Effect of memory on current fluctuations in interacting-particle systems Rosemary Harris Large Fluctuations in Non-Equilibrium Systems, Dresden, July 13th 2011
Traffic fluctuations
Outline • Introduction • General approach for current-dependent rates – “Temporal additivity principle” [RJH and H. Touchette: J. Phys. A: Math. Theor. 42 , 342001 (2009)] ∗ Toy example: random walk – Expansion about fixed-points • Application to many-particle systems – Example 1: Totally Asymmetric Simple Exclusion Process ∗ Modified phase diagram, (super-)diffusive fluctuations, simulation – Example 2: Zero-Range Process ∗ Exact numerics, validity of fluctuation symmetry • Summary and outlook
Introduction: Memoryless systems • Discrete-space, continuous-time Markov process – Configurations σ ( t ) – Transition rates w σ ′ ,σ – Non-equilibrium systems characterized by (time-integrated) currents J t – Typically have large deviation principle Prob ( J t /t = j ) ∼ e − I w ( j ) t • Toy example: Single particle hopping rightwards on an infinite lattice v 1.4 – Let J t count the number of jumps up to time t I v ( j ) 1.2 1 – Large deviation function given by 0.8 0.6 I v ( j ) = v − j + j ln j 0.4 0.2 v 0 0 0.5 1 1.5 2
Introduction: Adding memory • Many ways to introduce memory • We consider current-dependent rates • Class of processes where w σ ′ ,σ depend explicitly on σ , σ ′ and J t /t (To avoid singularities, assume initial time t 0 , where 0 ≪ t 0 ≪ t ) • Includes analogues of “elephant random walk” [Sch¨ utz and Trimper ’04] • Non-Markovian process but Markovian in joint current/configuration space • Back to toy example: v ( j ) • How does memory effect the current large deviation principle? I ( j ) t ?) (i.e., do we still have form Prob ( J t /t = j ) ∼ e − ˜
Temporal additivity principle • Claim: [RJH and Touchette ’09] � t � � I w ( j ) ( j + τj ′ ) dτ Prob ( J t /t = j ) ∼ exp − min j ( τ ) t 0 where integral is minimized over all j ( τ ) with j ( t 0 ) = j 0 and j ( t ) = j • General idea: Look for most probable path j ( τ ) satisfying boundary conditions • Temporal analogue of additivity principle of [Bodineau and Derrida ’04]
Temporal additivity principle • To make t -dependence more explicit write Prob ( J t /t = j ) ∼ e − t α ˜ I ( j ) , If ˜ I ( j ) exists and is not everywhere zero then have large deviation principle with � t 1 ˜ I w ( j ) ( j + τj ′ ) dτ. I ( j ) = lim t →∞ min t α j ( τ ) t 0 • If Markovian rate function is known, can find large deviation principle for system with current-dependent rates by minimizing relevant integral • But very few analytically solvable cases so... – Toy example (random walk) – Approximation (TASEP) – Numerics (ZRP)
Toy example: Uni-directional random walk v ( j ) • Euler-Lagrange equation: 2 τj ′ τ 2 j ′′ dv dj − jdv/dj − j + τj ′ − j + τj ′ = 0 v • Consider case v ( j ) = aj (rate proportional to average velocity so far) • Results depend on a : – a > 1 , escape regime: no large deviation principle – a < 1 , localized regime: ∗ System approaches state where particle has zero velocity ∗ Large deviation principle with “speed” t 1 − a Prob ( J t /t = j ) ∼ e − jt a 0 t 1 − a , for j > 0 ∗ Transition from subdiffusive regime to superdiffusive regime at a = 1 / 2 Var [ J t ] ∼ ( t/t 0 ) 2 a
Fixed points, stability • Mean current in memoryless case, given by ¯ j = f ( w ) • Fixed-point in current-dependent case at j ∗ = f ( w ( j ∗ )) • Two possible scenarios: j j f ( w ( j )) f ( w ( j )) • Stability determined by slope � A ∗ = ∂f � � ∂j � j = j ∗ A ∗ < 1 = A ∗ > 1 = ⇒ stable ⇒ unstable
Expansion about fixed point • Assume only one stable fixed point j ∗ • Expanding to second order about this fixed point, E-L equations have solution j ( τ ) = K 1 τ − A ∗ + K 2 τ A ∗ − 1 • ...fixing boundary conditions and integrating gives (1 − 2 A ∗ )( j − j ∗ ) 2 � � for A ∗ < 1 exp t 2 D ∗ 2 Prob ( J t /t = j ) ∼ � (2 A ∗ − 1)( j − j ∗ ) 2 t 2 − 2 A ∗ � for A ∗ > 1 t 2 A ∗ − 1 exp 2 D ∗ 0 2 with � − 1 � � D ∗ = I ′′ w ( j ) ( j ) � � j = j ∗ • Transition at A ∗ = 1 2 – For A ∗ < 1 2 have diffusive behaviour with modified diffusion coefficient – For A ∗ > 1 2 have superdiffusive behaviour
Example 1: Totally Asymmetric Exclusion Process α β • Well-known phase diagram ( p = 1 ): 1 β LD MC 1 2 HD 0 1 0 1 2 α • Current large deviations known in all phases [Lazarescu & Mallick ’11]... ...but can already get some information by expanding about fixed points
Current-dependent TASEP α ( j ) β • Consider current-dependent input rate α ( j ) • Fixed points given by 1 for α ( j ∗ ) > 1 2 , β > 1 4 2 j ∗ = α ( j ∗ )(1 − α ( j ∗ )) for α ( j ∗ ) < 1 2 , β > α ( j ∗ ) for α ( j ∗ ) > β, β < 1 β (1 − β ) 2 • For example, set α ( j ) = α 0 + aj (with a > 0 ): – Get modified phase diagram in ( α 0 , β ) plane – LD–MC transition at β = 1 2 − a 4 − (1 − a )+ √ (1 − a ) 2 +4 aα 0 – LD–HD transition at β = . 2 a
Current-dependent TASEP, phase diagram α ( j ) = α 0 + aj 1 1 0.8 0.8 0.6 0.6 β β 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 α 0 0.6 0.8 1 0 0.2 0.4 α 0 0.6 0.8 1 a = 0 . 8 a = 1 . 2 0.3 0.3 0.25 0.25 0.2 0.2 0.25 0.25 ¯ ¯ ¯ ¯ j j j j 0.15 0.15 0.2 0.2 0.1 0.1 0.05 0.05 0.15 0.15 0 0 0.1 0.1 0.05 0.05 1 1 0 0 0.8 0.8 0.6 0.6 β β 0.4 0.4 0.2 0.2 1 1 0.8 0.8 0 0.6 0 0.6 0.4 0.4 0.2 0.2 0 0 α 0 α 0
Current-dependent TASEP, mean current 1 • Fixed point j ∗ determines mean current in different phases 0.8 − (2 α 0 +1 − a )+ √ (1 − a ) 2 +4 α 0 a • In LD phase have j ∗ = 0.6 β 2 a 2 0.4 • Simulation for β = 0 . 6 , a = 0 . 8 : 0.2 0 0 0.2 0.4 α 0 0.6 0.8 1 0.3 0.25 0.2 0.15 ¯ j 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 α 0
Current-dependent TASEP, fluctuations 1 • In LD phase, have A ∗ = 1 − (1 − a ) 2 + 4 aα 0 � 0.8 • Fluctuations superdiffusive for α 0 < α c = 1 / 4 − (1 − a ) 2 0.6 β 4 a 0.4 • Simulation for β = 0 . 6 , a = 0 . 8 , α c ≈ 0 . 66 : 0.2 0 0 0.2 0.4 α 0 0.6 0.8 1 1 ( � J 2 � − � J � 2 ) /t ( � J 2 � − � J � 2 ) /t 2 A ∗ D ∗ / | 1 − 2 A ∗ | 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 α 0
Example 2: Zero-Range Process • 1d open-boundary ZRP [Levine et al. ’05]: δ α qw n pw n qw n βw n γw n pw n L − 1 1 2 3 4 5 L • No condensation if w n → ∞ as n → ∞ • Current rate function known in Markovian case ( p − q )[ αβ ( p/q ) L − 1 + γδ ] � 4 αβγδ ( p/q ) L − 1 ( p − q ) 2 j 2 + I ( j ) = γ ( p − q − β ) + β ( p − q + γ )( p/q ) L − 1 − [ γ ( p − q − β ) + β ( p − q + γ )( p/q ) L − 1 ] 2 � � � 2 αβ ( p/q ) L − 1 ( p − q ) 4 αβγδ ( p/q ) L − 1 ( p − q ) 2 � � j 2 + − j ln + j ln j + . γ ( p − q − β ) + β ( p − q + γ )( p/q ) L − 1 [ γ ( p − q − β ) + β ( p − q + γ )( p/q ) L − 1 ] 2 [RJH, R´ akos and Sch¨ utz, ’05]
Current-dependent ZRP • Choose current-dependent input rates δ ( j ) α ( j ) qw n pw n qw n βw n γw n pw n L − 1 1 2 3 4 5 L • Solve Euler-Lagrange equations numerically with α ( j ) = αe a ( j − j c ) δ ( j ) = δe − a ( j − j c ) and • For all values of a have fixed point at j ∗ = j c = αβ − γδ β + γ • Numerical parameters: α = 1 , b = 1 . 5 , c = 1 , d = 1 , p = 1 . 1 , q = 1 , L = 5
Current-dependent ZRP, rate function • Numerical solution beyond Gaussian regime: 0.18 a = 0 0.16 a = 0 . 25 a = − 0 . 25 0.14 0.12 0.1 I ( j ) ˜ 0.08 0.06 0.04 0.02 0 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 j
Current-dependent ZRP, fluctuation symmetry • Test of fluctuation symmetry ˜ I ( − j ) − ˜ I ( j ) = Ej 0.25 0.2 I ( j ) 0.15 I ( − j ) − ˜ 0.1 a = 0 ˜ a = 0 . 25 a = − 0 . 25 0.05 E 0 E + E − 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 j
Fluctuation symmetry for currents • Second-order expansion about fixed point yields 2(1 − 2 A ∗ ) j ∗ � � for A ∗ < 1 exp × jt Prob ( J t /t = − j ) D ∗ 2 Prob ( J t /t = j ) ∼ � 2(2 A ∗ − 1) j ∗ × jt 2 − 2 A ∗ � t 2 A ∗ − 1 for A ∗ > 1 exp D ∗ 0 2 • Cf. modified symmetry for anomalous dynamics found in [Chechkin & Klages ’09] • Open question: does symmetry still hold in tails of distribution? – Conjecture: necessary symmetry condition on current-dependence is f ( w ( j )) + f ( w ( − j )) = const j – Heuristic argument from fixed-point picture f ( w ( j )) – Proof from structure of E-L equations?
Recommend
More recommend
