An n component face-cubic model on the complete graph Zongzheng - PowerPoint PPT Presentation

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model An n − component face-cubic model on the complete graph Zongzheng (Eric) Zhou School of Mathematical Sciences Monash University

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Collaborators ◮ Tim Garoni (Monash University)

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Outline Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model on the complete graph Conclusion

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Outline Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model on the complete graph Conclusion

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Outline Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model on the complete graph Conclusion

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Outline Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model on the complete graph Conclusion

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Outline Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model on the complete graph Conclusion

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Outline Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model on the complete graph Conclusion

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Ising model ◮ Graph G = ( V, E ) ◮ Assign a random variable W i on i , for i ∈ V ◮ W i takes values in a state space Σ = { 1 , − 1 } ◮ Configuration ω = { W 1 = ω 1 , W 2 = ω 2 , · · · , W N = ω N } ∈ Σ N , where N = | V | . ◮ The Ising model is defined by choosing configurations ω randomly via Gibbs measure π ( ω ) = e − H ( ω ) /T , ω ∈ Σ N Z N ( T ) ◮ Hamiltonian (energy) H ( ω ) � H ( ω ) = − ω i · ω j ij ∈ E ◮ Partition sum Z N ( T ) � e − H ( ω ) /T Z N ( T ) = ω ∈ Σ N

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model High and low temperature phases ◮ Recall Gibbs measure π ( ω ) = e − H ( ω ) /T � , H ( ω ) = − ω i · ω j Z N ( T ) ij ∈ E

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model High and low temperature phases ◮ Recall Gibbs measure π ( ω ) = e − H ( ω ) /T � , H ( ω ) = − ω i · ω j Z N ( T ) ij ∈ E ◮ Relative weight for two configurations ω , ω ′ π ( ω ) π ( ω ′ ) = e − ( H ( ω ) − H ( ω ′ ) ) /T

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model High and low temperature phases ◮ Recall Gibbs measure π ( ω ) = e − H ( ω ) /T � , H ( ω ) = − ω i · ω j Z N ( T ) ij ∈ E ◮ Relative weight for two configurations ω , ω ′ π ( ω ) π ( ω ′ ) = e − ( H ( ω ) − H ( ω ′ ) ) /T ◮ If T is low, spins prefer to like their neighbours, which is called ordered phase or low temperature phase. ◮ If T is high, spins are independent of each other, which is called disordered phase or high temperature phase. ◮ A critical point at T = T c .

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Order parameter ◮ Order parameter is used to quantitatively characterise phase transitions. ◮ For Ising model, the order parameter is the magnetisation, �� � � � N i =1 W i � � M = � � N � � � � M T T 0 c ◮ Critical behaviors ◮ If T ≥ T c , M = 0 ◮ If T → T − c , M ∼ ( T c − T ) β ◮ The other independent critical exponent is defined from correlation length ξ ∼ | T − T c | − ν

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Phase transitions classification ◮ Phase transitions are classified by the continuity of the order parameter. ◮ First order phase transition (discontinuous): ice-liquid-gas transition, phase coexistence. ◮ Continuous phase transition: ferromagnetic-paramagnetic transition, superconducting transition, Kosterlitz-Thouless transition.

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Phase transitions classification ◮ Phase transitions are classified by the continuity of the order parameter. ◮ First order phase transition (discontinuous): ice-liquid-gas transition, phase coexistence. ◮ Continuous phase transition: ferromagnetic-paramagnetic transition, superconducting transition, Kosterlitz-Thouless transition.

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Other Important concepts ◮ Phase transitions happen only in thermodynamic limit ◮ Ensemble hypothesis: approximate time average by ensemble average ◮ Universality class: various continuous phase transitions fall into several universality class, in which all models have the same critical phenomena, and share same critical exponents.

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Outline Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model on the complete graph Conclusion

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Cram´ er’s theorem ◮ Consider a sequence of identically and independently distributed random variables: X 1 , X 2 , · · · , X N ◮ State space Σ = { a 1 , a 2 , · · · , a m } , a i ∈ R d , d ∈ N + ◮ X i is distributed according to a law µ and E ( X i ) = X . ◮ Sample mean S N = 1 � N i =1 X i N ◮ Law of large numbers tells S N → X as N → + ∞ . ◮ What’s the probability that S N = x with x deviating far from X ?

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model ◮ Cram´ er’s theorem P N ( S N = x ) ∼ e − NI ( x ) , as N → + ∞ ◮ Logarithmic generating function λ ( k ) , for any k ∈ R d , λ ( k ) = log E [ e k · S N ] ◮ Rate function from Legendre-Fenchel transform I ( x ) = sup k ∈ R d {� k, x � − λ ( k ) } ◮ I ( x ) is convex, non-negative and min x I ( x ) = 0 ◮ Set { x : I ( x ) = 0 } is called the most probable macroscopic states

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Outline Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model on the complete graph Conclusion

Brief introduction to lattice models and phase transitions Large deviations theory An n -component face-cubic model Limit theorems for the face-cubic model Face-cubic model ◮ Given G = ( V, E ) . ◮ Assign a random variable W i on i , for i ∈ V . ◮ W i takes values in a state space Σ . ◮ State space Σ = { ( ± 1 , 0 , 0 , · · · , 0) , (0 , ± 1 , 0 , · · · , 0) , . . . ⊂ R n (0 , 0 , · · · , 0 , ± 1) } ◮ E.g. If n = 3 , Σ = { ( ± 1 , 0 , 0) , (0 , ± 1 , 0) , (0 , 0 , ± 1) } ◮ Configuration ω = { W 1 = ω 1 , W 2 = ω 2 , · · · , W N = ω N } ∈ Σ N , where N = | V | .