Boundary-driven interacting particle systems Cristian Giardina - PowerPoint PPT Presentation

Boundary-driven interacting particle systems Cristian Giardina’ Joint work with J. Kurchan (Paris), F . Redig (Delft) K.Vafayi (Eindhoven), G. Carinci, C. Giberti (Modena). Cristian Giardin` a (UniMoRe)

J = κ ∇ T Fourier law Pure-carbon materials have extremely high thermal conductivity. 1D Hamiltonian models: Oscillators chains (Lebowitz, Lieb, Rieder, 1967): κ ∼ N . Non-linear oscillators chains (Lepri, Livi, Politi, Phys. Rep. 2003): κ ∼ N α , 0 < α < 1 Non-linear fluctuating hydrodynamics (van Beijeren 2012, Spohn 2013) Cristian Giardin` a (UniMoRe)

Stochastic energy exchange models Kipnis, Marchioro, Presutti (1982): z = ( z 1 , . . . , z N ) ∈ R N Observables: Energies at every site + Dynamics: Select a bond at random and uniformly redistribute the energy under the constraint of conserving the total energy. L KMP f ( z ) = � 1 N � dp [ f ( z 1 , . . . , p ( z i + z i + 1 ) , ( 1 − p )( z i + z i + 1 ) , . . . , z N ) − f ( z )] 0 i = 1 Cristian Giardin` a (UniMoRe)

Outline From Hamiltonian to stochastics: a simple model. 1 Duality Theory: 2 Brownian Momentum Process (BMP). Symmetric Inclusion Process (SIP). Self-duality (SIP). 3 Boundary driven systems. 4 A larger picture & “redistribution” models. 5 Cristian Giardin` a (UniMoRe)

From Hamiltonian to stochastics Cristian Giardin` a (UniMoRe)

A simple Hamiltonian model (G., Kurchan, 05) N 1 � 2 � � H ( q , p ) = p i − A i 2 i = 1 A = ( A 1 ( q ) , . . . , A N ( q )) “vector potential” in R N . dq i = v i dt N dv i � = B ij v j dt j = 1 where − ∂ A j ( q ) B ij ( q ) = ∂ A i ( q ) ∂ q j ∂ q i antisymmetric matrix containing the “magnetic fields” Cristian Giardin` a (UniMoRe)

Conservation laws Conservation of Energy: Even if the forces depend on velocities and positions, the model conserves the total (kinetic) energy �� � d 1 2 v 2 � = B ij v i v j = 0 i dt i i , j Conservation of Momentum: If we choose the A i ( x ) such that they are left invariant by the simultaneous translations x i → x i + δ , then the quantity � i p i is conserved. Cristian Giardin` a (UniMoRe)

Example: discrete time dynamics with “magnetic kicks” q ( t + 1 ) = q ( t ) + v ( t ) v ( t + 1 ) = R ( t + 1 ) · v ( t ) with R ( t ) a rotation matrix � cos ( B ( q ( t + 1 ))) � sin ( B ( q ( t + 1 ))) R ( t + 1 ) = − sin ( B ( q ( t + 1 ))) cos ( B ( q ( t + 1 ))) Cristian Giardin` a (UniMoRe)

Chaoticity properties of the map on T 2 q ( 1 ) q ( 1 )  t + 1 = + v cos ( β t ) t   Figure: Poincare section with plane q ( 2 ) = 0 of the map q ( 2 ) q ( 2 ) t + 1 = + v sin ( β t ) t  β t + B ( q ( 1 ) , q ( 2 )  β t + 1 = ) t t � B ( q ( 1 ) , q ( 2 ) ) = q ( 1 ) + q ( 2 ) − 2 π . v 2 1 + v 2 with v = 2 , β = arctan ( v 2 / v 1 ) , Cristian Giardin` a (UniMoRe)

Numerical result 1.00 k N=128 0.10 N=512 N=2048 −3 −2 −1 0 1 2 3 4 10 10 10 10 10 10 10 10 T Thermal conductivity Cristian Giardin` a (UniMoRe)

Duality theory Cristian Giardin` a (UniMoRe)

Duality Definition ( η t ) t ≥ 0 Markov process on Ω with generator L , ( ξ t ) t ≥ 0 Markov process on Ω dual with generator L dual ξ t is dual to η t with duality function D : Ω × Ω dual → R if ∀ t ≥ 0 E η ( D ( η t , ξ )) = E ξ ( D ( η, ξ t )) ∀ ( η, ξ ) ∈ Ω × Ω dual η t is self-dual if L dual = L . Cristian Giardin` a (UniMoRe)

Duality Condition LD ( · , ξ )( η ) = L dual D ( η, · )( ξ ) Indeed E η ( D ( η t , ξ )) = e tL D ( · , ξ )( η ) = e tL dual D ( η, · )( ξ ) = E ξ ( D ( η, ξ t )) Cristian Giardin` a (UniMoRe)

How to find a dual process? Write the generator in abstract form , i.e. as an element of a Lie 1 algebra, using creation and annihilation operators. Duality is related to a change of representation, i.e. new 2 operators that satisfy the same algebra. Self-duality is associated to symmetries, i.e. conserved 3 quantities. Cristian Giardin` a (UniMoRe)

The method at work Brownian momentum process ↓ SU(1,1) algebra ↓ Inclusion process Cristian Giardin` a (UniMoRe)

Brownian momentum process (BMP) on two sites Given ( x i , x j ) ≡ velocities of the couple ( i , j ) � 2 � ∂ ∂ L BMP f ( x i , x j ) = x i − x j f ( x i , x j ) i , j ∂ x j ∂ x i = ∂ 2 polar coordinates L BMP i , j ∂θ 2 ij Brownian motion for angle θ i , j = arctan ( x j / x i ) r 2 i , j = x 2 i + x 2 total kinetic energy conserved: j Cristian Giardin` a (UniMoRe)

Brownian momentum process (BMP) For a graph G = ( V , E ) let Ω = ⊗ i ∈ V Ω i = R | V | . Configuration x = ( x 1 , . . . , x | V | ) ∈ Ω Generator BMP � 2 � ∂ ∂ L BMP = � L BMP � = x i − x j i , j ∂ x j ∂ x i ( i , j ) ∈ E ( i , j ) ∈ E Stationary measures: Gaussian product measures x 2 | V | i e − 2 T � √ d µ ( x ) = dx i 2 π T i = 1 Cristian Giardin` a (UniMoRe)

Symmetric Inclusion Process (SIP) Ω dual = ⊗ i ∈ V Ω dual = { 0 , 1 , 2 , . . . } | V | i Configuration ξ = ( ξ 1 , . . . , ξ | V | ) ∈ Ω dual Generator SIP L SIP f ( ξ ) = � L SIP i , j f ( ξ ) ( i , j ) ∈ E � ξ j + 1 � � ξ i + 1 � � [ f ( ξ i , j ) − f ( ξ )] + ξ j [ f ( ξ j , i ) − f ( ξ )] = ξ i 2 2 ( i , j ) ∈ E Stationary (rever.) measures: product of Negative Binomial( r , p ) with r = 2 | V | p n i ( 1 − p ) r Γ( r + n i ) � P r ( ξ 1 = n 1 , . . . , ξ | V | = n | V | ) = n i ! Γ( r ) i = 1 Cristian Giardin` a (UniMoRe)

Duality between BMP and SIP Theorem 1 The process { x ( t ) } t ≥ 0 with generator L = L BMP and the process { ξ ( t ) } t ≥ 0 with generator L dual = L SIP are dual on x 2 ξ i � i D ( x , ξ ) = ( 2 ξ i − 1 )!! i ∈ V Proof: An explicit computation gives L BMP D ( · , ξ )( x ) = L SIP D ( x , · )( ξ ) Cristian Giardin` a (UniMoRe)

Duality explained SU ( 1 , 1 ) ferromagnetic quantum spin chain Abstract operator � � j + 1 � K + i K − + K − i K + − 2 K o i K o L = j j 8 ( i , j ) ∈ E with { K + i , K − i , K o i } i ∈ V satisfying SU ( 1 , 1 ) commutation relations: i , K ± j ] = ± δ i , j K ± [ K − i , K + [ K o j ] = 2 δ i , j K o i i Duality between L BMP e L SIP corresponds to two different representations of the operator L . Duality fct is the intertwiner. Cristian Giardin` a (UniMoRe)

SU ( 1 , 1 ) structure Continuous representation ∂ 2 = 1 = 1 K + K − 2 x 2 i i i 2 ∂ x 2 i i = 1 � ∂ + ∂ � K o x i x i 4 ∂ x i ∂ x i satisfy commutation relations of the SU ( 1 , 1 ) Lie algebra [ K o i , K ± i ] = ± K ± [ K − i , K + i ] = 2 K o i i In this representation � 2 � ∂ ∂ L = L BMP = � x i − x j ∂ x j ∂ x i ( i , j ) ∈ E Cristian Giardin` a (UniMoRe)

SU ( 1 , 1 ) structure Discete representation � � ξ i + 1 K + i | ξ i � = | ξ i + 1 � 2 K − i | ξ i � = ξ i | ξ i − 1 � � ξ i + 1 � K o i | ξ i � = | ξ i � 4 In this representation L f ( ξ ) = L SIP f ( ξ ) � ξ j + 1 � � ξ i + 1 � � [ f ( ξ i , j ) − f ( ξ )] + ξ j [ f ( ξ j , i ) − f ( ξ )] = ξ i 2 2 ( i , j ) ∈ E Cristian Giardin` a (UniMoRe)

SU ( 1 , 1 ) structure Intertwiner K + i D i ( · , ξ i )( x i ) = K + i D i ( x i , · )( ξ i ) K − i D i ( · , ξ i )( x i ) = K − i D i ( x i , · )( ξ i ) K o i D i ( · , ξ i )( x i ) = K o i D i ( x i , · )( ξ i ) From the creation operators x 2 � ξ i + 1 � i 2 D i ( x i , ξ i ) = D ( x , ξ i + 1 ) 2 Therefore x 2 ξ i i D i ( x i , ξ i ) = ( 2 ξ i − 1 )!! D i ( x i , 0 ) Cristian Giardin` a (UniMoRe)

Self-duality Cristian Giardin` a (UniMoRe)

Markov chain with finite state space 1. Matrix formulation of self-duality ( L dual = L ) LD = DL T Indeed � L ( η, η ′ ) D ( η ′ , ξ ) = LD ( · , ξ )( η ) = LD ( η, · )( ξ ) = � L ( ξ, ξ ′ ) D ( η, ξ ′ ) η ′ ξ ′ Cristian Giardin` a (UniMoRe)

Self-Duality 2. trivial self-duality ⇐ ⇒ reversible measure µ 1 d ( η, ξ ) = µ ( η ) δ η,ξ Indeed L ( η, ξ ) = Ld ( η, ξ ) = dL T ( η, ξ ) = L ( ξ, η ) µ ( ξ ) µ ( η ) Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd = SdL T = DL T Self-duality is related to the action of a symmetry. Cristian Giardin` a (UniMoRe)

Self-duality of the SIP process Theorem 2 The process with generator L SIP is self-dual on functions � 1 � Γ η i ! � 2 D ( η, ξ ) = � 1 ( η i − ξ i )! � Γ 2 + ξ i i ∈ V Proof: [ L SIP , � K o i ] = [ L SIP , � K + i ] = [ L SIP , � K − i ] = 0 i i i i K + � Self-duality fct related to the simmetry S = e i Cristian Giardin` a (UniMoRe)

Boundary driven systems. Cristian Giardin` a (UniMoRe)

Brownian Momentum Process with reservoirs 2 2 x x i j BMP L     2 2     res res L T x L T x     R R N 2 L L 1 2 x x x x N N 1 1 Cristian Giardin` a (UniMoRe)

Inclusion Process with absorbing reservoirs j i SIP L         abs L       L abs 1 , 0 f 2 f f N 1 1 Cristian Giardin` a (UniMoRe)