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: P eq ( 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. J R2 R1 K. Mallick Multispecies Exclusion Processes and Current Fluctuations
ASEP q p p q p 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 on 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 overtake holes (= 0-th class particles). First-class particles have highest priority etc... There are P I particles of class I . Total number of configurations: L ! Ω = P 0 ! P 1 ! P 2 ! . . . P N ! 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 ( D 2 EA 3 ) = Tr (( D 2 + D + E ) A 3 ) ∝ 3 Tr ( A 3 ) 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 = | 1 �� 1 | = [ δ, ǫ ] projector on first coordinate. 1 1 0 0 . . . 1 0 0 . . . 0 1 1 0 0 0 0 . . . , E = D † , A = ... D = 0 0 1 1 0 0 0 . . . ... ... . . . . 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.) 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 P 1 First Class Particles and P 2 Second Class Particles starting from two independent configurations of the 1 species TASEP. P 1 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 P 1 First Class Particles and P 2 Second Class Particles starting from two independent configurations of the 1 species TASEP. P 1 1 1 1 1 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 P 1 First Class Particles and P 2 Second Class Particles starting from two independent configurations of the 1 species TASEP. P 1 1 1 1 1 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 P 1 First Class Particles and P 2 Second Class Particles starting from two independent configurations of the 1 species TASEP. P 1 1 1 1 1 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 P 1 First Class Particles and P 2 Second Class Particles starting from two independent configurations of the 1 species TASEP. P 1 1 1 1 1 P + P 1 2 2 1 1 1 1 2 2 K. Mallick Multispecies Exclusion Processes and Current Fluctuations
Summary of the construction FROM 2 LINES OF TASEP TO 2−TASEP P 1 P + P 1 2 P 1 1 1 1 1 P + P 1 2 2 1 1 1 1 2 2 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). K. Mallick Multispecies Exclusion Processes and Current Fluctuations
From 3 lines of TASEP to the 3-TASEP P 1 1 1 1 1 P + P 1 2 1 1 1 1 2 2 P + P + P 3 2 1 2 3 1 1 1 1 3 2 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. K. Mallick Multispecies Exclusion Processes and Current Fluctuations
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