Solution to a problem arising from Mayers theory of cluster - PowerPoint PPT Presentation

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 CRM, Barcelona Olivier Bernardi – p.1/37

Content of the talk Mayer’s theory of cluster integrals Pressure = Generating function of weighted connected graphs. CRM, Barcelona ▽ Olivier Bernardi – p.2/37

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. CRM, Barcelona ▽ Olivier Bernardi – p.2/37

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] CRM, Barcelona ▽ Olivier Bernardi – p.2/37

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 CRM, Barcelona Olivier Bernardi – p.2/37

Mayer’s theory of cluster integrals CRM, Barcelona Olivier Bernardi – p.3/37

Statistical physics Gas of n particules in a box Ω . x 2 Ω x 3 x 1 CRM, Barcelona ▽ Olivier Bernardi – p.4/37

Statistical physics Gas of n particules in a box Ω . x 2 Ω x 3 x 1 The energy of a configuration x 1 , . . . , x n is � � ǫ ( x 1 , . . . , x n ) = µ ( x i ) + φ ( x i , x j ) . i i<j CRM, Barcelona ▽ Olivier Bernardi – p.4/37

Statistical physics Gas of n particules in a box Ω . x 2 Ω x 3 x 1 The energy of a configuration x 1 , . . . , x n is � � ǫ ( x 1 , . . . , x n ) = µ ( x i ) + φ ( x i , x j ) . i i<j No external field : µ ( x i ) = µ . CRM, Barcelona Olivier Bernardi – p.4/37

Statistical physics x 2 Ω x 3 x 1 Energy : ǫ ( x 1 , . . . , x n ) = nµ + � i<j φ ( x i , x j ) . CRM, Barcelona ▽ Olivier Bernardi – p.5/37

Statistical physics x 2 Ω x 3 x 1 Energy : ǫ ( x 1 , . . . , x n ) = nµ + � i<j φ ( x i , x j ) . The partition function is � � 1 �� − ǫ ( x 1 , . . . , x n ) Z (Ω , T, n ) = Ω n exp dx 1 ..dx n . n ! kT CRM, Barcelona ▽ Olivier Bernardi – p.5/37

Statistical physics x 2 Ω x 3 x 1 Energy : ǫ ( x 1 , . . . , x n ) = nµ + � i<j φ ( x i , x j ) . The partition function is � � 1 �� − ǫ ( x 1 , . . . , x n ) Z (Ω , T, n ) = Ω n exp dx 1 ..dx n n ! kT � � 1 �� − φ ( x i , x j ) � = exp dx 1 ..dx n . λ n n ! kT Ω n i<j CRM, Barcelona Olivier Bernardi – p.5/37

Example Hard particules in Ω = { 1 , . . . , q } . + ∞ λ = 1 and φ ( x, y ) = if x = y 0 otherwise . CRM, Barcelona ▽ Olivier Bernardi – p.6/37

Example Hard particules in Ω = { 1 , . . . , q } . + ∞ λ = 1 and φ ( x, y ) = if x = y 0 otherwise . The partition function : � � 1 − φ ( x i , x j ) � � Z (Ω , T ) ≡ exp n ! kT Ω n i<j CRM, Barcelona ▽ Olivier Bernardi – p.6/37

Example Hard particules in Ω = { 1 , . . . , q } . + ∞ λ = 1 and φ ( x, y ) = if x = y 0 otherwise . The partition function : � � � q � 1 − φ ( x i , x j ) � � Z (Ω , T ) ≡ exp = n ! kT n Ω n i<j CRM, Barcelona Olivier Bernardi – p.6/37

Mayer’s Idea (1940) � − φ ( x i , x j ) � exp = 1 + f ( x i , x j ) . kT CRM, Barcelona ▽ Olivier Bernardi – p.7/37

Mayer’s Idea (1940) � − φ ( x i , x j ) � � � � � exp = 1+ f ( x i , x j ) = f ( x i , x j ) . kT G ⊆ K n i<j i<j ( i,j ) ∈ G CRM, Barcelona ▽ Olivier Bernardi – p.7/37

Mayer’s Idea (1940) � − φ ( x i , x j ) � � � � � exp = 1+ f ( x i , x j ) = f ( x i , x j ) . kT G ⊆ K n i<j i<j ( i,j ) ∈ G ⇒ Partition function can be written as a sum over graphs : � � 1 �� − φ ( x i , x j ) � ≡ Z (Ω , T, n ) exp dx 1 ..dx n λ n n ! kT Ω n i<j 1 � = W ( G ) , λ n n ! G ⊆ K n �� � where W ( G ) = f ( x i , x j ) dx 1 ..dx n Ω n ( i,j ) ∈ G is the Mayer’s weight of G . CRM, Barcelona Olivier Bernardi – p.7/37

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] CRM, Barcelona Olivier Bernardi – p.8/37

Example Hard particules in Ω = { 1 , . . . , q } . + ∞ − 1 φ ( x, y ) = if x = y f ( x, y ) = if x = y 0 otherwise . 0 otherwise . CRM, Barcelona ▽ Olivier Bernardi – p.9/37

Example Hard particules in Ω = { 1 , . . . , q } . + ∞ − 1 φ ( x, y ) = if x = y f ( x, y ) = if x = y 0 otherwise . 0 otherwise . Mayer’s weight of G : f ( x i , x j ) = ( − 1) e ( G ) q c ( G ) . � � W ( G ) = Ω n ( i,j ) ∈ G CRM, Barcelona ▽ Olivier Bernardi – p.9/37

Example Hard particules in Ω = { 1 , . . . , q } . + ∞ − 1 φ ( x, y ) = if x = y f ( x, y ) = if x = y 0 otherwise . 0 otherwise . Mayer’s weight of G : f ( x i , x j ) = ( − 1) e ( G ) q c ( G ) . � � W ( G ) = Ω n ( i,j ) ∈ 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 CRM, Barcelona Olivier Bernardi – p.9/37

Allowing any number of particules The grand canonical partition function is � Z (Ω , T, n ) λ n z n . Z gr (Ω , T, z ) = n In terms of Mayer’s weights : � � W ( G ) z | G | 1 λ n z n = � � � Z gr (Ω , T, z ) = W ( G ) . λ n n ! | G | ! n G ⊆ K n G CRM, Barcelona Olivier Bernardi – p.10/37

Pressure The pressure of the system is given by P (Ω , T, z ) = kT | Ω | log ( Z gr (Ω , T, z )) . CRM, Barcelona ▽ Olivier Bernardi – p.11/37

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 W ( G ) z | G | P (Ω , T, z ) = kT | Ω | log ( Z gr (Ω , T, z )) = kT � . | Ω | | G | ! G connected CRM, Barcelona Olivier Bernardi – p.11/37

Example Hard particules in Ω = { 1 , . . . , q } . Grand canonical partition function : � q � Z (Ω , T, n ) z n = z n = (1 + z ) q . � � Z gr (Ω , T, z ) = n n n CRM, Barcelona ▽ Olivier Bernardi – p.12/37

Example Hard particules in Ω = { 1 , . . . , q } . Grand canonical partition function : � q � Z (Ω , T, n ) z n = z n = (1 + z ) q . � � Z gr (Ω , T, z ) = n n n Pressure : P (Ω , T, z ) = kT | Ω | log ( Z gr (Ω , T, z )) = kT log(1 + z ) . CRM, Barcelona Olivier Bernardi – p.12/37

Example Mayer’s weights : W ( G ) = ( − 1) e ( G ) q c ( G ) . Pressure : W ( G ) z | G | ( − 1) e ( G ) z | G | P (Ω , T, z ) = kT � � = kT . | Ω | | G | ! | G | ! G connected G connected CRM, Barcelona ▽ Olivier Bernardi – p.13/37

Example Mayer’s weights : W ( G ) = ( − 1) e ( G ) q c ( G ) . Pressure : W ( G ) z | G | ( − 1) e ( G ) z | G | P (Ω , T, z ) = kT � � = kT . | Ω | | G | ! | G | ! G connected G connected Comparing the two expressions of the pressure yields : ( − 1) e ( G ) z | G | � = log(1 + z ) . | G | ! G connected CRM, Barcelona ▽ Olivier Bernardi – p.13/37

Example Mayer’s weights : W ( G ) = ( − 1) e ( G ) q c ( G ) . Pressure : W ( G ) z | G | ( − 1) e ( G ) z | G | P (Ω , T, z ) = kT � � = kT . | Ω | | G | ! | G | ! G connected G connected Comparing the two expressions of the pressure yields : ( − 1) e ( G ) z | G | � = log(1 + z ) . | G | ! G connected ( − 1) e ( G ) = ( − 1) n − 1 ( n − 1)! . � In other words : G ⊆ K n connected CRM, Barcelona Olivier Bernardi – p.13/37

How did we get there ? Mayer W ( G ) z | G | � Z (Ω , T, z ) | G | ! G log log = A B CRM, Barcelona Olivier Bernardi – p.14/37

A killing involution ( − 1) e ( G ) = ( − 1) n − 1 ( n − 1)! � G ⊆ K n connected CRM, Barcelona ▽ Olivier Bernardi – p.15/37

A killing involution ( − 1) e ( G ) = ( − 1) n − 1 ( n − 1)! � G ⊆ K n connected 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 } , if E ∗ ( G ) = ∅ Φ( G ) = G G ⊕ min( E ∗ ( G )) otherwise . CRM, Barcelona ▽ Olivier Bernardi – p.15/37

A killing involution ( − 1) e ( G ) = ( − 1) n − 1 ( n − 1)! � G ⊆ K n connected 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 } , if E ∗ ( G ) = ∅ Φ( G ) = 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 } .) CRM, Barcelona Olivier Bernardi – p.15/37

Increasing trees 4 3 4 3 4 3 1 2 1 2 1 2 4 3 4 3 4 3 1 2 1 2 1 2 CRM, Barcelona Olivier Bernardi – p.16/37