Multispecies Exclusion Processes and Current Fluctuations K. - - PowerPoint PPT Presentation

Multispecies Exclusion Processes and Current Fluctuations

• K. Mallick

Institut de Physique Th´ eorique, CEA Saclay (France)

Dresden, Germany, July 2011

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Introduction

The statistical mechanics of a system at thermal equilibrium is encoded in the Boltzmann-Gibbs canonical law: Peq(C) = e−E(C)/kT Z the Partition Function Z being related to the Thermodynamic Free Energy F: F = −kTLog Z This provides us with a well-defined prescription to analyze systems at equilibrium: (i) Observables are mean values w.r.t. the canonical measure. (ii) Statistical Mechanics predicts fluctuations (typically Gaussian) that are out of reach of Classical Thermodynamics.

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Systems far from equilibrium

No fundamental theory is yet available.

• What are the relevant macroscopic parameters?
• Which functions describe the state of a system?
• Do Universal Laws exist? Can one define Universality Classes?
• Can one postulate a general form for the microscopic measure?
• What do the fluctuations look like (‘non-gaussianity’)?

Example: Stationary driven systems in contact with reservoirs.

R1

J

R2

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

ASEP

q p p p q

Asymmetric Exclusion Process. A paradigm for non-equilibrium Statistical Mechanics.

• EXCLUSION: Hard core-interaction; at most 1 particle per site.
• ASYMMETRIC: External driving; breaks detailed-balance
• PROCESS: Stochastic Markovian dynamics; no Hamiltonian
• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

ORIGINS

• Interacting Brownian Processes (Spitzer, Harris, Liggett).
• Driven diffusive systems (Katz, Lebowitz and Spohn).
• Transport of Macromolecules through thin vessels.

Motion of RNA templates.

• Hopping conductivity in solid electrolytes.
• Directed Polymers in random media. Reptation models.

APPLICATIONS

• Traffic flow.
• Sequence matching.
• Brownian motors.
• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Outline

1.Algebraic Structures in Multispecies Exclusion Processes (C. Arita, A. Ayyer, P. Ferrari, M. R. Evans and S. Prolhac)

• 2. Large deviations of the current in the Open TASEP

(A. Lazarescu)

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

1. Multispecies Models

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

The dynamical rules

N classes of particles and holes with hierarchical priority rules. During an infinitesimal time step dt, the following processes take place

• n each bond with probability dt:

I 0 → 0 I for I = 0 I J → J I for 1 ≤ I < J ≤ N This defines the N-TASEP model on a RING: Particles can always

• vertake holes (= 0-th class particles). First-class particles have highest

priority etc... There are PI particles of class I. Total number of configurations: Ω = L! P0!P1!P2! . . . PN! Description of the Stationary Measure ?

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Matrix Ansatz for Two Species

Algebraic description of the Stationary Measure (Derrida, Evans, Hakim and Pasquier, 1993; Derrida, Janowski, Lebowitz and Speer, 1993). A Configuration is represented by a string e.g. 01220211. The corresponding Stationary Weight is given by p(01220211) = 1 Z Tr(EDAAEADD) 0 → E, 1 → D and 2 → A, operators belong to a quadratic algebra DE = D + E DA = A AE = A e.g. p(01220211) ∝ Tr(D2EA3) = Tr((D2 + D + E)A3) ∝ 3Tr(A3) The Matrix Ansatz allows to calculate stationary state properties such as currents, correlations, fluctuations...

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Representations of the quadratic algebra

The algebra encodes combinatorial recursion relations. Infinite dimensional Representation: D = 1 + δ where δ =right-shift. E = 1 + ǫ where ǫ =left-shift. A = |11| = [δ, ǫ] projector on first coordinate. D =       1 1 . . . 1 1 1 1 ... ... ...       , E = D†, A =     1 . . . . . . . . . . . . .    

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Properties of the Matrix Ansatz

• Matrix Ansatz: Stationary state properties (currents, correlations,

fluctuations).

• Proof that the stationary measure is not given by a

Boltzmann-Gibbs measure (E. Speer).

• Combinatorial Interpretation of these operators?
• No Matrix Ansatz was known for N-TASEP models (for N ≥ 3.)
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Multispecies Exclusion Processes and Current Fluctuations

Geometric Construction of the 2-TASEP stationary measure (O. Angel, P. Ferrari, J. Martin)

A procedure to construct a configuration of the 2-TASEP with P1 First Class Particles and P2 Second Class Particles starting from two independent configurations of the 1 species TASEP.

P1 P + P

1 2

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Geometric Construction of the 2-TASEP stationary measure (O. Angel, P. Ferrari, J. Martin)

A procedure to construct a configuration of the 2-TASEP with P1 First Class Particles and P2 Second Class Particles starting from two independent configurations of the 1 species TASEP.

P1 P + P

1 2

1 1 1 1

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Geometric Construction of the 2-TASEP stationary measure (O. Angel, P. Ferrari, J. Martin)

A procedure to construct a configuration of the 2-TASEP with P1 First Class Particles and P2 Second Class Particles starting from two independent configurations of the 1 species TASEP.

P1 P + P

1 2

1 1 1 1 1 1 1 1

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Geometric Construction of the 2-TASEP stationary measure (O. Angel, P. Ferrari, J. Martin)

A procedure to construct a configuration of the 2-TASEP with P1 First Class Particles and P2 Second Class Particles starting from two independent configurations of the 1 species TASEP.

P1 P + P

1 2

1 1 1 1 1 1 1 1

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Geometric Construction of the 2-TASEP stationary measure (O. Angel, P. Ferrari, J. Martin)

A procedure to construct a configuration of the 2-TASEP with P1 First Class Particles and P2 Second Class Particles starting from two independent configurations of the 1 species TASEP.

P1 P + P

1 2

1 1 1 1 1 1 1 2 2 1 2

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Summary of the construction

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 construction is NOT one-to one: the weight of a 2-TASEP configuration is proportional to the total number of ways you can generate it by this construction.

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Relation to the Matrix Ansatz

Characterization of the stationary weights:

• A 1 (on the 1st line) can not be located above a 2 (on the 2nd line).
• Factorisation Property: All the 1’s (on the 2nd line) situated

between two 2’s MUST be linked to 1’s (on the 1st line) that are located between the positions of the two 2’s (No Crossing Condition).

• ‘Pushing’ Procedure: The ‘ancestors’ of a string of the type

210102 are the strings obtained by pushing the 1’s to the right i.e., 210102, 210012, 201102, 201012, 200112.

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Relation to the Matrix Ansatz

Characterization of the stationary weights:

• A 1 (on the 1st line) can not be located above a 2 (on the 2nd line).
• Factorisation Property: All the 1’s (on the 2nd line) situated

between two 2’s MUST be linked to 1’s (on the 1st line) that are located between the positions of the two 2’s (No Crossing Condition).

• ‘Pushing’ Procedure: The ‘ancestors’ of a string of the type

210102 are the strings obtained by pushing the 1’s to the right i.e., 210102, 210012, 201102, 201012, 200112. This Geometric Construction is encoded by the Matrix Ansatz:

• Factorisation Property: A is a PROJECTOR.
• Pushing Procedure: D and E are SHIFT OPERATORS

(right-shift and left-shift, respectively).

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Multispecies Exclusion Processes and Current Fluctuations

From 3 lines of TASEP to the 3-TASEP

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 weight of a 3-TASEP configuration is proportional to the total number of ways you can generate it by this construction.

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Weights of the 3-TASEP

• REVERT the graphical procedure → ALGORITHM for constructing

all ancestors of a given N-TASEP configuration.

• ENCODE this reverse algorithm into operators → ALGEBRA.
• CALCULATE the stationary weights → TRACES over this algebra.
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Multispecies Exclusion Processes and Current Fluctuations

Nested Matrix Ansatz for the N-TASEP

Tensor Products of Quadratic Algebras Hierarchical construction of representations of ‘nested algebras’ using the D, A and E matrices and the shift operators δ = D − 1 and ǫ = E − 1. For the 3-species TASEP case: 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

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Explicit representations

The operators have a bidiagonal block structure. Defining matrices F, G, H and K as F =           D . . . D ... D ... D ... . . . ... ... ...           G =           E . . . A E ... A E ... A E ... . . . ... ... ...           H =           A E . . . A E ... A E ... A ... . . . ... ... ...           K =           E . . . ... ... ... . . . ... ... ...          

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Explicit representations

The matrix representation of the 3-TASEP is given by ˆ P1 =           F G . . . F G ... F G ... F ... . . . ... ... ...           ˆ P2 =           H . . . ... ... ... . . . ... ... ...           ˆ P3 =           K . . . ... ... ... . . . ... ... ...           ˆ P0 =           G . . . F G ... F G ... F G ... . . . ... ... ...           Triply infinite dimensional matrices: the coefficients are infinite dimensional matrices with elements D, A and E (also infinite matrices).

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Queueing Theory Interpretation

1 1 1 1 1 1 2 2 1 2 1

Time the queue length of

queue arrivals services

The matrices D and E act on |n the length of the queue: Service Time: D|n = |n + |n − 1 Non-Service Time: E|n = |n + |n + 1 This queueing process can be generalized to the N-TASEP. The matrices act on the queue at each time step: they are constructed by inspection of the different possible arrivals at a given time.

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Generalization to the N-ASEP

If backward jumps are allowed (rate q = 0) DE − qED = (1 − q)(D + E) DA − qAD = (1 − q)A AE − qEA = (1 − q)A → Replace the previous shift-operators by deformed shift-operators: δǫ = 1 → δǫ − qǫδ = 1 Recursive Matrix Ansatz: X (N)

J

=

N−1

• M=0

a(N)

JM ⊗ X (N−1) M

for 0 ≤ J ≤ N with X (1) = X (1)

1

= 1 The auxiliary matrices a(N)

JM can be written as a rectangular tableau

a(2) =  

1

1 ǫ δ

1

1 A   and a(3) =    

1

1 ⊗ 1 1 ǫ ⊗ 1 1

1

1 ⊗ ǫ δ ⊗ 1 1

1

1 ⊗ 1 1 δ ⊗ ǫ A ⊗ δ A ⊗ 1 1 A ⊗ A    

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Multispecies Exclusion Processes and Current Fluctuations

Conjugation Relation

The Recursive Matrix Ansatz allows us to define a ‘Transfer Matrix’ between the (N-1)-species ASEP and the N-ASEP. ΩN−1 ΩN−1 ΩN ΩN TN−1,N TN−1,N MN MN−1 ✲ ✲ ❄ ❄ The operator TN−1,N lifts the (N-1)-ASEP into the N-ASEP and allows us to construct whole sectors of the spectrum. It plays a role opposite to that of the identification operator.

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Hasse Diagram of the N-ASEP

(0) (1,1,1,1) 1,2,3 (1,2,1) 1,3 (2,1,1) 2,3 (1,1,2) 1,2 (2,2) 2 (3,1) 3 (1,3) 1 f g f g f g f g f g f g f g

To each arrow corresponds a different quadratic algebra that leads to different lifting operators and to generalized Ferrari-Martin constructions. These algebras have been studied in C. Arita et al. (2011).

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

• 2. Current Fluctuations

in the open TASEP

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Multispecies Exclusion Processes and Current Fluctuations

The total current

The fundamental paradigm

R1

J

R2

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Multispecies Exclusion Processes and Current Fluctuations

The total current

The fundamental paradigm

R1

J

R2

The totally asymmetric exclusion model with open boundaries

α β

1 1

1 L

RESERVOIR RESERVOIR

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Multispecies Exclusion Processes and Current Fluctuations

The Matrix Ansatz (DEHP, 1993)

The stationary probability of a configuration C is given by P(C) = 1 ZL α|

L

• i=1

(τiD + (1 − τi)E) |β . where τi = 1 (or 0) if the site i is occupied (or empty). The normalization constant is ZL = α| (D + E)L |β The operators D and E, the vectors α| and |β satisfy D E = D + E D |β = 1 β |β α| E = 1 αα|

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Multispecies Exclusion Processes and Current Fluctuations

Phase Diagram

= β (1 −β) ρ = 1 − β

J

ρ = α = α(1−α)

J

ρ = 1/2 J = 1/4

HIGH DENSITY LOW DENSITY

α β

MAXIMAL CURRENT

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Multispecies Exclusion Processes and Current Fluctuations

Large Deviations of the Current: Framework

Let Nt be the TOTAL (time-integrated) current through the system between 0 and t. When a particle enters the system: Nt = Nt + 1 Expectation value: limt→∞

Nt t

= J(α, β, L) = ZL−1

ZL

Variance: limt→∞

N2

t −Nt2

t

= ∆(α, β, L) Cumulant Generating Function: exp(γ Nt) ≃ exp(E(γ)t) The Large-Deviation Function F(j) of the total current P Nt t = j

• ∼e−tF(j)

is the Legendre transform of the Cumulant Generating Function E(γ).

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Multispecies Exclusion Processes and Current Fluctuations

Full Current Statistics

In the case α = β = 1, a parametric representation of the cumulant generating function E(γ): γ = −

∞

• k=1

(2k)! k! [2k(L + 1)]! [k(L + 1)]! [k(L + 2)]! Bk 2k , E = −

∞

• k=1

(2k)! k! [2k(L + 1) − 2]! [k(L + 1) − 1]! [k(L + 2) − 1]! Bk 2k . First cumulants of the current Mean Value : J =

L+2 2(2L+1)

Variance : ∆ = 3

2 (4L+1)![L!(L+2)!]2 [(2L+1)!]3(2L+3)!

Skewness : E3 = 12 [(L+1)!]2[(L+2)!]4

(2L+1)[(2L+2)!]3

• 9 (L+1)!(L+2)!(4L+2)!(4L+4)!

(2L+1)![(2L+2)!]2[(2L+4)!]2 − 20 (6L+4)! (3L+2)!(3L+6)!

• For large systems: E3 → 2187−1280

√ 3 10368

π ∼ −0.0090978...

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Multispecies Exclusion Processes and Current Fluctuations

Full Current Statistics

For arbitrary (α, β), the parametric representation of E(γ) is γ = −

∞

• k=1

Ck(α, β)Bk 2k E = −

∞

• k=1

Dk(α, β)Bk 2k with Ck(α, β) =

• {0,a,b}

dz 2iπ F(z)k z and Dk(α, β) =

• {0,a,b}

dz 2iπ F(z)k (1 + z)2 where F(z) = −(1 + z)2L(1 − z2)2 zL(1 − az)(z − a)(1 − bz)(z − b) , a = 1 − α α , b = 1 − β β

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Multispecies Exclusion Processes and Current Fluctuations

Some explicit expressions

Mean Current: (Same expression as in DEHP) J = D1(α, β) C1(α, β) Fluctuations: (An expression more compact than the one previously

• btained)

∆ = D1 C2 − D2 C1 C 3

1

Saddle point analysis in the low density phase: (ρ = α) E1 = ρ(1 − ρ) E2 = ρ(1 − ρ)(1 − 2ρ) E3 = ρ(1 − ρ)(1 − 6ρ + 6ρ2) E4 = ρ(1 − ρ)(1 − 2ρ)(1 − 12ρ + 12ρ2) E5 = ρ(1 − ρ)(1 − 30ρ + 150ρ2 − 240ρ3 + 120ρ4) ...

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Multispecies Exclusion Processes and Current Fluctuations

• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations

Behaviour in the TASEP Phase Diagram

In the limit L → ∞ of systems of large size, we have Maximal Current phase α > 1/2 and β > 1/2: Cumulants are independent from α and β and are the same as for α = β = 1: Ek ∼ π(πL)k/2−3/2 for k ≥ 2 Low Density phase α < min(β, 1/2): Use saddle-point and Lagrange Inversion Formula to obtain E(γ) = a a + 1 eγ − 1 eγ + a Agrees with Bethe Ansatz and Macroscopic Fluctuation Theory. High Density phase is symmetrical to Low Density via α ↔ β. Along the shock line α = β ≤ 1/2: Ek ≃ ǫkα(1 − α)(1 − 2α)k−1Lk−2 for k ≥ 2 The coefficients ǫ2 = 2/3, ǫ3 = −1/30, ǫ4 = 2/315, ǫ5 = −1/1890..., can be calculated by Domain Wall Theory.

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Multispecies Exclusion Processes and Current Fluctuations

DMRG Results ( M. Gorissen, C. Vanderzande)

• 0.03
• 0.028
• 0.026
• 0.024
• 0.022
• 0.02
• 0.018
• 0.016
• 0.014
• 0.012
• 0.01

10 20 30 40 50 60 70 80 90 100 C3

*(L)

L α = 0.50, β = 0.65 DMRG results exact results

• 0.009
• 0.008
• 0.007
• 0.006
• 0.005
• 0.004
• 0.003
• 0.002

10 20 30 40 50 60 70 80 90 100 C3

*(L)

L α = 0.65, β = 0.65 DMRG results exact results

SKEWNESS

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Multispecies Exclusion Processes and Current Fluctuations

Fourth Cumulant (DMRG)

0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 10 20 30 40 50 60 70 80 90 100 C4

*(L)

L α = 0.65, β = 0.65 DMRG results exact results

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Multispecies Exclusion Processes and Current Fluctuations

Outline of the calculation

• The function E(γ) is expressed as the dominant eigenvalue of a

deformation of the Markov Matrix: M(γ) = M + (eγ − 1)M1

• E(γ) and its corresponding eigenvector are developed perturbatively

w.r.t. γ.

• Construction of a Matrix Ansatz at each order k with (2k + 1)

Tensor Products of quadratic algebras as in the multispecies exclusion process.

• k-th Matrix Ansatz → k-th term in the expansion of E(γ).
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Multispecies Exclusion Processes and Current Fluctuations

Outline of the calculation

• The function E(γ) is expressed as the dominant eigenvalue of a

deformation of the Markov Matrix: M(γ) = M + (eγ − 1)M1

• E(γ) and its corresponding eigenvector are developed perturbatively

w.r.t. γ.

• Construction of a Matrix Ansatz at each order k with (2k + 1)

Tensor Products of quadratic algebras as in the multispecies exclusion process.

• k-th Matrix Ansatz → k-th term in the expansion of E(γ).
• The formula is checked against exact calculations on systems of

sizes ≤ 10 for arbitrary rational values of (α, β).

• Large system size limits and known special cases are recovered.
• DMRG results of M. Gorissen and C. Vanderzande.
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Multispecies Exclusion Processes and Current Fluctuations

Conclusion

Exact solutions of the asymmetric exclusion process are paradigms for the behaviour of systems far from equilibrium in low dimensions: Dynamical phase transitions, Non-Gibbsean measures, Large deviations, Fluctuations Theorems... Tensor products of quadratic algebras provides us with an efficient tool to solve very challenging problems: multispecies models; current fluctuations in the open TASEP. They also allow us to calculate current fluctuations in the PASEP with

• pen boundaries (in progress). They may be useful to solve other difficult
• pen problems such as the bridge model or the ABC model.
• C. Arita, A. Ayyer, M. Evans, P. Ferrari, O. Golinelli, A. Lazarescu,
• S. Prolhac and M. Woelki.
• K. Mallick

Multispecies Exclusion Processes and Current Fluctuations