, ----] 27 64 125 216 343 512 729 1000 that obviously fits 2 r (2 r + 1) ------------ 3 (r + 1) Hence we discovered the recurrence for the coefficients: 2 b[2 r] r (2 r + 1) b[2 r + 2] = -------------------, , 3 (r + 1) that immediately implies the closed-form expression for, b[2 r], r>=1 2 binomial(2 r, r) b[2 r] = - ----------------- (r + 1) r 4 and indeed the first few terms are -9 -25 -1225 -3969 -17787 -184041 -41409225 -147744025 -2133423721 -1/4, --, ---, -----, -----, ------, -------, ---------, ----------, ----------- 32 48 1024 1280 2048 7168 524288 589824 2621440 Hence we discovered Theorem: Let G(z) be infinity ----- / 2 (2 r)\ \ | binomial(2 r, r) z | ) |- ------------------------| / | (r + 1) | ----- \ r 4 / r = 1 Then psi sinh(2 v) - --- = ln(2 cosh(2 v)) + G(----------) k T 2 cosh(2 v) ---------------------------- this ends our ab initio, non-rigorous, but absolutely certain, derivation of Onsager's famous formula, by purely computational means and elementary combinatorial reasonings due to van der Waerden and Kramers and Wannier.