Transition probabilities of Bethe ansatz solvable interacting particle systems Eunghyun Lee University of California, Davis September 8, 2011 Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 1 / 19
Outline 1 Introduction to coordinate Bethe ansatz for ASEP 2 Borodin-Ferrari’s PushASEP and its generalized model 3 Transition probability of the PushASEP by Bethe ansatz Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 2 / 19
Bethe ansatz technique in interacting particle systems Bethe ansatz : technique to obtain eigenvalues and eigenvectors of the Hamiltonian of 1D quantum spin- 1 2 chains A generator of ASEP is a similarity transformation of the quantum spin chain Hamiltonian ( XXZ model). ASEP with N particles on Z – Bethe ansatz solvable Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 3 / 19
ASEP - a model in KPZ universality class Early works regarding RMT - TASEP Current fluctuation of TASEP with step initial condition – GUE TW distribution ; Johansson, Nagao and Sasamoto, Sch¨ utz and Rakos Limiting processes – A 2 -process (step initial), A 1 -process (flat initial) ; Pr¨ ahofer and Spohn; Borodin, Ferrari, Pr¨ ahofer and Sasamoto TASEP with particle dependent hopping rates ; Sch¨ utz and Rakos (via Bethe ansatz), Baik, Ben Arous and Peche (Last passage percolation, RMT) Bethe ansatz in TASEP Quite successful ! Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 4 / 19
Bethe ansatz Quantity of interest : P Y ( X ; t ) where X = ( x 1 , · · · , x N ) with x 1 < · · · < x N . Time evolution − ˆ ∂ t P Y ( X ; t ) = HP Y ( X ; t ) = P Y ( x 1 − 1 , x 2 , · · · , x N ; t ) + (1 − δ x 2 − 1 ,x 1 ) P Y ( x 1 , x 2 − 1 , x 3 , · · · , x N ; t ) . . . + (1 − δ x N − 1 ,x N − 1 ) P Y ( x 1 , x 2 , · · · , x N − 1; t ) − ( N − δ x 2 − 1 ,x 1 − · · · − δ x N − 1 ,x N − 1 ) P Y ( x 1 , · · · , x N ; t ) Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 5 / 19
Idea Extend the physical region of definition of P Y ( X ; t ) { ( x 1 , · · · , x N ) : x i < x i +1 } to Z N by imposing a boundary condition P Y ( · · · , x k , x k , · · · ; t ) = P Y ( · · · , x k , x k +1 , · · · ; t ) . Problem to solve − ˆ ∂ t P Y ( X ; t ) = HP Y ( X ; t ) � = P Y ( x 1 , · · · , x i − 1 , x i − 1 , x i +1 , · · · , x N ; t ) i − NP Y ( x 1 , · · · , x N ; t ) with P Y ( · · · , x k , x k , · · · ; t ) = P Y ( · · · , x k , x k +1 , · · · ; t ) . and P Y ( X ; 0) = δ Y ( X ) . Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 6 / 19
Bethe ansatz solution A solution of DE is � x j e ε ( ξ j ) t ξ j j with ε ( ξ ) = 1 ξ − 1 and ξ ( � = 0) ∈ C . Important observation : For σ ∈ S N , ξ x j � σ ( j ) e ε ( ξ j ) t j is another solution and � ε ( ξ j ) j is invariant under permutation. Any integral of a linear combination is also a solution. Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 7 / 19
Essence of Bethe ansatz Take a linear combination of N ! solutions and determine coefficients so that the linear combination satisfies BC. Bethe ansatz solution � ξ x j � � � e ε ( ξ j ) t d N ξ A σ σ ( j ) σ ∈ S N j j with A σ = � S βα where S βα = 1 − ξ β 1 − ξ α and the product is over all pairs ( β, α ) in σ such that α < β but σ − 1 ( α ) > σ − 1 ( β ) Theorem (Sch¨ utz) Let F Y ( X ; t ) be an N × N matrix with entries F ij = f i − j ( x i − y j ; t ) where 1 � e − (1 − ξ − 1 ) t (1 − ξ ) − p ξ n − 1 dξ f p ( n ; t ) = 2 πi | ξ | =1 − 0 Then P Y ( X ; t ) = det F Y ( X ; t ) . Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 8 / 19
Determinantal structure and special properties of f p ( n ; t ) are essential for further works of Current distribution for step/flat initial conditions One-point distribution -TW(GUE, GOE) Limiting processes - A 2 , A 1 In ASEP, S βα = p + qξ β ξ α − ξ β p + qξ β ξ α − ξ α non-determinantal structure Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 9 / 19
ASEP -Tracy and Widom’s formula by Bethe ansatz Theorem � � x i − y σ ( i ) − 1 � � � i ε ( ξ i ) t dξ 1 · · · dξ N P Y ( X ; t ) = · · · A σ ξ e σ ( i ) C r C r σ ∈ S N i where C r is a circle centered at zero with sufficiently small radius r so that all the poles of the integrand except at the origin lie outside C r and A σ = � S βα where S βα = p + qξ β ξ α − ξ β p + qξ β ξ α − ξ α and the product is over all pairs ( β, α ) in σ such that α < β but σ − 1 ( α ) > σ − 1 ( β ) and ε ( ξ i ) = pξ − 1 + qξ i − 1 . i Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 10 / 19
Based on the transition probability, Probability distribution of a single particle’s position – time-integrated current (TW) Fredholm determinant representation of the distribution for step initial condition (TW) Asymptotic analysis – KPZ universality class (TW) x m ( t ) − c 1 t → F 2 c 2 t 1 / 3 in distribution as m, t → ∞ One-point probability distribution of Hopf-Cole solution to KPZ equation with narrow edge initial data (Amir-Corwin-Quastel, and Sasamoto-Spohn) Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 11 / 19
Borodin-Ferrari’s model (2008) - PushASEP Dynamics Each particle has two Poisson clocks; one for left jump with rate L and the other for right jump with rate R . When the i th right clock rings, the i th particle jumps according to TASEP dynamics. When the i th left clock rings, the i th particle jumps to the nearest vacant site on its left – Pushing dynamics Interpolates between TASEP and Drop-push model which are in KPZ universality class. Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 12 / 19
Borodin-Ferrari’s model - PushASEP Totally asymmetric dynamics. S -matrices are separable ; determinantal transition probability by Bethe ansatz. For step and flat initial conditions; GUE TW and GOE TW, further, A 2 , A 1 processes, respectively. Question : Partially asymmetric dynamics ? Clock rates depending on the environment ? Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 13 / 19
Two-sided PushASEP pr n ql n n n p + q = 1 Assume that pushing rates depend on the number of particles to be pushed. Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 14 / 19
Boundary Condition for N = 2 pr 1 P Y ( x 1 , x 2 − 1; t ) − pr 2 P Y ( x 1 − 1 , x 2 − 1; t ) − p ( r 1 − r 2 ) P Y ( x 1 , x 2 ; t ) = − ql 1 P Y ( x 1 + 1 , x 2 ; t ) + ql 2 P Y ( x 1 + 1 , x 2 + 1; t ) + q ( l 1 − l 2 ) P Y ( x 1 , x 2 ; t ) In the original B.C. Set both sides to be zero, λ = l 2 /l 1 , µ = r 2 /r 1 Set r 1 = l 1 = 1 , λ + µ = 1 Then after letting x 1 = x , we have much simpler BC P Y ( x, x ; t ) = µP Y ( x − 1 , x ; t ) + λP Y ( x, x + 1; t ) . Alimohammadi et al.(1999) suggested this BC directly as a combined version of TASEP and Drop-push model. Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 15 / 19
Constraints for Bethe ansatz solvability Alimohammadi et al. (1999) 1 r n = 1 + λ/µ + ( λ/µ ) 2 + · · · + ( λ/µ ) n − 1 , ( λ + µ = 1) 1 l n = 1 + µ/λ + ( µ/λ ) 2 + · · · + ( µ/λ ) n − 1 , ( r 1 = l 1 = 1) This model generalizes Borodin-Ferrari model ( µ → 0 ). Question : Is this model in KPZ universality class ? Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 16 / 19
S -matrices for two-particle interaction αβ := − ξ α · µ + λξ α ξ β − ξ β := ξ α S † · S ( µ ) βα ξ β µ + λξ α ξ β − ξ α ξ β Remarks Non-determinantal q = 0 but µ, λ � = 0 ; totally asymmetric (to the right only) but not determinantal. Eunghyun Lee (University of California, Davis) Prague School on Mathematical statistical physics September 8, 2011 17 / 19
Recommend
More recommend
