Current Fluctuations in the Exclusion Process K. Mallick Institut de Physique Th´ eorique, CEA Saclay (France) GGI Workshop in Non-equilibrium Physics (Florence, June 20, 2014) K. Mallick Current Fluctuations in the Exclusion Process
Classical Transport in 1d: ASEP A paradigmatic picture of a non-equilibrium system J R2 R1 K. Mallick Current Fluctuations in the Exclusion Process
Classical Transport in 1d: ASEP A paradigmatic picture of a non-equilibrium system J R2 R1 The asymmetric exclusion model with open boundaries α β q 1 RESERVOIR RESERVOIR 1 L γ δ Our aim is to study the statistics of the current and its large deviations starting from this microscopic model. K. Mallick Current Fluctuations in the Exclusion Process
Elementary Model for Protein Synthesis C. T. MacDonald, J. H. Gibbs and A.C. Pipkin, Kinetics of biopolymerization on nucleic acid templates, Biopolymers (1968). K. Mallick Current Fluctuations in the Exclusion Process
The Grotthuss Mechanism for proton transfer (b) (a) A proton hops along an oxygen backbone of a line of water molecules transiently converting each water molecule it visits into H 3 O + . K. Mallick Current Fluctuations in the Exclusion Process
The Kardar-Parisi-Zhang equation in 1d The height of an interface h ( x , t ) satisfies the generic KPZ equation � 2 ∂ t = ν ∂ 2 h ∂ h ∂ x 2 + λ � ∂ h + ξ ( x , t ) 2 ∂ x The ASEP is a discrete version of the KPZ equation in one-dimension. K. Mallick Current Fluctuations in the Exclusion Process
Various Boundary Conditions for the ASEP The pure ASEP can be studied on a periodic chain (a), on the infinite lattice (b) or on a finite lattice connected to two reservoirs (c). K. Mallick Current Fluctuations in the Exclusion Process
Statistics of the Integrated Current J R2 R1 Let Y t be the total charge transported through (a bond of) the system (Integrated or total current) between time 0 and time t . Y t In the stationary state: a non-vanishing mean-current t → J The fluctuations of Y t obey a Large Deviation Principle: � Y t � ∼ e − t Φ( j ) P t = j Φ( j ) being the large deviation function of the integrated current e µ Y t � ≃ e E ( µ ) t for t → ∞ , � Equivalently, use the generating function: They are related by Legendre transform: E ( µ ) = max j ( µ j − Φ( j )) K. Mallick Current Fluctuations in the Exclusion Process
Steady State Properties of the ASEP K. Mallick Current Fluctuations in the Exclusion Process
Steady-State of the PERIODIC ASEP 1 x L SITES N PARTICLES ( N ) L Ω = CONFIGURATIONS x asymmetry parameter In the stationary state all configurations have the same probability. If Y t denotes the total number of particles having crossed any bond, then Y t t → J = (1 − x ) N ( L − N ) L ( L − 1) where J is the mean-current in the steady state. What are the current fluctuations? K. Mallick Current Fluctuations in the Exclusion Process
Steady-State of the OPEN-BOUNDARY ASEP Consider first the TASEP on a finite lattice with open boundaries. 1 1 α β RESERVOIR RESERVOIR 1 L In a system of size L , there are 2 L configurations. Each configuration can be represented by a binary string. In the steady state, the configurations appear with some stationary probability: how are these weights be calculated? K. Mallick Current Fluctuations in the Exclusion Process
Exact Solution (1993) The weights of the system satisfy recursion relations: the probability of a configuration of size L can be written as a linear combination of (at most 2) weights of configurations of size L − 1: there is combinatorial structure between systems of different sizes. These recursions can be encoded using generating functions (DDM, 1992: α = β = 1; Sch¨ utz and Domany, 1993: General case). K. Mallick Current Fluctuations in the Exclusion Process
The Matrix Ansatz (DEHP, 1993) The totally asymmetric exclusion model with open boundaries 1 1 α β RESERVOIR RESERVOIR 1 L The stationary probability of a configuration C is given by L P ( C ) = 1 � � α | ( τ i D + (1 − τ i ) E ) | β � . Z L i =1 where τ i = 1 (or 0) if the site i is occupied (or empty). The normalization constant is Z L = � α | ( D + E ) L | β � = � α | C L | β � The operators D and E , the vectors � α | and | β � satisfy D E = D + E D | β � = 1 � α | E = 1 β | β � and α � α | K. Mallick Current Fluctuations in the Exclusion Process
Representations of the quadratic algebra The algebra encodes combinatorial recursion relations between systems of different sizes. The matrices D and E commute whenever they are finite-dimensional: ( D − 1)( E − 1) = 1 . Infinite dimensional Representation: D = 1 + d where d =right-shift. E = 1 + e where e =left-shift. 1 1 0 0 . . . 1 0 0 0 . . . 0 1 1 0 . . . 1 1 0 0 . . . E = D † = D = and 0 0 1 1 . . . 0 1 1 0 . . . ... ... ... ... We also have � α | = (1 , a , a 2 , a 3 . . . ) and | β � = (1 , b , b 2 , b 3 . . . ) with a = (1 − α ) /α and b = (1 − β ) /β. K. Mallick Current Fluctuations in the Exclusion Process
Phase Diagram of the TASEP The matrix Ansatz allows one to calculate Stationary State Properties (currents, correlations, fluctuations). In particular, the following Phase Diagram is found in the infinite size limit (DEHP, 1993; Sch¨ utz and Domany, 1993). β MAXIMAL CURRENT ρ = 1/2 LOW DENSITY J = 1/4 ρ = α J = α(1−α) HIGH DENSITY ρ = 1 − β = β (1 −β) J α K. Mallick Current Fluctuations in the Exclusion Process
Equal-time Steady State Correlations More generally, the Matrix Ansatz gives access to all equal time correlations in the steady-state. Density Profile: ρ i = � τ i � = � α | C i − 1 D C L − i | β � � α | C L | β � Average Stationary Current: J = � τ i (1 − τ i +1 ) � = � α | C i − 1 D E C L − i − 1 | β � = � α | C L − 1 | β � = Z L − 1 � α | C L | β � � α | C L | β � Z L Explicit formulae either by using purely combinatorial/algebraic techniques or via a specific representation (e.g., C can be chosen as a discrete Laplacian): β − p − 1 − α − p − 1 L p (2 L − 1 − p )! � � α | C L | β � = β − 1 − α − 1 L ! ( L − p )! p =1 L +2 For α = β = 1 , J = 2(2 L +1) K. Mallick Current Fluctuations in the Exclusion Process
The General Case (DEHP,1993) α β q 1 RESERVOIR RESERVOIR 1 L γ δ The operators D and E , the vectors � W | and | V � now satisfy D E − qED = (1 − q )( D + E ) ( β D − δ E ) | V � = | V � � W | ( α E − γ D ) = � W | K. Mallick Current Fluctuations in the Exclusion Process
Infinite dimensional Representations The representations are now related to q -deformed oscillators. √ 1 − q 1 0 0 . . . � 1 − q 2 0 1 0 . . . E = D † D = and � 0 0 1 1 − q 3 . . . ... ... K. Mallick Current Fluctuations in the Exclusion Process
The Phase Diagram of the open ASEP 1 − ρ b MAXIMAL CURRENT LOW DENSITY 1/2 HIGH DENSITY 1/2 ρ a 1 ρ a = a + +1 : effective left reservoir density. b + ρ b = b + +1 : effective right reservoir density. (1 − q − α + γ ) 2 + 4 αγ � a ± = (1 − q − α + γ ) ± 2 α (1 − q − β + δ ) 2 + 4 βδ � b ± = (1 − q − β + δ ) ± 2 β K. Mallick Current Fluctuations in the Exclusion Process
The PASEP average current The Matrix Ansatz again gives access to all equal time correlations in the steady-state of PASEP but the calculation are much harder. Average Stationary Current: F ( z ) � dz � Y t � Γ 2 i π z J = lim = (1 − q ) F ( z ) t t →∞ � dz ( z +1) 2 Γ 2 i π (cf. T. Sasamoto, 1999.) • The function F ( z ) is the generating function of Askey-Wilson Polynomials: (1+ z ) L (1+ z − 1 ) L ( z 2 ) ∞ ( z − 2 ) ∞ F ( z ) = ( a + z ) ∞ ( a + z − 1 ) ∞ ( a − z ) ∞ ( a − z − 1 ) ∞ ( b + z ) ∞ ( b + z − 1 ) ∞ ( b − z ) ∞ ( b − z − 1 ) ∞ where ( x ) ∞ = � ∞ k =0 (1 − q k x ) and a ± , b ± depend on the boundary rates. • The complex contour Γ encircles 0, q k a + , q k a − , q k b + , q k b − for k ≥ 0. K. Mallick Current Fluctuations in the Exclusion Process
Current Fluctuations in the periodic ASEP K. Mallick Current Fluctuations in the Exclusion Process
Current Fluctuations on a ring 1 x L SITES N PARTICLES ( N ) L Ω = CONFIGURATIONS x asymmetry parameter Total integrated current Y t , total distance covered by all the N particles, hopping on a ring of size L, between time 0 and time t. WHAT IS THE STATISTICS of Y t ? Let P t ( C , Y ) be the joint probability of being at time t in configuration C with Y t = Y . The time evolution of this joint probability can be deduced from the original Markov equation, by splitting the Markov operator M = M 0 + M + + M − into transitions for which ∆ Y = 0, +1 or -1. K. Mallick Current Fluctuations in the Exclusion Process
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