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

SLIDE 1 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 Thorique, CEA Sa la y (F ran e)

SLIDE 2 ASEP

q p p p q

Asymmetri Ex lusion Pro ess. A paradigm for non-equilibrium Statisti al Me hani s. EX CLUSION : Hard

re-in

tera tion ; at most 1 parti le p er site. ASYMMETRIC : External driving ; breaks detailed-balan e PR OCESS : Sto hasti Mark

vian

dynami s ; no Hamiltonian

SLIDE 3 ORIGINS

In

tera ting Bro wnian Pro esses (Spitzer, Harris, Liggett).

Driv

en diusiv e systems (KLS).

T

ransp

rt
f

Ma romole ules through thin v essels. Motion

f

RNA templates.

Hopping
ndu tivit

y in solid ele trolytes.

Dire ted

P

lymers

in random media. Reptation mo dels. APPLICA TIONS

T

ra

w.
Sequen e

mat hing. Bro wnian motors.

SLIDE 4 1. Sp e tral Prop erties

f

the Mark

v

Matrix (O. Golinelli) 2. Flu tuations

f

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 )

SLIDE 5 Mark

v

Equation for the ASEP

L N)

(

Ω =

N PARTICLES L SITES

x asymmetry parameter

1

x

CONFIGURATIONS

Pt(x1, . . . , xN)

: Prob.

f
ng. 1 ≤ x1 < . . . < xN ≤ L

at time t .

dPt dt =

i

[Pt(x1, . . . , xi − 1, . . . , xN) − Pt(x1, . . . , xi, . . . xN)] = MPt (x = 0)

The sum is restri ted to xi−1 < xi − 1 .

SLIDE 6 ASEP : An In tegrable System MAPPING TO A NON-HERMITIAN SPIN CHAIN

M =

L

l=1
S+

l S− l+1 + xS− l S+ l+1 + 1 + x

4 Sz

l Sz l+1 − 1 + x

4

Complex

Eigen v alues Mψ = Eψ :

Ground

State : E = 0 , P = Ω−1 (non-degenerate).

Ex ited

States : ℜ(E) < 0 (P erron-F rob enius). Ex itations

rresp
nd

to relaxation times T ASEP : x = 0

SLIDE 7 1. T ASEP

n

a ring : Sp e tral Prop erties

SPECTRAL

GAP : Largest relaxation time T . Ho w do es it dep end

n

the size L

f

the system : T ∼ Lz ?

DEGENERA

CIES in the Mark

v

Matrix : Hidden symmetries.

SLIDE 8 Example

f

a sp e trum

SLIDE 9 Bethe Ansatz for T ASEP Eigen v e tors

f M

as linear

m

binations

f

plane w a v es, with pseudo-momen ta giv en b y z1, . . . zN :

ψ(x1, . . . , xN) = det 2xj(zi + 1)j−xj (zi − 1)j

for

1 ≤ i, j ≤ N

ψ

is an eigenfun tion with eigen v alue E = 1

2(−N + j zj).

Can ellation
f

the t w

-parti le
llision

terms (xk−1 = xk − 1).

Bethe

Equations

(1 − zi)N(1 + zi)L−N = −2L

N

j=1

zj − 1 zj + 1

for

i = 1, . . . N

Note that the r.h.s. is a

nstant

indep endent

f i

.

SLIDE 10 Pro edure for solving the Bethe Equations

F
r

an y giv en v alue

f Y

, SOL VE

(1 − zi)N(1 + zi)L−N = Y .

The ro

ts

are lo ated

n

Cassini Ov als

CHOOSE N

r

ts zc(1), . . . zc(N)

amongst the L a v ailable ro

ts,

with a hoi e set c : {c(1), . . . , c(N)} ⊂ {1, . . . , L} .

SOL

VE the self- onsisten t equation Ac(Y) = Y where

Ac(Y ) = −2L

N

j=1

zc(j) − 1 zc(j) + 1 .

DEDUCE

from the v alue

f Y

, the zc(j) 's and the energy

rresp
nding

to the hoi e set c :

2Ec(Y ) = −N +

N

j=1

zc(j).

SLIDE 11 Lab elling the ro

ts
f

the Bethe Equations The lo i

f

the ro

ts

are remarquable urv es : Cassini Ov als

−1 1 Z1 Z2 ZN−1 ZN ZN+1 ZL−1 ZL 1−2ρ

SLIDE 12 Cal ulation

f

the GAP A n

riginal

metho d : EXA CT

m

binatorial form ulae for A0(Y ) and E0(Y ) for an y nite v alues

f L

and N :

log A0(Y ) Y =

∞

k=1

  kL kN   Y k k2kL E0(Y ) = −

∞

k=1

  kL − 2 kN − 1   Y k k2kL

These expressions are analyti ally

n

tin ued in C − [1, ∞). When

L → ∞

, A0(Y ) and E0(Y ) b e ome the p

lylogarithm

fun tions Li3/2 and Li5/2 , resp e tiv ely .

SLIDE 13 Cal ulation

f

the rst ex ited state b y solving trans enden tal equations. F

r

a densit y ρ :

E1 = −2

ρ(1 − ρ)6.509189337 . . .

L3/2 ± 2iπ(2ρ − 1) L . RELAXATION OSCILLATIONS

Higher ex itations. Opp

site

side

f

the sp e trum. T agged parti le.

SLIDE 14 SPECTRAL DEGENERA CIES NA TURAL SYMMETRIES OF T ASEP :

T

ranslation T : MT = TM . Momen tum k

Charge- onjugation C

+ Ree tion R : M(CR) = (CR)M . These natural symmetries do not

mm

ute (CR)T = T −1(CR) → The sp e trum

f M

is

mp
sed
f

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

f

ertain

rders

with sp e i n um b ers

f

m ultiplets app ear. The highest degenera y

rder ∼ 2L/6

(at half-lling). Can w e al ulate these n um b ers ? Can w e explain their

rigin

?

SLIDE 15

L N m(1) m(2) m(6) m(20) m(70)

2 1 2 4 2 4 1 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.

m(d)

is the numb er

f

multiplets with de gener a y d .

SLIDE 16

ρ L N m(1) m(2) m(3) m(4) m(5) m(15)

1/3 9 3 81 1 12 4 459 12 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

f

sp e tr al de gener a ies in the T ASEP at l ling ρ = 1/2.

SLIDE 17 A symmetry

f

the Bethe equations Let us all δ = gcd(L, N). The L Bethe ro

ts

form δ pa k ages, ea h

f

ardinalit y L/δ. The ro

ts
mp
sing

the pa k age Ps ha v e the indi es

{s, s + δ, s + 2δ, . . . , s + (L/δ − 1) δ}

with 1 ≤ s ≤ δ . Consider a hoi e set c (i.e., a hoi e

f N

ro

ts

amongst the L a v ailable

nes).

Supp

se

there exist pa k ages Ps and Pt su h that

Ps ⊂ c and Pt ∩ c = ∅ .

The hoi e set ˆ

c = (c\Ps) ∪ Pt

btained

from c b y ex hanging Ps and

Pt

rresp
nds

to the same self- onsisten t equation and to the same eigen v alue as c . Equiv alen e lasses

f

hoi e sets b y `P a k age-sw apping'.

SLIDE 18

c c

L = 10 and N = 5 : 5 PACKAGES EACH OF 2 ROOTS

c

AND

c

CHOOSE 5 ROOTS AMONGST THE 10 AVAILABLE

THE SAME EIGENVALUE

SLIDE 19 Cal ulation

f

the degenera ies The n um b er Ω

f

p

ssible

hoi e sets is the same as the dimension

f

the matrix M . W e supp

se

that there is a

ne

to

ne
rresp
nden e

: hoi e sets ↔ solutions

f

the Bethe Equations.

`pa

k age sw apping' equiv alen e lasses ↔ m ultiplets in sp e trum

ardinalit

y

f

a lass ↔

rder
f

the m ultiplet

#
f

lasses

f

ardinalit y d ↔ #

f

m ultiplets

f
rder d .

Cal ulation

f

the de gener a ies : a pr

blem

in

mbinatori s.

All the n um b ers in the tables an b e determined. Half-lling : dr =

  2r r   , m(dr) =   N 2r   2N−2r, 0 ≤ r ≤ N

2

SLIDE 20 2. Curren t Flu tuations and Large Deviations

TRANSPOR

T PR OPER TIES : mo died b e ause

f

in tera tions

DT ASEP = DF ree

LONG

RANGE CORRELA TIONS : Non-Gaussian b eha viour, non-v anishing higher um ulan ts.

SLIDE 21 Curren t statisti s as an eigen v alue problem Statisti s

f Yt

, total distan e

v

ered b y all the parti les b et w een and t . Deformation

f

the Mark

v

Matrix M b y adding a jump- oun ting fuga it y γ : M(γ) = M0 + eγM+ + e−γM− In the long time limit, t → ∞

eγYt

≃ eE(γ)t E(γ)

eigen v alue

f M(γ)

with maximal real part. Equiv alen tly , F(j), the large-deviation fun tion

f

the urren t

P Yt t = j

∼e−tF (j)

is the Legendre transform

f E(γ).

SLIDE 22 Bethe Ansatz for urren t statisti s The Bethe Equations are giv en b y

zL

i = (−1)N−1 N

j=1

xe−γzizj − (1 + x)zi + eγ xe−γzizj − (1 + x)zj + eγ

The eigen v alues

f M(γ)

are

E(γ; z1, z2 . . . zN) = eγ

N

i=1

1 zi + xe−γ

N

i=1

zi − N(1 + x) .

The Bethe equations do not de ouple unless x = 0 .

SLIDE 23 T ASEP CASE x = 0 (Derrida Leb

witz

1998)

E(γ)

is al ulated b y Bethe Ansatz to all

rders

in γ , thanks to the de oupling prop ert y

f

the Bethe equations. Mean T

tal

urren t :

J = lim

t→∞

Yt t = n(L − n) L − 1

Diusion Constan t :

D = lim

t→∞

Y 2

t − Yt2

t = Ln(L − n) (L − 1)(2L − 1) C2n

2L

(Cn

L)2

Exa t form ula for the large deviation fun tion.

SLIDE 24 In the general ase x = 0 , NO DECOUPLING. After a hange

f

v ariable, yi =

1−e−γzi 1−xe−γzi

, the Bethe equations read

eLγ 1 − yi 1 − xyi L = −

N

j=1

yi − xyj xyi − yj for i = 1 . . . N .

Let T b e auxiliary v ariable pla ying a symmetri role w.r.t. all the yi :

eLγ 1 − T 1 − xT L = −

N

j=1

T − xyj xT − yj for i = 1 . . . N . i.e. P(T) = eLγ(1 − T)L

N

j=1

(xT − yj) + (1 − xT)L

N

j=1

(T − xyj) = 0.

But P(yi) = 0 (Bethe Eqs.). Th us, Q(T) =

N

i=1

(T − yi)

divides P(T) :

Q(T) DIVIDES eLγ(1 − T)LQ(xT) + (1 − xT)LxNQ(T/x).

SLIDE 25 There exists a p

lynomial R(T)

su h that

Q(T)R(T) = eLγ(1 − T)LQ(xT) + xN(1 − xT)LQ(T/x)

F un tional Bethe Ansatz (Baxter's TQ equation). This equation is solv ed p erturbativ ely w.r.t. γ .

Mean

Curren t : J = (1 − x) N(L−N)

L−1

∼ (1 − x)Lρ(1 − ρ)

for L → ∞

Diusion

Constan t :

D = (1 − x) 2L L − 1

k>0

k2 CN+k

L

CN

L

CN−k

L

CN

L

1 + xk 1 − xk

D ∼ 4φLρ(1 − ρ)

∞ du u2 tanh φue−u2

when L → ∞ and x → 1 with xed v alue

f φ =

(1−x)√ Lρ(1−ρ) 2

.

SLIDE 26 Third um ulan t When time t → ∞ , Y 3

t −3Y 2 t Yt+2Yt3

t

→ E3

Non-vanishing Skewness E3 → Non Gaussian u tuations. When L → ∞ and x → 1 k eeping φ =

(1−x)√ Lρ(1−ρ) 2

xed,

E3 φ(ρ(1 − ρ))3/2L5/2 ≃ − 4π 3 √ 3 + 12 ∞ dudv (u2 + v2)e−u2−v2 − (u2 + uv + v2)e−u2−uv−v2 tanh φu tanh φv

F

r φ → ∞

, w e re o v er the kno wn T ASEP limit :

E3 ≃ 3 2 − 8 3 √ 3

π(ρ(1 − ρ))2L3

SLIDE 27

E3 6L2 = 1 − x L − 1

i>0
j>0

CN+i

L

CN−i

L

CN+j

L

CN−j

L

(CN

L )4

(i2 + j2)1 + xi 1 − xi 1 + xj 1 − xj − 1 − x L − 1

i>0
j>0

CN+i

L

CN+j

L

CN−i−j

L

(CN

L )3

i2 + ij + j2 2 1 + xi 1 − xi 1 + xj 1 − xj − 1 − x L − 1

i>0
j>0

CN−i

L

CN−j

L

CN+i+j

L

(CN

L )3

i2 + ij + j2 2 1 + xi 1 − xi 1 + xj 1 − xj − 1 − x L − 1

i>0

CN+i

L

CN−i

L

(CN

L )2

i2 2 1 + xi 1 − xi 2 + (1 − x) N(L − N) 4(L − 1)(2L − 1) C2N

2L

(CN

L )2

− (1 − x) N(L − N) 6(L − 1)(3L − 1) C3N

3L

(CN

L )3

SLIDE 28 The w eakly symmetri ase x = 1 − ν

L

Odd momen ts, su h as the mean urren t v anish. F

r L → ∞

,

E γ L

∼ ρ(1 − ρ)(γ2 + γν)

L − ρ(1 − ρ)γ2ν 2L2 + 1 L2 φ[ρ(1 − ρ)(γ2 + γν)] φ(z) =

∞

k=1

B2k−2 k!(k − 1)!zk

Leading
rder

(in 1/L ) : Gaussian u tuations.

Subleading

(in 1/L2) : Non-Gaussian

rre tion.
Phase

transition when ν ≥ νc =

2π

√

ρ(1−ρ)

SLIDE 29 3. Multisp e ies Ex lusion Mo dels.

Stationary

state

f

generalized ex lusion pro esses.

Relation

to the Matrix Ansatz for the stationary measure.

SLIDE 30 Denition

f

the N-T ASEP N lasses

f

parti les and holes with hierar hi al priorit y rules. During an innitesimal time step dt, the follo wing pro esses tak e pla e

n

ea h b

nd

with probabilit y dt :

I 0 → 0 I

for

1 ≤ I ≤ N I J → J I

for

1 ≤ I < J ≤ N

P arti les an alw a ys

v

ertak e holes (= 0-th lass parti les). First- lass parti les ha v e highest priorit y et ... There are PI parti les

f

lass I . T

tal

n um b er

f
ngurations

:

Ω = L! P0!P1!P2! . . . PN!

Stationary Measure ?

SLIDE 31 Matrix Ansatz for the 2-T ASEP Algebrai des ription

f

the Stationary Measure (DEHP , DJLS '93). Conguration represen ted b y a string e.g. 01220211. Stationary w eigh t :

p(01220211) = 1 Z Tr(EDAAEADD)

Repla e b y E, 1 b y D and 2 b y A. The

p

erators A, D and E satisfy the quadr ati algebr a

DE = D + E DA = A AE = A

e.g. p(01220211) ∝ Tr(D2EA3) = Tr((D2 + D + E)A3) ∝ 3Tr(A3)

SLIDE 32 Represen tations

f

the quadrati algebra Innite dimensional : D = 1 + δ where δ =righ t-shift.

E = 1 + ǫ

where ǫ =left-shift.

A = |11| = [δ, ǫ]

pro je tor

n

rst

rdinate.

D =         1 1 . . . 1 1 1 1

. . . . . . . . .

        , E = D†, A =        1 . . . . . . . . . . . . .       

SLIDE 33

Matrix

Ansatz : Stationary state prop erties ( urren ts,

rrelations,

u tuations).

Pro
f

that the stationary measure is not given by a Boltzmann-Gibbs me asur e (E. Sp eer).

Com

binatorial In terpretation

f

these

p

erators ?

No

Matrix Ansatz w as kno wn for N-T ASEP mo dels (for N ≥ 3.)

SLIDE 34 Geometri Constru tion

f

the 2-T ASEP stationary measure (Omer Angel, P ablo F errari, James Martin) A pro edure to

nstru t

a

nguration
f

the 2-T ASEP with P1 First Class P arti les and P2 Se ond Class P arti les starting from t w

indep

enden t

ngurations
f

the 1 sp e ies T ASEP .

P1 P + P

1 2

SLIDE 35 Geometri Constru tion

f

the 2-T ASEP stationary measure (Omer Angel, P ablo F errari, James Martin) A pro edure to

nstru t

a

nguration
f

the 2-T ASEP with P1 First Class P arti les and P2 Se ond Class P arti les starting from t w

indep

enden t

ngurations
f

the 1 sp e ies T ASEP .

P1 P + P

1 2

1 1 1 1

SLIDE 36 Geometri Constru tion

f

the 2-T ASEP stationary measure (Omer Angel, P ablo F errari, James Martin) A pro edure to

nstru t

a

nguration
f

the 2-T ASEP with P1 First Class P arti les and P2 Se ond Class P arti les starting from t w

indep

enden t

ngurations
f

the 1 sp e ies T ASEP .

P1 P + P

1 2

1 1 1 1 1 1 1 1

SLIDE 37 Geometri Constru tion

f

the 2-T ASEP stationary measure (Omer Angel, P ablo F errari, James Martin) A pro edure to

nstru t

a

nguration
f

the 2-T ASEP with P1 First Class P arti les and P2 Se ond Class P arti les starting from t w

indep

enden t

ngurations
f

the 1 sp e ies T ASEP .

P1 P + P

1 2

1 1 1 1 1 1 1 1

SLIDE 38 Geometri Constru tion

f

the 2-T ASEP stationary measure (Omer Angel, P ablo F errari, James Martin) A pro edure to

nstru t

a

nguration
f

the 2-T ASEP with P1 First Class P arti les and P2 Se ond Class P arti les starting from t w

indep

enden t

ngurations
f

the 1 sp e ies T ASEP .

P1 P + P

1 2

1 1 1 1 1 1 1 2 2 1 2

SLIDE 39

P1 P + P

1 2

P1 P + P

1 2

1 1 1 1 1 1 2 2 1 2 1

FROM 2 LINES OF TASEP TO 2−TASEP

This

nstru tion

is NOT

ne-to
ne

: the w eigh t

f

a 2-T ASEP

nguration

is prop

rtional

to the total n um b er

f

w a ys y

u

an generate it b y this

nstru tion.

SLIDE 40 F undamen tal Remarks :

A

1 (on the 1st line) an not b e lo ated ab

v

e a 2 (on the 2nd line).

F

a torisation Prop ert y : All the 1's (on the 2nd line) situated b et w een t w

2's

MUST b e link ed to 1's (on the 1st line) that are lo ated b et w een the p

sitions
f

the t w

2's

(No Cr

ssing

Condition).

`Pushing'

Pro edure : The `an estors'

f

a string

f

the t yp e

210102

are the strings

btained

b y pushing the 1's to the righ t i.e.,

210102, 210012, 201102, 201012, 200112.

These prop erties uniquely hara terize the stationary w eigh ts.

SLIDE 41 THE MA TRIX ANSA TZ PERF ORMS A UTOMA TICALL Y THE COMBINA TORICS UNDERL YING THE GEOMETRIC CONSTR UCTION OF THE WEIGHTS.

F

a torisation Prop ert y : A is a PR OJECTOR.

Pushing

Pro edure : D and E are SHIFT OPERA TORS (righ t-shift and left-shift, resp e tiv ely).

SLIDE 42 F rom 3 lines

f

T ASEP to a 3-T ASEP

P1 P + P

1 2

P + P + P

1 2 3

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

The w eigh t

f

a 3-T ASEP

nguration

is prop

rtional

to the total n um b er

f

w a ys y

u

an generate it b y this

nstru tion.

SLIDE 43

REVER

T the graphi al pro edure → ALGORITHM for

nstru ting

all an estors

f

a giv en N

T

ASEP

nguration.
ENCODE

this rev erse algorithm in to

p

erators → ALGEBRA.

CALCULA

TE the stationary w eigh ts → TRA CES

v

er this algebra.

SLIDE 44 NESTED MA TRIX ANSA TZ Hierar hi al

nstru tion

based up

n

tensor pro du ts

f

the

riginal

algebra, using the D , A and E matri es and the shift

p

erators. F

r

the 3-T ASEP :

ˆ P0 = 1 ⊗ 1 ⊗ E + 1 ⊗ ǫ ⊗ A + ǫ ⊗ 1 ⊗ D . ˆ P1 = 1 ⊗ 1 ⊗ D + δ ⊗ ǫ ⊗ A + δ ⊗ 1 ⊗ E ˆ P2 = A ⊗ 1 ⊗ A + A ⊗ δ ⊗ E ˆ P3 = A ⊗ A ⊗ E

SLIDE 45 F

r

the N

T

ASEP :

EXPLICIT
nstru tion
f

all the matri es.

DIRECT

PR OOF that the Matrix Ansatz leads to the stationary measure : indep enden t and purely algebrai pro

f.
F

A CTORISA TION prop erties

f

the stationary measure. EXA CT SOLUTION OF THE N SPECIES ASEP : Ba kw ard jumps allo w ed (rate x = 0 )

→

T ensor pro du ts

f

a deformation

f

the initial quadrati algebra. Repla e the shift-op erators b y deformed shift-op erators :

δǫ = 1 → δǫ − xǫδ = 1.

The stationary me asur e was not known in this ase (NO GRAPHICAL CONSTR UCTION).

SLIDE 46 CONCLUSIONS The asymmetri ex lusion pro ess an b e studied b y using a v ariet y

f

te hniques : Bethe Ansatz, Matrix Pro du t Ansatz, Com binatori s, Orthogonal p

lynomials,

Random Matrix Theory ... Relev an t for mathemati s (in tera ting parti le pro esses, generalization

f

the Bro wnian Motion) and for Statisti al Me hani s ( lassi al N-Bo dy problem

ut
f

equilibrium). Can b e used as a paradigm to study the b eha viour

f

systems far from equilibrium in lo w dimensions : Dynami al phase transitions, Non-Gaussian u tuations, Non-Gibbs measures, Flu tuations Theorems.