Solution to a problem arising from Mayer s theory of cluster integrals Olivier Bernardi, C.R.M. Barcelona
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1 Solution to a problem arising from Mayer s theory of cluster integrals Olivier Bernardi, C.R.M. Barcelona October 2006, 57 th Seminaire Lotharingien de Combinatoire Olivier Bernardi p.1/37
2 Content of the talk Mayer s theory of cluster integrals Pressure = Generating function of weighted connected graphs. Olivier Bernardi p.2/37
3 Content of the talk Mayer s theory of cluster integrals Pressure = Generating function of weighted connected graphs. Hard-core continuum gas Pressure = Generating function of Cayley trees. Olivier Bernardi p.2/37
4 Content of the talk Mayer s theory of cluster integrals Pressure = Generating function of weighted connected graphs. Hard-core continuum gas Pressure = Generating function of Cayley trees. Why? [Labelle, Leroux, Ducharme : SLC 54] Olivier Bernardi p.2/37
5 Content of the talk Mayer s theory of cluster integrals Pressure = Generating function of weighted connected graphs. Hard-core continuum gas Pressure = Generating function of Cayley trees. Why? [Labelle, Leroux, Ducharme : SLC 54] Combinatorial explaination Olivier Bernardi p.2/37
6 Mayer s theory of cluster integrals Olivier Bernardi p.3/37
7 Statistical physics Gas of n particules in a box Ω. Ω x 2 x 3 x 1 Olivier Bernardi p.4/37
8 Statistical physics Gas of n particules in a box Ω. Ω x 2 x 3 x 1 The energy of a configuration x 1,..., x n is ɛ(x 1,..., x n ) = i µ(x i ) + i<j φ(x i, x j ). Olivier Bernardi p.4/37
9 Statistical physics Gas of n particules in a box Ω. Ω x 2 x 3 x 1 The energy of a configuration x 1,..., x n is ɛ(x 1,..., x n ) = i µ(x i ) + i<j φ(x i, x j ). No external field : µ(x i ) = µ. Olivier Bernardi p.4/37
10 Statistical physics Ω x 2 x 3 x 1 Energy : ɛ(x 1,..., x n ) = nµ + i<j φ(x i, x j ). Olivier Bernardi p.5/37
11 Statistical physics Ω x 2 x 3 x 1 Energy : ɛ(x 1,..., x n ) = nµ + i<j φ(x i, x j ). The partition function is Z(Ω, T, n) = 1 exp n! Ω n ( ɛ(x ) 1,..., x n ) dx 1..dx n. kt Olivier Bernardi p.5/37
12 Statistical physics Ω x 2 x 3 x 1 Energy : ɛ(x 1,..., x n ) = nµ + i<j φ(x i, x j ). The partition function is Z(Ω, T, n) = 1 exp n! Ω n = 1 λ n n! Ω n ( ɛ(x 1,..., x n ) kt i<j exp ( φ(x i, x j ) kt ) dx 1..dx n ) dx 1..dx n. Olivier Bernardi p.5/37
13 Example Hard particules in Ω = {1,..., q}. λ = 1 and φ(x, y) = + if x = y 0 otherwise. Olivier Bernardi p.6/37
14 Example Hard particules in Ω = {1,..., q}. λ = 1 and φ(x, y) = + if x = y 0 otherwise. The partition function : Z(Ω, T ) 1 exp n! Ω n i<j ( φ(x ) i, x j ) kt Olivier Bernardi p.6/37
15 Example Hard particules in Ω = {1,..., q}. λ = 1 and φ(x, y) = + if x = y 0 otherwise. The partition function : Z(Ω, T ) 1 exp n! Ω n i<j ( φ(x ) i, x j ) kt = ( ) q n Olivier Bernardi p.6/37
16 Mayer s Idea (1940) exp ( φ(x ) i, x j ) kt = 1 + f(x i, x j ). Olivier Bernardi p.7/37
17 Mayer s Idea (1940) exp i<j ( φ(x ) i, x j ) kt = i<j 1+f(x i, x j ) = G K n (i,j) G f(x i, x j ). Olivier Bernardi p.7/37
18 Mayer s Idea (1940) exp i<j ( φ(x ) i, x j ) kt = i<j 1+f(x i, x j ) = G K n (i,j) G f(x i, x j ). Partition function can be written as a sum over graphs : Z(Ω, T, n) 1 ( exp φ(x ) i, x j ) dx λ n 1..dx n n! Ω kt n i<j = 1 W (G), λ n n! G K n where W (G) = Ω n (i,j) G is the Mayer s weight of G. f(x i, x j )dx 1..dx n Olivier Bernardi p.7/37
19 For those familiar with the Tutte Polynomial Mayer s tranformation is the analogue (for general partition function) of the correspondence Partition function of the Potts model Tutte polynomial (coloring expansion) (subgraph expansion) [Fortuin & Kasteleyn 72] Olivier Bernardi p.8/37
20 Example Hard particules in Ω = {1,..., q}. φ(x, y) = + if x = y f(x, y) = 1 if x = y 0 otherwise. 0 otherwise. Olivier Bernardi p.9/37
21 Example Hard particules in Ω = {1,..., q}. φ(x, y) = + if x = y f(x, y) = 1 if x = y 0 otherwise. 0 otherwise. Mayer s weight of G : W (G) = Ω n (i,j) G f(x i, x j ) = ( 1) e(g)qc(g). Olivier Bernardi p.9/37
22 Example Hard particules in Ω = {1,..., q}. φ(x, y) = + if x = y f(x, y) = 1 if x = y 0 otherwise. 0 otherwise. Mayer s weight of G : W (G) = Ω n (i,j) G f(x i, x j ) = ( 1) e(g)qc(g). Mayer s correspondence W (G) = n!z(ω, n) shows : G K n ( ) q ( 1) e(g) q c(g) = n! = q(q 1)... (q n + 1). n G K n Olivier Bernardi p.9/37
23 Allowing any number of particules The grand canonical partition function is Z gr (Ω, T, z) = n Z(Ω, T, n)λ n z n. In terms of Mayer s weights : Z gr (Ω, T, z) = n ( 1 λ n n! G K n W (G) ) λ n z n = G W (G)z G. G! Olivier Bernardi p.10/37
24 Pressure The pressure of the system is given by P (Ω, T, z) = kt Ω log (Z gr(ω, T, z)). Olivier Bernardi p.11/37
25 Pressure The pressure of the system is given by P (Ω, T, z) = kt Ω log (Z gr(ω, T, z)). Since Mayers weights are multiplicative P (Ω, T, z) = kt Ω log (Z gr(ω, T, z)) = kt Ω G connected W (G)z G. G! Olivier Bernardi p.11/37
26 Example Hard particules in Ω = {1,..., q}. Grand canonical partition function : Z gr (Ω, T, z) = Z(Ω, T, n)z n = n n ( ) q z n = (1 + z) q. n Olivier Bernardi p.12/37
27 Example Hard particules in Ω = {1,..., q}. Grand canonical partition function : Z gr (Ω, T, z) = Z(Ω, T, n)z n = n n ( ) q z n = (1 + z) q. n Pressure : P (Ω, T, z) = kt Ω log (Z gr(ω, T, z)) = kt log(1 + z). Olivier Bernardi p.12/37
28 Example Mayer s weights : W (G) = ( 1) e(g) q c(g). Pressure : P (Ω, T, z) = kt Ω G connected W (G)z G G! = kt G connected ( 1) e(g) z G. G! Olivier Bernardi p.13/37
29 Example Mayer s weights : W (G) = ( 1) e(g) q c(g). Pressure : P (Ω, T, z) = kt Ω G connected W (G)z G G! = kt G connected ( 1) e(g) z G. G! Comparing the two expressions of the pressure yields : ( 1) e(g) z G G connected G! = log(1 + z). Olivier Bernardi p.13/37
30 Example Mayer s weights : W (G) = ( 1) e(g) q c(g). Pressure : P (Ω, T, z) = kt Ω G connected W (G)z G G! = kt G connected ( 1) e(g) z G. G! Comparing the two expressions of the pressure yields : ( 1) e(g) z G G connected G! = log(1 + z). In other words : ( 1) e(g) = ( 1) n 1 (n 1)!. G K n connected Olivier Bernardi p.13/37
31 How did we get there? Z(Ω, T, z) Mayer G W (G)z G G! log log A = B Olivier Bernardi p.14/37
32 A killing involution G K n connected ( 1) e(g) = ( 1) n 1 (n 1)! Olivier Bernardi p.15/37
33 A killing involution G K n connected ( 1) e(g) = ( 1) n 1 (n 1)! We define an involution Φ on the set of connected graphs : - Order the edges of K n lexicographicaly. - Define E (G) = {e = (i, j) / i and j are connected by G >e }, Φ(G) = G if E (G) = G min(e (G)) otherwise. Olivier Bernardi p.15/37
34 A killing involution G K n connected ( 1) e(g) = ( 1) n 1 (n 1)! We define an involution Φ on the set of connected graphs : - Order the edges of K n lexicographicaly. - Define E (G) = {e = (i, j) / i and j are connected by G >e }, Φ(G) = G if E (G) = G min(e (G)) otherwise. Prop [B.] : The only remaining graphs are the increasing spanning trees. (Known to be in bijection with the permutations of {1,.., n 1}.) Olivier Bernardi p.15/37
35 Increasing trees Olivier Bernardi p.16/37
36 For those familiar with the Tutte Polynomial The sum of the Mayer s weight correspond to the evaluations of T Kn (1, 0). This is the number of internal spanning trees. Subgraph expansion Spanning tree expansion Olivier Bernardi p.17/37
37 Hard-core continuum gas Olivier Bernardi p.18/37
38 Hard-core continuum gas Hard particules in Ω = [0, q]. Ω x 2 x 1 x 3 φ(x, y) = + if x y < 1 f(x, y) = 1 if x y < 1 0 otherwise. 0 otherwise. Olivier Bernardi p.19/37
39 Hard-core continuum gas Hard particules in Ω = [0, q]. Ω x 2 x 1 x 3 φ(x, y) = + if x y < 1 f(x, y) = 1 if x y < 1 0 otherwise. 0 otherwise. W (G) = Ω n P (Ω, T, z) = kt Ω (i,j) G f(x i, x j )dx 1..dx n. G connected W (G)z G. G! Olivier Bernardi p.19/37
40 Thermodynamical limit ( Ω ) x 2 x 1 x 3 P (T, z) lim P (Ω, T, z) = kt Ω G connected W (G)z G G! where, W (G) lim Ω W Ω (G) Ω = n 1 ; x 1 =0 (i,j) G f(x i, x j )dx 2..dx n. Olivier Bernardi p.20/37
41 Mayer s weight for the hard-core gas f(x, y) = 1 if x y < 1 0 otherwise. W (G) = n 1 ; x 1 =0 (i,j) G f(x i, x j )dx 2..dx n. Olivier Bernardi p.21/37
42 Mayer s weight for the hard-core gas f(x, y) = 1 if x y < 1 0 otherwise. W (G) = n 1 ; x 1 =0 (i,j) G f(x i, x j )dx 2..dx n. P (T, z) = kt G connected W (G)z G G! Olivier Bernardi p.21/37
43 Mayer s diagram for the hard-core gas Z(T, z) Mayer G W (G)z G G! log log ( 1) n 1 n n 1 = G K n W (G) Cayley trees!? [Labelle, Leroux, Ducharme : SLC 54] Olivier Bernardi p.22/37
44 Slicing W (G) [Lass] W (G) = n 1 ; x 1 =0 f(x i, x j )dx 2..dx n, (i,j) G = ( 1) e(g) Volume(Π G ), where Π G R n 1 is the polytope x i x j 1. (i,j) G Olivier Bernardi p.23/37
45 Slicing W (G) [Lass] W (G) = n 1 ; x 1 =0 f(x i, x j )dx 2..dx n, (i,j) G = ( 1) e(g) Volume(Π G ), where Π G R n 1 is the polytope x i x j 1. (i,j) G Example : x 3 G : x 3 Π G : x 2 x 1 x 2 Olivier Bernardi p.23/37
46 Slicing W (G) [Lass] W (G) = n 1 ; x 1 =0 f(x i, x j )dx 2..dx n, (i,j) G = ( 1) e(g) Volume(Π G ), where Π G R n 1 is the polytope x i x j 1. (i,j) G Example : x 3 G : x 3 Π G : x 2 x 1 x 2 Olivier Bernardi p.23/37
47 Slicing W (G) [Lass] Fractional representation x i = h(x i ) + ɛ(x i ). 3 2 h(x i ) = 1 x i 0 1 ɛ = 0 ɛ(x i ) ɛ = 1 Olivier Bernardi p.24/37
48 Slicing W (G) [Lass] Fractional representation x i = h(x i ) + ɛ(x i ). x i x j < x j 0 1 x i x j 1 x i Olivier Bernardi p.24/37
49 Slicing W (G) [Lass] Fractional representation x i = h(x i ) + ɛ(x i ). Prop [Lass] : (x 2,.., x n ) Π G? only depends on the integer parts h(x 2 ),.., h(x n ) and the order of the fractional parts ɛ(x 2 ),.., ɛ(x n ). Olivier Bernardi p.24/37
50 Slicing W (G) [Lass] Fractional representation x i = h(x i ) + ɛ(x i ). Prop [Lass] : (x 2,.., x n ) Π G? only depends on the integer parts h(x 2 ),.., h(x n ) and the order of the fractional parts ɛ(x 2 ),.., ɛ(x n ). h 1 = 0 3 h 2 = 1 h 3 = 0 2 h 4 = 2 h 5 = 1 1 h 6 = 0 0 0=ɛ 1 <ɛ 4 <ɛ 6 <ɛ 2 <ɛ 5 <ɛ 3 1 x 4 x 2 x 6 x 1 x 3 x 5 Olivier Bernardi p.24/37
51 Slicing W (G) [Lass] Fractional representation x i = h(x i ) + ɛ(x i ). Prop [Lass] : (x 2,.., x n ) Π G? only depends on the integer parts h(x 2 ),.., h(x n ) and the order of the fractional parts ɛ(x 2 ),.., ɛ(x n ). 3 x 6 x 5 2 x 4 x 1 x x 2 x 2 x 3 1 x 6 x 1 x 3 x 5 Olivier Bernardi p.24/37
52 Slicing W (G) [Lass] Fractional representation x i = h(x i ) + ɛ(x i ). Prop [Lass] : (x 2,.., x n ) Π G? only depends on the integer parts h(x 2 ),.., h(x n ) and the order of the fractional parts ɛ(x 2 ),.., ɛ(x n ). Each subpolytope defined by h 2,.., h n and an order on 1 ɛ(x 2 ),.., ɛ(x n ) has volume (n 1)!. Olivier Bernardi p.24/37
53 Counting labelled schemes x 3 G : x 1 x 2 Π G : x 3 x 2 x 1 x 1 x 2 x 1 x 2 x 3 x 3 x 2 x 3 x 1 x 1 x 1 x 3 x 2 x 2 x 3 x 2 Each labelled scheme has weight ( 1)e(G) (n 1)!. Olivier Bernardi p.25/37
54 Rearanging the sum G Kn connected W (G) = G Kn connected ( 1) e(g) (n 1)! #{S labelled scheme containing G} Olivier Bernardi p.26/37
55 Rearanging the sum G Kn connected W (G) = G Kn = connected ( 1) e(g) (n 1)! S labelled scheme #{S labelled scheme containing G} 1 (n 1)! G contained in S ( 1) e(g) Olivier Bernardi p.26/37
56 Rearanging the sum G Kn connected W (G) = G Kn = connected ( 1) e(g) (n 1)! S labelled scheme = S scheme #{S labelled scheme containing G} 1 (n 1)! G contained in S G contained in S ( 1) e(g) ( 1) e(g) Olivier Bernardi p.26/37
57 Rearranging the sum G contained in S ( 1) e(g) Olivier Bernardi p.27/37
58 A killing involution We define an involution Φ on the set of connected graphs contained in S : - Order the edges of K n lexicographicaly. - Define E (G) = {e = (i, j) / i and j are connected by G >e }, Φ(G) = G if E (G) = G min(e (G)) otherwise. Olivier Bernardi p.28/37
59 A killing involution We define an involution Φ on the set of connected graphs contained in S : - Order the edges of K n lexicographicaly. - Define E (G) = {e = (i, j) / i and j are connected by G >e }, Φ(G) = G if E (G) = G min(e (G)) otherwise. Proposition [B.] : The only remaining graphs are the increasing spanning trees. Corrolary : G contained in S ( 1) e(g) = ( 1) n 1 #{increasing tree on S}. Olivier Bernardi p.28/37
60 A killing involution Olivier Bernardi p.29/37
61 A killing involution Olivier Bernardi p.29/37
62 Bijection with Cayley trees Theorem [B.] : {increasing tree} are in bijection with S scheme rooted Cayley trees. Olivier Bernardi p.30/37
63 Bijection with Cayley trees Theorem [B.] : {increasing tree} are in bijection with S scheme rooted Cayley trees. Olivier Bernardi p.30/37
64 Bijection with Cayley trees Theorem [B.] : {increasing tree} are in bijection with S scheme rooted Cayley trees Olivier Bernardi p.30/37
65 Bijection with Cayley trees Theorem [B.] : {increasing tree} are in bijection with S scheme rooted Cayley trees Olivier Bernardi p.30/37
66 Bijection with Cayley trees Corollary [B.] : W (G) = G Kn connected S scheme = ( 1) n 1 G contained in S S scheme = ( 1) n 1 n n 1. ( 1) e(g) #{increasing tree on S} Olivier Bernardi p.31/37
67 Bijection with Cayley trees Olivier Bernardi p.32/37
68 Concluding remarks Olivier Bernardi p.33/37
69 Mayer s transformation Producing graph weights Z(Ω, T, z) Mayer G W (G)z G G! Olivier Bernardi p.34/37
70 Mayer s transformation Producing graph weights Producing nasty identities Z(Ω, T, z) Mayer G W (G)z G G! log log A = B Olivier Bernardi p.34/37
71 Discrete hard-core gas (colorings) Z(Ω, T, z) Mayer G W (G)z G G! log log ( 1) n 1 (n 1)! = G Kn connected ( 1) e(g) Olivier Bernardi p.35/37
72 Discrete hard-core gas (colorings) Z(Ω, T, z) log Potts model Subgraph expansion Mayer G W (G)z G G! log ( 1) n 1 (n 1)! = ( 1) e(g) Spanning tree expansion G Kn connected Subgraph expansion Olivier Bernardi p.35/37
73 Continuous hard-core gas Z(T, z) Mayer G W (G)z G G! log log ( 1) n 1 n n 1 = G K n W (G) Olivier Bernardi p.36/37
74 Thanks. Olivier Bernardi p.37/37
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