Matrix ansatz for exclusion processes - Bernard Derrida.pdf

MATRIX ANSATZ FOR EXCLUSION PROCESSES

Bernard DERRIDACollège de France, Paris

Matrix ansatz and exclusion processes

Phase diagram of the TASEP

Correlation functions and Brownian excursions

Infinite line and shocks

Additivity and large deviations function of the density

1993 → 2013M. Evans, V. Hakim, V. PasquierS. Janowsky, J.L. Lebowitz, E.R. SpeerK. Mallick, D. MukamelC. Enaud, M. Retaux

Edinburgh 2015

Reviews

Richard A. Blythe and Martin R. Evans (2007)Nonequilibrium steady states of matrix-product form: a solver’s guideJournal of Physics A: Mathematical and Theoretical, 40(46), R333

Alexandre Lazarescu (2015)The Physicist’s Companion to Current Fluctuations: One-Dimensional

Bulk-Driven Lattice Gasespreprint arXiv:1507.04179

NON-EQUILIBRIUM STEADY STATES

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T a

ReservoirReservoir

System T b

Equilibrium Ta = Tb = T

P(C) = Z−1e−E(C)kT

No phase transition in one dimension for short-range interactions

Non-equilibrium Ta 6= Tb

P(C) = ?

EXCLUSION PROCESSES

SSEP (Symmetric simple exclusion process)

1 L

1 1 1 1

γ

αβ

δ

ρa =α

α+ γ, τi =

{1 occupied0 empty

, ρb =δ

β + δ

SSEP

1 L

1 1 1 1

γ

αβ

δ

ρa =α

α+ γ, τi =

{1 occupied0 empty

, ρb =δ

β + δ

ASEP (Asymmetric simple exclusion process)

1 L

q 1 q 1

γ

αβ

δ

SSEP

1 L

1 1 1 1

γ

αβ

δ

ρa =α

α+ γ, τi =

{1 occupied0 empty

, ρb =δ

β + δ

ASEP

1 L

q 1 q 1

γ

αβ

δ

TASEP (Totally ASEP)

1 L

1 1α

β

STEADY STATE

1 L

1 1 1 1

γ

αβ

δ

ρa =α

α+ γ, τi =

{1 occupied0 empty

, ρb =δ

β + δ

Equilibrium ρa = ρb = ρ

P(τ1, ...τL) =L∏

i=1

[ ρ τi + (1− τi )(1− ρ) ]

Non-equilibrium ρa 6= ρb

P(τ1, ...τL) given by the matrix ansatz

Alternative expressions: D. Domany Mukamel 1992, Schutz Domany 1993, Liggett 99

MATRIX ANSATZFadeev 1980, ...., D. Evans Hakim Pasquier 1993

SSEP

1 L

1 1 1 1

γ

αβ

δ

P(τ1, . . . , τL) =〈W |X1 . . .XL|V 〉〈W |(D + E )L|V 〉

where Xi =

{D if site i occupied

E if site i empty

〈W |(αE − γD) = 〈W |DE − ED = D + E

(βD − δE )|V 〉 = |V 〉

PROOF (SSEP)

Gain =α〈W |ED3E 2|V 〉+ 〈W |D3EDE |V 〉+ β〈W |ED4E |V 〉

〈W |(D + E)6|V 〉

Loss =(γ + 1+ δ)〈W |D4E 2|V 〉〈W |(D + E)6|V 〉

Gain− Loss =〈W |(αE − γD)D3E 2 − D3(DE − ED)E + D4E(βD − δE)|V 〉

〈W |(D + E)6|V 〉

〈W |(αE − γD) = 〈W | DE − ED = D + E (βD − δE)|V 〉 = |V 〉

PROOF (SSEP)


〈W |(D + E)6|V 〉

Loss =(γ + 1+ δ)〈W |D4E 2|V 〉〈W |(D + E)6|V 〉


〈W |(D + E)6|V 〉


PROOF (SSEP)


〈W |(D + E)6|V 〉

Loss =(γ + 1+ δ)〈W |D4E 2|V 〉〈W |(D + E)6|V 〉


〈W |(D + E)6|V 〉


PHASE DIAGRAM FOR THE TASEP

1 L

1 1α

β

P(τ1, . . . , τL) =〈W |X1 . . .XL|V 〉〈W |(D + E)L|V 〉

where Xi = D (site i occupied) and Xi = E (empty)

〈W |αE = 〈W |DE = D + E

βD|V 〉 = |V 〉

Current through bond i , i + 1

J =〈W |(D + E )i−1DE (D + E )L−i |V 〉

〈W |(D + E )L|V 〉=

〈W |(D + E )L−1|V 〉〈W |(D + E )L|V 〉

〈W |αE = 〈W |DE = D + EβD|V 〉 = |V 〉

If F (E) is a polynomial, one has DF (E) = F (1)D + E F (E)−F (1)E−1

(D + E)N =N∑

p=1

p(2N − 1− p)!N!(N − p)!

(Ep + Ep−1D + . . .+ Dp)

and〈W |EmDn|V 〉〈W |V 〉 =

1αm

1βn

〈W |(D + E )N |V 〉〈W |V 〉

=N∑

p=1

p(2N − 1− p)!N!(N − p)!

1αp+1 −

1βp+1

1α− 1β

PHASE DIAGRAM FOR THE TASEP

Krug 1991D. Domany Mukamel 1992

D. Evans Hakim Pasquier 1993Schutz Domany 1993

1 L

1 1α

β

α

β

.5

.5

ρ= .5

ρ= 1−β

ρ=α

NON-GAUSSIAN DENSITY FLUCTUATIONS FOR THE ASEP

D. Enaud Lebowitz 2004

1 1α

β

1 LL x L x’

N(x , x ′) number of particles between Lx and Lx ′

N(x , x ′)L

− 12=

B(x ′)− B(x) + Y (x ′)− Y (x)2√

L

B is a Brownian path and Y is a Brownian excursion

x

Y(x)

B(x)

x

TASEP: α = β = 1〈W |αE = 〈W |

DE = D + EβD|V 〉 = |V 〉

D =

1 1 0 0 · · ·0 1 1 0 · · ·0 0 1 1 · · ·

. . .. . .

E =

1 0 0 0 · · ·1 1 0 0 · · ·0 1 1 0 · · ·0 0 1 1 · · ·

. . .. . .

with 〈W | = (1, 0, 0 . . .) |V 〉 =

100...

〈W |(D + E )N |V 〉〈W |V 〉

=∑

Paths

Weight(Path)

〈W | = (1, 0, 0 . . .) D =

1 1 0 0 · · ·0 1 1 0 · · ·0 0 1 1 · · ·

. . .. . .

E =

1 0 0 0 · · ·1 1 0 0 · · ·0 1 1 0 · · ·0 0 1 1 · · ·

. . .. . .

|V 〉 =

100...

〈W |(D + E )N |V 〉〈W |V 〉

=∑

Paths

Weight(Path)

〈W | = (1, 0, 0 . . .) D =

1 1 0 0 · · ·0 1 1 0 · · ·0 0 1 1 · · ·

. . .. . .

E =

1 0 0 0 · · ·1 1 0 0 · · ·0 1 1 0 · · ·0 0 1 1 · · ·

. . .. . .

|V 〉 =

100...

〈W |(D + E )N |V 〉〈W |V 〉

=∑

Paths

Weight(Path)

TASEP and Brownian excursions⟨(τLx1 −

12

). . .

(τLxk −

12

)⟩=

1(4L)k/2

dk 〈y1 . . . yk〉dx1 . . . dxk

where y(x) is a Brownian excursion and yi = y(xi )

small

P(y1 . . . yk) =

hx1(y1) gx2−x1(y1, y2) . . . gxk−xk−1(yk−1, yk) h1−xk (yk)√π

andhx(y) =

2yx3/2 e−y2/x

gx(y , y ′) =1√πx

(e−(y−y ′)2/x − e−(y+y ′)2/x

)

SECOND CLASS PARTICLE

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First class

Second class

1

1

1

tr(X1X2 . . .XN)

First class = DSecond class = AEmpty = E

DE = D + E

DA = A

AE = A

SECOND CLASS PARTICLE

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First class

Second class

1

1

1

tr(X1X2 . . .XN)

First class = DSecond class = AEmpty = E

DE = D + E

DA = A

AE = A

Shocks

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Shock

density

ρ

ρ

−

+

Weight= 〈w |X−k · · ·X−1 A X1 · · ·Xk′ |v〉

〈w |(D + E ) = 〈w | ; (D + E )|v〉 = |v〉AE = (1− ρ−)(1− ρ+)A ; DA = ρ−ρ+A

DE = (1− ρ−)(1− ρ+)D + ρ−ρ+)E

PHASE TRANSITION

0.2 0.21

LARGE DEVIATION FUNCTIONAL F [{ρ(x)}]

(x)ρ

ρb

ρ a

0 1x

ρ*(x)

Pro({ρ(x)}) ∼ exp[−LF({ρ(x)})]

For equilibrium systems, F({ρ(x)}) is the free energy

For the typical profile ρ∗(x), one has F({ρ∗(x)}) = 0

LARGE DEVIATION FUNCTIONAL F [{ρ(x)]}(x)ρ

ρb

ρ a

0 1x

ρ*(x)


For equilibrium systems, F [{ρ(x)}] is the free energy

Procedure:

1. Cut the system into k boxes of length L/k

2. Pro(ρ1, ρ2, ...ρk) is the sum of the weights of all microscopicconfigurations with densities ρ1 in the first box, .... ρk in the kth box

3. For large L, Pro(ρ1, ρ2, ...ρk) ∼ exp[−LF(ρ1, ρ2, ...ρk)]

4. Take k →∞ with (k � L)

LARGE DEVIATION FUNCTIONAL FOR THE SSEP


Equilibrium ρa = ρb = F

F({ρ(x)}) =∫ 1

0dx[(1− ρ(x)) log 1− ρ(x)

1− F+ ρ(x) log

ρ(x)F

]

Non-equilibrium (ρa 6= ρb)

D. Lebowitz Speer 2001-2002Bertini De Sole Gabrielli Jona-Lasinio Landim 2002

F({ρ(x)}) = supF (x)

∫ 1

0dx[(1− ρ(x)) log 1− ρ(x)

1− F (x)+ ρ(x) log

ρ(x)F (x)

+log F ′(x)ρb − ρa

]

with F (x) monotone, F (0) = ρa and F (1) =ρb

Non-equilibrium (ρa 6= ρb)

F({ρ(x)}) = supF (x)

∫ 1

0dx[(1− ρ(x)) log 1− ρ(x)

1− F (x)+ ρ(x) log

ρ(x)F (x)

+ logF ′(x)ρb − ρa

]with F (x) monotone, F (0) = ρa and F (1) =ρb

Consequences:

F is non-local: for example for small ρa − ρb

F({ρ(x)}) =∫ 1

0dx (1− ρ(x)) log

1− ρ(x)1− ρ∗(x)

+ ρ(x) logρ(x)ρ∗(x)

+(ρa − ρb)

2

(ρa − ρ2a)

2

∫ 1

0dx∫ 1

xdy x(1− y)

(ρ(x)− ρ∗(x)

)(ρ(y)− ρ∗(y)

)+ O(ρa − ρb)

3

where ρ∗(x) = 〈ρ(x)〉 =(1− x)ρa + xρb

Long-range correlations Spohn 82

〈ρ(x)ρ(y)〉 − 〈ρ(x)〉〈ρ(y)〉 ' 1L

G(x , y) = − (ρa − ρb)2

Lx(1− y)

ADDITIVITY FOR THE SSEP

( x ) ( x )( x )ρ1ρ c

0

ρa

1

ρ b

x x

ρ 2

0

ρa

1

ρ b

x

ρ


Try to find ρc such that

F({ρ(x)}|ρa, ρb) = x F({ρ1(x)}|ρa, ρc) + (1− x) F({ρ2(x)}|ρc , ρb)

Idea

P(τ1, . . . , τL) =〈W |X1 . . .XL|V 〉〈W |(D + E)L|V 〉

Try to insert a complete basis

〈W |X1 . . .XL|V 〉 =∫

dU 〈W |X1 . . .XL′ |U〉K (U)〈U| XL′+1 . . .XL|V 〉

In practise

Define the eigenvectors 〈ρ, a| and |ρ, b〉

〈ρ, a| [ρE − (1− ρ)D] = a〈ρ, a|[(1− ρ)D − ρE ] |ρ, b〉 = b|ρ, b〉

Then 〈W | = 〈ρa, (α+ γ)−1| and |V 〉 = |ρb, (β + δ)−1〉

Then one can prove that:

〈ρa, a|Y1Y2|ρb, b〉〈ρa, a|ρb, b〉

=

∮ρb<|ρc |<ρa

dρc2iπ

(ρa − ρb)a+b

(ρa − ρc )a+b(ρc − ρb)

〈ρa, a|Y1|ρc , b〉〈ρa, a|ρc , b〉

〈ρc , 1− b|Y2|ρb, b〉〈ρc , 1− b|ρb, b〉

LARGE DEVIATIONS OF THE DENSITY: EXTENSIONS

Large deviations of the density profileD. Lebowitz Speer 2002-2003 ASEPEnaud D. 2004 WASEP

Macroscopic fluctuation theoryBertini De Sole Gabrielli Jona-Lasinio Landim 2001 →· · · KMP model· · · tagged particle· · · current fluctuations· · · · · ·

CONCLUSION

Matrix ansatz for the steady state

Phase diagram

Correlation functions

Several species

Large deviation function

Finite size effects

Matrix ansatz for the current fluctuations

Talk by Kirone Mallick

Review by A. Lazarescu

Matrix ansatz for exclusion processes - Bernard Derrida.pdf

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