Simple Exclusion Process conditioned on extreme flux V. Popkov - PowerPoint PPT Presentation

Effective Dynamics of Asymmetric Simple Exclusion Process conditioned on extreme flux V. Popkov Università di Salerno, Italy with G. Schütz , D. Simon MPI, Drezden, July 2011

Plan • Introduction • TASEP on a ring as an integrable model • Large deviations • Modified rate matrix: eigenvalues and maximal eigenvector • Effective dynamics and conditional probabilities • Connection to quantum free fermion problem • Connection to Dyson ´ s Circular Unitary Ensemble

Introduction j j 0 Extremely rare event τ 0 t ( ) F j Large deviation function Q t exp ( ) P j F j t j j 0 Our goal: to investigate typical dynamics inside the interval , where extreme event happened

ASEP on a ring sites, particles L N P C Master Equation W P W P ' ' '' C C C C C C t ' '' C C C 1 Equiprobable stationary state P C L N Counting jumps across a bond up to time t / t N N ( ) ( ) 1 J t J t J t t k stat L L 0 k ( ) are correlated random variables (not i.i.d.) J t k

ASEP – reference model of stochastic many-body systems • Solvable by Bethe Ansatz : spectrum, finite size scaling, universality • Conditional probabilities can be found exactly in determinantal form; connections to random matrices ensembles • Minimal model for traffic processes • A base for traffic applications for pedestrian dynamics, vehicle dynamics, growth processes • Biological applications (RNA synthesis) • ASEP with open boundaries – reference model for boundary driven phase transitions • A simplest example of exclusion processes

Applications of ASEP and its generalizations I. Protein synthesis RNA β Ribosome protein J.T. MacDonald, J.H. Gibbs, and A.C. Pipkin, Biopolymers 6 , 1 (1968). 2. Vehicular traffic (Nagel-Schreckenberg model) α Off-ramp β

Generating function for the flux ( ) ( ) sJ t sJ s t ( , , | ) e e e P C J t C P 0 C 0 C C J 0 By steepest descent can be shown to reduce to maximal eigenvalue problem s ( ) eigen value equation for Modified Rate Matrix s Y e W Y W Y ' ' '' C C C C C C C ' '' C C C j ( ) , W s C Y S 1 1 1 C j ( ) Left eigenvector W s 0 1 1 S '( ) ; '(0) s j j 0 0 s

ASEP conditioned on reduced flux (s<0) B. Derrida and C. Appert, J. Stat. Phys 94 , 1 (1999) 1) Found analytically ( ) for 0 and arbitrary particle density s s 2) Showed that major contribution is given by TASEP configurations with a single shock-antishock structure Typical Density profile if conditioned on reduced flux Usual homogeneous density profile Stable shock unstable shock What are typical configurations and particle dynamics for large atypical fluxes (s>0) ??

Effective particle dynamics of a TASEP particles conditioned on high flux j j 0 Extremely rare event τ 0 t Particles should (1) move faster (2) formation of jams should be suppressed Quantifying the problem : ( ) , W s 1 1 S EFFECTIVE DYNAMICS ( D. Simon, J. Stat. Mech. 20 09 ) ' C 1 Eff S Effective rates W e W C C ' ' C C C 1 Eff Stationary state ( ) P C C C 1 1

1 particle 1 S j e 1 1 j W 0 1 1 0 s S 1 1 e S ; ; 1 1 ; ( ) 1 W s e 1 1 S S 1 1 e S '( ) j s e ' C 1 eff S S . ( , ') eff rates W C C e W e ' C C C 1 eff Stationary probabilities ( ) 1/ 2 P C C C 1 1

3 particles, restricted on high flux 1  n 1  n 2  n 3  L   n 3 | n 1 n 2 n 3   n 1 z   2  n 2 z   3  |  1   A  z   1    n 3  A 213 z 2 n 3  ...  | n 1 n 2 n 3   A 123 z 1 n 1 z 2 n 2 z 3 n 1 z 1 n 2 z 3 1  n 1  n 2  n 3  L z k  exp i  2  k  2  1  3 e  S , k  1,2,3 L L   1  z j e  S A   ij  k 1  z i e  S A ijk Limit of large flux e S L  1 Y 123   C |  1     1 | C   sin   n 2  n 1  sin   n 3  n 2  sin   n 3  n 1   O  e  s  L L L   s   e s  N  1  N  e s 3 1  36   O  1   O  e  s  z k 24 L 2 k  1

3 particles, restricted on high flux 6 2 2 eff ( , , ) P n n n Y 1 2 3 123 3 L eff log( ) U P eff n n j k 2 logsin U eff L j k , Long distance pair particle potential 2 6 2 n n n n n n 2 1 3 2 3 1 EFF 2 s ( ) sin sin sin ( ) P Y O e n , n , n 123 3 1 2 3 L L L L Cluster of two particles Single cluster state Most probable state 8 6 2 6 2 2 L 1 2 24 EFF LP 4 eff EFF sin . P L P x n 1, n 2, n 3 8 2 1, 2, n n n 2 2 3 6 1 2 3 L 1 2 3 2 9 L L L L L 4 EFF 6 n P sin 3 /3, 2 /3, L L L 3 2 3 3 3 1 L L 1 L 6 6 L TASEP 1 TASEP P L ( 4) P L L L 2 L 2 TASEP 3 L P L 3 2 3 3 L

3 particles, restricted on high flux Effective hopping rates ( 1) ( 1) l l 1 2 sin sin L L s s eff ( ) . W e O e l 1, l 1, l l l l | l l 1 2 3 1 2 3 1 2 sin sin L L Effective hopping rates for particles at short distances d L d eff s , 1, 2,... W e d .. ..,.. 1.. d d 1 d 1 , d eff s 1, 2,.. W e d .. d 1..,.. .. d d Pair of particles at short distances tend to increase the d istance W W 1 1 d d d d

N particles, conditioned on infinite flux n n n ... | ... | ... A z z z n n n Y n n n 1 2 N 1 (1) (2) ( ) 1 2 12.. 1 2 N N N N 1 1 1 1 n n n L S n n n L 2 3 N 2 3 S in the limit of e ( ) ( ) ( ) n n n sgn( ) ... Y z z z 1 2 N 12.. 1 2 N s N 1 e S N n n n ... z z z 1 2 N 1 1 1 n n n ... z z z 1 2 N 2 2 2 det Y 12.. N e s 1 . .. ... ... ... n n n ... z z z 1 2 N N N N 1 2( N ) k 2 exp( ), where , 1, 2,... z i k N k k k L n n k p sin , Y 12... N L 1 p k N

N particles, conditioned on infinite flux n n k p sin , Y 12... P eff  C    Y 12... N  2 / K N N L 1 p k N m n N N ( 1) N 2 2 sin K L , N L L 1 n m L L max is reached for equdistant configuration | | for all Y n n k 12... 1 N k k N Effective Potential ( , ,..., ) log | sin ( ) |. U x x x x x 1 2 N i j i j Probability of most probable state at half-filling N=L/2 L 1 eff (..0101..) 2 / 2 P K K 2 /2, /2 /2, L L L L L equidist L (..0101..) 2/ 2 P TASEP L /2 eff equidist (..0101..) (..0101..) P P TASEP

N particles, conditioned on infinite flux Probability of most probable state at fillings , 1/2, 1/3, 1/4, …. 1 1 , ,... particle density F 2 3 1 ln eff L Lv ( ) /( ) P K K e F F F F L , L F L , L F F F F 1 L L v ln (1 v )ln(1 ) 1 equidist (100...100...1 00..) P e F F F F TASEP F 1/ L F F equidist (100...100...1 00..) P TASEP 1/ L (1 v )ln(1 ) F e F F ef f ( ) P F

Clustering probabilities d Suppose, in an infinite system we have a sparse finite size cluster of particle with distances ; N d ij Probability to observe a cluster of half-size / 2 d 2 ij sin ( / 2) P d L cluster of size 1 eff i j N ( 1) N N 2 d ( ) P d similar cluster of si ze 2 ij d/2 eff sin L 1 i j N ( / 2) P d cluster of size ideal gas N For ideal gas 2 ( ) P d similar cluster of size ideal gas

Connection to Free Fermions L H XX 0 n n 1 n 1 n n 1 Wave function for N fermions on a ring of L sites is given by ikn Slater determinant, constructed from single-particle wave functions e n n n ... z z z 1 2 N 1 1 1 n n n ... z z z 1 2 N L 2 2 2 imL det , where 1 z e 0 m ... ... ... ... n n n ... z z z 1 2 N N N N 3 n n 2 k p Ground state | ... , where sin Y n n n Y 0 12.. 1 2 12... N N N 3/2 L L 1 n n ... n L 1 p k N 1 2 N 2 Quantum Expectation values of a configuration ( ) P C C 0 Stochastic Classical TASEP conditioned on large flux 2 2 EFF , conditioned probabilities of a configuration ( ) P C C C 1 0 1 0 L S ( ) (1 ) H s e n n TASEP m m 1 m m 1 1 m ( ), 0 H s H TASEP XX 0

Using connection to Free Fermions Pair correlation function is given by Quantum Expectation value n n k k m 2 1 1 sin m FF z z 2 eff (1 )(1 ) n n k k m k k m 2 2 4 m 2 ASEP Note: for ASEP = n n k k m Conditional probability ( , ; ,0) P t 0 FF H t is given by the respective Quantum Expectation valu e e XX 0 0 in imaginary t i m e eff Time dependent correlation functi on ( ) (0) n t n k m k is given by the respective Quantum FF Expectation value in imaginary time V.P., G.M. Schütz, J. Stat. Phys. 142 , 627 (2011)

Connection to Circular Unitary Ensemble Random unitary matrices , eigenvalues exp( ) U i k k Dyson 1962 Joint Probability distribution of the ei genvalues in CUE 2 k j Prob( , ,..., ) sin 1 2 N 2 k j Compare to steady state probabilities of configu r a tions in effective ASEP dynamic s n n k p 2 ( , ,... , ) s in , P n n n 1 2 eff N L 1 p k N