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

Large deviation function exp ( )

t

Q P j F j t

j

( ) F j

j

j t Extremely rare event

j

τ

Our goal: to investigate typical dynamics inside the interval , where extreme event happened

ASEP on a ring

' ' '' ' ''

sites, particles Master Equation 1 Equiprobable stationary state

C C C C C C C C C C C

L N P W P W P t P L N Counting jumps across a bond up to time t

/

( ) ( ) 1 ( ) are correlated random variables (not i.i.d.)

t k stat k k

N N J t J t J t t L L J t

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

RNA

• I. Protein synthesis

β

Ribosome protein

• 2. Vehicular traffic (Nagel-Schreckenberg model)

Off-ramp

β

J.T. MacDonald, J.H. Gibbs, and A.C. Pipkin, Biopolymers 6, 1 (1968).

α

( ) ( )

( , , | ) e

sJ t sJ s t C C C J

e e P C J t C P

By steepest descent can be shown to reduce to maximal eigenvalue problem

' ' '' ' ''

( ) eigen value equation for Modified Rate Matrix

C C C C C C C C C C

s W Y s Y e Y W

1 1 1 1 1

( ) , ( ) Left eigenvector '( ) ; '(0)

S C S

W s C Y W s s j j

j j s

Generating function for the flux

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 2) Showed that major contribution is given by TASEP configurations with a single shock-antishock structure s s

Usual homogeneous density profile Typical Density profile if conditioned on reduced flux 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 t Extremely rare event

j

τ

1 1 '

Particles should (1) move faster (2) formation of jams should be suppressed Quantifying the problem

• D. Simon, J. Stat. Mech. 20

: ( ) , EFFECTIVE DYNAMICS ( ) Effective rates 09

S Eff S C C C C

W s W e W

1 ' 1 1 1

' Stationary state ( )

Eff

C C P C C C

1 particle

1 1 1 ' 1 1 1

1 1 1 1 1 1 ; ; 1 1 ; ( ) 1 1 1 '( ) ' . ( , ') Stationary probabilities ( ) 1/ 2

S S S S S eff S S C C eff

W e W s e e j s e C eff rates W C C e W e C P C C C

1 j

j s

S

e

3 particles, restricted on high flux

|1 



1n1n 2n 3L 

Az1

n 1 z2 n 2 z3 n 3 |n1n2n3 



1n1n 2n 3L

A123z1

n 1z2 n 2z3 n 3  A213z2 n 1z1 n 2z3 n 3 ... |n1n2n3

zk  exp i 2k  2 L 1  3eS L , k  1,2,3

Aijk Aijk   1  zjeS 1  zieS

Limit of large flux eSL  1

s  es 

k1 N

zk

1  N  es3 1  36

24L2  O1  Oes

Y123  C|1  1|C  sin n2  n1  L sin n3  n2 L sin n3  n1 L  Oes

3 particles, restricted on high flux

1 2 3

2 6 2 1 3 2 3 1 2 , , 123 3

2 ( ) sin sin sin ( )

EFF s n n n

n n n n n n P Y O e L L L L

6 2 1 2 3 123 3 ,

2 ( , , ) log( ) 2 logsin Long distance pair particle potential

eff eff eff j k eff j k

P n n n Y L U P n n U L

6 6 /3, 2 /3, 3 2 1 2

2 9 sin 3 3 3 2 3 3

EFF L L L TASEP

L L P L L L L P L

1 2 3

8 6 1, 2, 3 1 2 8

2 6 3

EFF n n n TASEP

L LP L P L L

1 2 3 3

6 2 2 1 4 2 1, 3 2, 2 2 4 1 3

2 24 sin . 6 ( 4)

EFF

L eff n n n n L TASEP

P L P x L L P L L L L

Single cluster state Most probable state Cluster of two particles

3 particles, restricted on high flux

1 2 3 1 2 3

1 2 1, 1, | 1 2

Effective hopping rates ( 1) ( 1) sin sin ( ) . sin sin

eff

s s l l l l l l

l l L L W e O e l l L L

.. ..,.. 1.. .. 1..,.. ..

Effective hopping rates for particles at short distances , 1, 2,... 1 1 , 1, 2,.. Pair of particles at short distances tend to increase the d

eff s d d eff s d d

d L d W e d d d W e d d

1 1

istance

d d d d

W W

N particles, conditioned on infinite flux

1 2 2 3 2 3 1 2 1 2 1 2

1 (1) (2) ( ) 1 2 12.. 1 2 1 1 1 1 ( ) ( ) ( ) 12.. 1 2 1 1 1 1 2 2 2 12.. 1

... | ... | ... in the limit of sgn( ) ... ... ... det .

N N N s N N N s

n n n N N N N n n n L S n n n L S n n n N N e S n n n n n n N e

A z z z n n n Y n n n e Y z z z z z z z z z Y

1 2

1 2

.. ... ... ... ... 2( ) exp( ), where , 1, 2,...

N

n n n N N N N k k k

z z z k z i k N L

12... 1

sin ,

k p N p k N

n n Y L

N particles, conditioned on infinite flux

12... 1

sin ,

k p N p k N

n n Y L

12... 1

max is reached for equdistant configuration | | for all

N k k

L Y n n k N

PeffC  Y12...N2/KN

( 1) 2 , 1

2 sin

N N N N L n m L

m n K L L

1 2

Effective Potential ( , ,..., ) log | sin ( ) |.

N i j i j

U x x x x x Probability of most probable state at half-filling N=L/2

2

1 /2, /2 /2, /2

(..0101..) 2 / 2 (..0101..) 2/ 2 (..0101..) (..0101..)

L

eff L L L L L equidist L TASEP L eff equidist TASEP

P K K P P P

N particles, conditioned on infinite flux

1 ln , , 1 ln (1 )ln(1 ) 1 1/ 1/

1 1 , ,... particle density 2 3 ( ) /( ) (100...100...1 00..) (100...100...1 00..)

F F F F F F F F F F F F F

F L Lv eff F L L F L L F L L v v equidist TASEP F L equidist TASEP ef

P K K e P e P P

(1 )ln(1 )

( )

F F

L v f F

e

Probability of most probable state at fillings , 1/2, 1/3, 1/4, ….

Clustering probabilities

cluster of size similar cluster of si

Suppose, in an infinite system we have a sparse finite size cluster of particle with distances ; Probability to observe a cluster of half-size ( / 2) (

ij eff eff

N d P d P

2 1 ( 1) 2 1 ideal gas ideal gas

ze cluster of size similar cluster of size

/ 2 sin 2 ) sin ( / 2) For ideal gas 2 ( )

ij i j N N N ij i j N N

d L d d L P d P d

d d/2

Connection to Free Fermions

1 1 1 L XX n n n n n

H

1 2 1 2 1 2

1 1 1 2 2 2

Wave function for N fermions on a ring of L sites is given by Slater determinant, constructed from single-particle wave functions ... ... det ... ... ... ... ...

N N N

ikn n n n n n n n n n N N N

e z z z z z z z z z

1 2

3 12.. 1 2 12... 3/2 1 ... 1

, where 1 2 Ground state | ... , where sin

N

L imL m k p N N N n n n L p k N

z e n n Y n n n Y L L

2 2 2 1 1

Quantum Expectation values of a configuration ( ) Stochastic Classical TASEP conditioned on large flux , conditioned probabilities of a configuration ( )

EFF

P C C P C C C

1 1 1

( ) (1 ) ( ),

S L TASEP m m m m m TASEP XX

H s n n H s H e

Using connection to Free Fermions

2 2 2 2 2

is given by Quantum Expectation value 1 Pair correlation function 1 sin (1 )(1 ) 4 Note: for ASEP =

k k m FF z z eff k k m k k m ASEP k k m

n n n n n n m m is given by the respective Quantum Expectation valu Conditional probability ( , ; ,0) imaginary e t e i in m

XX

FF H t

e P t ( ) (0) is given by the respective Time dependent correlation functi Quantum FF Expectation value in

• n

imaginary time

eff k m k

n t n

V.P., G.M. Schütz, J. Stat. Phys. 142 , 627 (2011)

Connection to Circular Unitary Ensemble

1 2 2

Random unitary matrices , eigenvalues exp( ) Joint Probability distribution of the ei Dyson 1962 sin 2 Compare to steady state probabilities of configu genvalues in CUE Prob( , ,..., r ) a

k j k j k k N

U i

2 1 2 1

( tions in , ,... effective ASEP dynamic , ) s , s in

k p eff N p k N

n n P n n n L

Connection to Circular Unitary Ensemble

2

One can define dynamics on , where requiring that elements of perform independent random walks, i.e. during time 1 0, 2 Independent random walks of elements indu inte ce

iH ii ii

U e H H H t H H t H

1 1 2 2

Brownian motions

• n

, ( ,..., ) ( ) , ( , ,..., ) log sin 2 , lim 0,... for which one can write Fokker Planck equation. Unique s ra t c a ting

k k j N k k N k j k k m j k t

W F t W t t

2 1 2

tionary solution of the FP equation ( , ,..., ) sin 2

k j stat N k j

P

Conclusions

• ASEP conditioned on producing an extreme current

during a long period of time, is described by a effective dynamics with long-range interactions

• Large set of expectation values for Quantum free

fermion system receive classical interpretation

• Steady state distribution for the effective dynamics

coincide with distribution of eigenvalues in Dyson´s Circular Unitary Ensemble

V.P., D. Simon, G.M. Schütz, J. Stat. Mech. (2010) P10007 V.P., G.M. Schütz, J. Stat. Phys. 142 , 627 (2011)

Generating function of the flux

Generating function of the flux Jt is defined by logesJt  log 

J

PJtesJt  log 

J

etFJ/tjesJt  st Here Fj is large deviation function. It can be shown that s  maxis minis is a highest eigenvalue of the modified Rate matrix Ws  esW  esW  W0

Calculation of the generating function

Let us once determine the generating function for the current esJt  

C  C0  J

esJPC,J,t|C0PC0



where PC,J,t|C0 is the probability to have a current J at moment t in configuration C , provided the initial configuration was C0 , and PC0

 is stationary probability of the configuration H|P  0 . Let us find the equation for

PC,s,t  J esJPC,J,t|C0 (in the following, sometimes we omit C0 for brevity ). First, we note that PC,J,t t  

C

WCC

 PC,J  1,t  WCC  PC,J  1,t  WCC 0 PC,J,t

 

C

WCCPC,J,t

where WCC is the transition rate from C  C , WCC



are transition rates of the processes which increase the flux by one unit (hops to the right ), WCC



are transition rates which decrease the flux by one unit (hops to the left), WCC are remaining rates.

summing both parts with J esJ... we obtain PC,s,t t  

C

esWCC



 esWCC



 WCC PC,s,t  PC,s,t 

C

WCC

• r, in vector form, PC,s,t;C0  |Ps,t;C0C

|Ps,t;C0 t  W  |Ps,t;C0 where W   esH  esH  H0 . The solution of the above equation is clearly |Ps,t;C0  eW

 t|Ps,0;C0 but since J0  0 ,

|Ps,0;C0C  

J

esJ0PC,J,0|C0  CC0 and |Ps,0;C0  |C0 simply. Substituting the solution for |Ps,t;C0  eW

 t|C0 and accounting for J esJPC,J,t|C0  |Ps,t;C0C  C|Ps,t;C0

we have esJt  

C  C0

C|eW

 t|C0PC0 

 

C

C|eW

 t|P  s|eW  t|P

where s| CC| is the left stationary vector, trivial for stochastic Hamiltonian s|H  1111....|H  0 . Finally, if t   , eW

 t  emaxt|maxmax| where max is the largest eigenvalue of the modified rate matrix W

 . Note that since W  is not hermitian, the left max| and the right eigenvectors |max are different. Subsituting in (genFunc), we have esJtt  emaxts|maxmax|P

( ) max

Generating function for the current log

sJ t

e t