The Asymmetri Exlusion Pro ess : An In tegrable Mo del for - PowerPoint PPT Presentation

The Asymmetri Ex lusion Pro ess : An In tegrable Mo del for Non-Equilibrium Statisti al Me hani s Kirone Malli k Institut de Ph ysique Th�orique, CEA Sa la y (F ran e)

ASEP q p p q p Asymmetri Ex lusion Pro ess. A paradigm for non-equilibrium Statisti al Me hani s. EX CLUSION : Hard ore-in tera tion ; at most 1 parti le p er site. ASYMMETRIC : External driving ; breaks detailed-balan e PR OCESS : Sto hasti Mark o vian dynami s ; no Hamiltonian

ORIGINS In tera ting Bro wnian Pro esses (Spitzer, Harris, Liggett). Driv en di�usiv e systems (KLS). T ransp ort of Ma romole ules through thin v essels. Motion of RNA templates. • Hopping ondu tivit y in solid ele trolytes. • Dire ted P olymers in random media. Reptation mo dels. • APPLICA TIONS T ra� �o w. • Sequen e mat hing. Bro wnian motors. • • •

1. Sp e tral Prop erties of the Mark o v Matrix (O. Golinelli) 2. Flu tuations of the urren t (S. Prolha ) 3. Multisp e ies ex lusion pro esses and Matrix Ansatz (M. Ev ans, P . F errari and S. Prolha )

Mark o v Equation for the ASEP 1 x L SITES N PARTICLES L Ω = ( N ) : Prob. of on�g. 1 ≤ x 1 < . . . < x N ≤ L at time t . CONFIGURATIONS x asymmetry parameter P t ( x 1 , . . . , x N ) The sum is restri ted to x i − 1 < x i − 1 . dP t � d t = [ P t ( x 1 , . . . , x i − 1 , . . . , x N ) − P t ( x 1 , . . . , x i , . . . x N )] = MP t i ( x = 0)

ASEP : An In tegrable System MAPPING TO A NON-HERMITIAN SPIN CHAIN Complex Eigen v alues Mψ = Eψ : L � l +1 + 1 + x l +1 − 1 + x � Ground State : E = 0 , P = Ω − 1 (non-degenerate). � S + l S − l +1 + x S − l S + S z l S z M = 4 4 Ex ited States : ℜ ( E ) < 0 (P erron-F rob enius). l =1 Ex itations orresp ond to relaxation times • • T ASEP : x = 0

1. T ASEP on a ring : Sp e tral Prop erties SPECTRAL GAP : Largest relaxation time T . Ho w do es it dep end on the size L of the system : T ∼ L z ? DEGENERA CIES in the Mark o v Matrix : Hidden symmetries. • •

Example of a sp e trum

Bethe Ansatz for T ASEP Eigen v e tors of M as linear om binations of plane w a v es, with pseudo-momen ta giv en b y z 1 , . . . z N : for is an eigenfun tion with eigen v alue E = 1 j z j ) . Can ellation of the t w o-parti le ollision terms ( x k − 1 = x k − 1) . � 2 x j ( z i + 1) j − x j � ψ ( x 1 , . . . , x N ) = det 1 ≤ i, j ≤ N Bethe Equations ( z i − 1) j 2 ( − N + � • ψ for • • Note that the r.h.s. is a onstant indep endent of i . N z j − 1 ( 1 − z i ) N ( 1 + z i ) L − N = − 2 L � i = 1 , . . . N z j + 1 j = 1

Pro edure for solving the Bethe Equations F or an y giv en v alue of Y , SOL VE The ro ots are lo ated on Cassini Ov als CHOOSE N r o ots z c (1) , . . . z c ( N ) amongst the L a v ailable ro ots, with a hoi e set c : { c (1) , . . . , c ( N ) } ⊂ { 1 , . . . , L } . (1 − z i ) N (1 + z i ) L − N = Y . • SOL VE the self- onsisten t equation A c ( Y ) = Y where • • DEDUCE from the v alue of Y , the z c ( j ) 's and the energy N orresp onding to the hoi e set c : z c ( j ) − 1 � A c ( Y ) = − 2 L z c ( j ) + 1 . j =1 • N � 2 E c ( Y ) = − N + z c ( j ) . j =1

Lab elling the ro ots of the Bethe Equations The lo i of the ro ots are remarquable urv es : Cassini Ov als Z N+1 Z N Z N−1 1−2 ρ −1 1 Z 2 Z 1 Z L Z L−1

Cal ulation of the GAP A n original metho d : EXA CT om binatorial form ulae for A 0 ( Y ) and E 0 ( Y ) for an y �nite v alues of L and N :   ∞  Y k  kL log A 0 ( Y ) � = k 2 kL Y kN k =1 These expressions are analyti ally on tin ued in C − [1 , ∞ ) . When   ∞  kL − 2  Y k , A 0 ( Y ) and E 0 ( Y ) b e ome the p olylogarithm fun tions Li 3 / 2 � E 0 ( Y ) = − k 2 kL kN − 1 and Li 5 / 2 , resp e tiv ely . k =1 L → ∞

Cal ulation of the �rst ex ited state b y solving trans enden tal equations. F or a densit y ρ : ± 2 iπ (2 ρ − 1) ρ (1 − ρ )6 . 509189337 . . . � − 2 E 1 = . L 3 / 2 L RELAXATION OSCILLATIONS Higher ex itations. Opp osite side of the sp e trum. T agged parti le.

SPECTRAL DEGENERA CIES NA TURAL SYMMETRIES OF T ASEP : T ranslation T : MT = TM . Momen tum k Charge- onjugation C + Re�e tion R : M ( CR ) = ( CR ) M . These natural symmetries do not omm ute ( CR ) T = T − 1 ( CR ) → The sp e trum of M is omp osed of singlets for ( k = ± 1) and • doublets ( k, k ⋆ ) for ( k � = ± 1) . • A NUMERICAL OBSER V A TION F OR T ASEP : Unexp e ted degenera ies of ertain orders with sp e i� n um b ers of m ultiplets app ear. The highest degenera y order ∼ 2 L/ 6 (at half-�lling). Can w e al ulate these n um b ers ? Can w e explain their origin ?

2 1 2 4 2 4 1 L N m (1) m (2) m (6) m (20) m (70) 6 3 8 6 8 4 16 24 1 10 5 32 80 10 12 6 64 240 60 1 14 7 128 672 280 14 16 8 256 1792 1120 112 1 18 9 512 4608 4032 672 18 Sp e tr al de gener a ies in the T ASEP at half �l ling. is the numb er of multiplets with de gener a y d . m ( d )

1/3 9 3 81 1 12 4 459 12 ρ L N m (1) m (2) m (3) m (4) m (5) m (15) 15 5 2673 90 15 18 6 15849 540 270 1 21 7 95175 2835 2835 189 21 1/4 16 4 1816 1 20 5 15424 20 24 6 133456 240 36 1/5 25 5 53125 1 2/5 15 6 4975 15 Examples of sp e tr al de gener a ies in the T ASEP at �l ling ρ � = 1 / 2 .

A symmetry of the Bethe equations Let us all δ = gcd( L, N ) . The L Bethe ro ots form δ pa k ages, ea h of ardinalit y L/δ. The ro ots omp osing the pa k age P s ha v e the indi es with 1 ≤ s ≤ δ . Consider a hoi e set c ( i.e. , a hoi e of N ro ots amongst the L a v ailable ones). Supp ose there exist pa k ages P s and P t su h that { s, s + δ, s + 2 δ, . . . , s + ( L/δ − 1) δ } The hoi e set ˆ obtained from c b y ex hanging P s and orresp onds to the same self- onsisten t equation and to the same P s ⊂ c P t ∩ c = ∅ . and eigen v alue as c . Equiv alen e lasses of hoi e sets b y `P a k age-sw apping'. c = ( c \P s ) ∪ P t P t

L = 10 and N = 5 : 5 PACKAGES EACH OF 2 ROOTS CHOOSE 5 ROOTS AMONGST THE 10 AVAILABLE c c c c THE SAME EIGENVALUE AND

Cal ulation of the degenera ies The n um b er Ω of p ossible hoi e sets is the same as the dimension of the matrix M . W e supp ose that there is a one to one orresp onden e : hoi e sets ↔ solutions of the Bethe Equations. `pa k age sw apping' equiv alen e lasses ↔ m ultiplets in sp e trum ardinalit y of a lass ↔ order of the m ultiplet of lasses of ardinalit y d ↔ # of m ultiplets of order d . Cal ulation of the de gener a ies : a pr oblem in ombinatori s. • All the n um b ers in the tables an b e determined. • • # Half-�lling : d r =      2 r  N  , m ( d r ) =  2 N − 2 r , 0 ≤ r ≤ N 2 r 2 r

2. Curren t Flu tuations and Large Deviations TRANSPOR T PR OPER TIES : mo di�ed b e ause of in tera tions • LONG RANGE CORRELA D T ASEP � = D F ree TIONS : Non-Gaussian b eha viour, non-v anishing higher um ulan ts. •

Curren t statisti s as an eigen v alue problem Statisti s of Y t , total distan e o v ered b y all the parti les b et w een 0 and t . Deformation of the Mark o v Matrix M b y adding a jump- oun ting fuga it y γ : M ( γ ) = M 0 + e γ M + + e − γ M − In the long time limit, t → ∞ eigen v alue of M ( γ ) with maximal real part. Equiv alen tly , F ( j ) , the large-deviation fun tion of the urren t e γY t � ≃ e E ( γ ) t � E ( γ ) is the Legendre transform of E ( γ ) . � Y t � ∼ e − tF ( j ) P t = j

Bethe Ansatz for urren t statisti s The Bethe Equations are giv en b y The eigen v alues of M ( γ ) are N x e − γ z i z j − (1 + x ) z i + e γ � z L i = ( − 1) N − 1 x e − γ z i z j − (1 + x ) z j + e γ j =1 The Bethe equations do not de ouple unless x = 0 . N N 1 � � E ( γ ; z 1 , z 2 . . . z N ) = e γ + x e − γ z i − N (1 + x ) . z i i =1 i =1

T ASEP CASE x = 0 (Derrida Leb o witz 1998) is al ulated b y Bethe Ansatz to all orders in γ , thanks to the de oupling prop ert y of the Bethe equations. Mean T otal urren t : E ( γ ) Di�usion Constan t : � Y t � = n ( L − n ) J = lim L − 1 t t →∞ Exa t form ula for the large deviation fun tion. � Y 2 t � − � Y t � 2 C 2 n Ln ( L − n ) 2 L D = lim = L ) 2 t ( L − 1)(2 L − 1) ( C n t →∞