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Two -Component Symmetric Exclusion Process with Open Boundaries

This paper discusses the Two-Component Symmetric Exclusion Process with Open Boundaries, including its definition, examples, and questions. It explores the concept of Single-File Diffusion and where it occurs in various systems such as biology, randomness, and diffusion in zeolites. The paper also delves into the behavior of these systems in equilibrium and far from equilibrium, as well as the hydrodynamic limit for open boundary conditions.

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Two -Component Symmetric Exclusion Process with Open Boundaries

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  1. Two-Component Symmetric Exclusion Process with Open Boundaries Andreas Brzank1,2 and Gunter M. Schütz1,3 1) Institut für Festkörperforschung, Forschungszentrum Jülich 2) Institut für Experimentalphysik, Universität Leipzig 3) Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn Diffusion Fundamentals 4, 7.1-7.12 (2006) J. Stat. Mech: Theory and Experiment (2007)

  2. Outline: • 1) Single-File Diffusion: Definition, Examples and Questions • 2) Symmetric Exclusion Process with Open Boundaries • Two-Component Symmetric Simple Exclusion Process • Hydrodynamic Limit for Open Boundaries • Steady State Behaviour • Conclusions

  3. Definition • What is Single-File Diffusion and where does it happen? • interacting diffusive particles • (Newtonian or generalized effective forces plus random part) • quasi one-dimensional motion • - confinement to a tube or channel • - attachment to a track • - motion on a lane, narrow passage or trail • no passing (hard core repulsion, size of order of channel width)

  4. Where does it happen? • Biology: ion channels (e.g. pumps: symporter, antiporter) • Randomness: • Diffusion • Thermal activation

  5. Where does it happen? • Colloidal systems: etched channels or optical lattices • Randomness: • Thermal activation

  6. Where does it happen? • Diffusion in zeolites: Automobile exhaust cold-start problem • significant hydrocarbon emission during cold-start period • suggestion: trap heavy HCs until light-off temperature is reached • use channel topology of certain zeolites to trap alsolight HC components • Fibrous zeolites: • quasi-one-dimensional channel network • channel length up to 100 cross sections • pronounced single-file effect MFI-type zeolite

  7. Experimental (Czaplewski et al (2002)): Loading of zeolite samples with model mixture of toluene and propane 1-D EUO zeolite: different single component desorption temperatures (40C,75C), binary mixture has single (toluene) desorption temperature  Trapping Effect Similar: Na-MOR (Mordenite), Cs-MOR (smaller pore size, less side pockets). Where does it happen? zeolite pore wall (quasi 1-D open system) Gas Gas Phase Phase Heavy HC molecules (toluene)) Light molecules (propane)

  8. Questions (1) Do these diverse single-file systems have anything in common? • Equilibrium: No phase transition (quasi one-dimensional, short range interactions) • Subdiffusive MSD <x2(t)> ~ t1/2 (infinite system, rigorous for SEP) • Longest relaxation time  ~ L3 (finite system, scaling and numerics) • More ??  Use lattice gas model to study generic large-scale behaviour!

  9. Questions (2) Stochastic particle systems as models for hydrodynamic behaviour: One-component systems (identical particles):Well-understood Two-component systems (two conservation laws): • hydrodynamics for infinite systems up to appearance of shock • some insight in shocks (Budapest group) • only numerical (but very interesting) results on open boundaries - pumping - boundary layers  Try to derive hydrodynamic limit for open boundary conditions!

  10. I. Two-component Symmetric Simple Exclusion Process (1) • Two-component Symmetric Exclusion Process (2c-SEP) • diffusive motion (random walk) • hard core repulsion (site exclusion) • two particle species (hopping rates DA, DB, “colour”) • non-equilibrium steady state (open boundaries)

  11. I. Two-component Symmetric Simple Exclusion Process (2) Physical interpretation of boundary processes: boundary chemical potentials -A,B = A,B / A,B, +A,B = A,B / A,B • boundary densities A,B = A,B/(1+A,B) (exclusion) • boundary processes = coupling to infinite reservoirs

  12. I. Two-component Symmetric Simple Exclusion Process (3) Equilibrium (reversible dynamics): equal reservoir chemical potentials -A,B = +A,B  equilibrium distribution: product measure with density A,B (bulk density equal to boundary density) • Far from equilibrium (finite reservoir gradients): • No exact results

  13. II. Hydrodynamic Limit for Open Boundaries (1) • Hydrodynamic Limit • Diffusive scaling: • scaling limit: lattice constant a  0, k, t 1 • macroscopic coordinates x = ka, t’ = ta2 • coarse-grained deterministic densityA,B(x,t’) (law of large numbers) • local stationarity (large microscopic time)

  14. II. Hydrodynamic Limit for Open Boundaries (2) • Ansatz (ignore boundary, rigorous for DA=DB [Quastel, 1992]): • Conservation law  macroscopic continuity equation • current • tS(x,t) = - x[ -xDself(x,t)S(x,t) + b(x,t)S(x,t) ] diffusive background • Diffusive motion of tracer particle, interacting with background • Relaxation of background

  15. II. Hydrodynamic Limit for Open Boundaries (3) • Background relaxation: • Introduce weighted density field  = A/DA + B/DB • Exact linear equation • d/dt  = x2 • Plug into ansatz • b = 1/x (Dself - )

  16. II. Hydrodynamic Limit for Open Boundaries (4) • Self-diffusion coefficient: Vanishes in infinite system (subdiffusion) • Finite system: Dself = 1/L (1-)/  Remarks: (i) vanishes in limit, (ii) equal for both species • Proof (Brzank, GMS, 2007): • Mapping to current fluctuations in zero-range process (ZRP) • (use finite ring with periodic boundary conditions) • Einstein relation and Green-Kubo formula • (relates diffusion coefficient with particle drift (linear response theory)) • Exact steady state of locally driven ZRP • (explicit computation)

  17. II. Hydrodynamic Limit for Open Boundaries (6) • Step 1) Self-diffusion in 2c-SEP and disordered ZRP: • Numerate particles  sites in 1-dim lattice • Empty interval length (i,i+1)  particle occupation number ni • bond-symmetric ZRP with bimodal quenched disorder w(ni) = DA, DB • Jumps of tagged particle 0  particle jumps across bond (-1,0) • Define displacement X(t) as net number of jumps until time t • Displacement X(t) of tagged particle  Integrated current across (-1,0)

  18. II. Hydrodynamic Limit for Open Boundaries (7) • Step 2) Einstein relation and locally driven ZRP: • Introduce hopping bias eE/2 of tagged particle (external driving field) •  stationary velocity v(E) • Define (for E=0) limt 1h X2(t) i/t = 2 Ds • Einstein relation (E=0): d/dE v(E) = D_s • ZRP: hopping asymmetry across bond (-1,0) (local driving field) • Velocity v(E)  stationary particle current j(E)

  19. Step 3) Stationary current in locally driven ZRP: • Invariant measure: (inhomogeneous) product measure [Benjamini et al (1996)] • with marginals • Prob[ni = n] = zin (1-zi) with local fugacity zi, • Here for finite lattice with L sites and periodic boundary conditions: • j(E) = Di+1 (zi – zi+1) i  -1 • = D0 (eE/2 z-1 – e-E/2 z0) p.b.c. 0=N • j(E,z0), z0 given by  in 2c-SEP • proves Dself = L-1 (1-)/ • Corollary: zk = z0 + i=1k Di-1 linear on large scales (LLN) with jump at 0 II. Hydrodynamic Limit for Open Boundaries (8)

  20. II. Hydrodynamic Limit for Open Boundaries (9) • Nonlinear diffusion equation t y = x (Dxy): • Diffusion matrix • AAB/DB - A/DA • D = 1/ + Dself • BB - B/DAA/DA • Boundary conditions: • Left boundary: A,B(0,t) = A,B- • Right boundary: A,B(L,t) = A,B+

  21. II. Hydrodynamic Limit for Open Boundaries (10) • Standard procedure for boundary conditions, • BUT • Vanishing self-diffusion coefficient  • Overdetermined boundary-value problem • Conjecture: • Keep Dself as regularization

  22. III. Steady State (1) • Steady State properties • Stationary density profiles in finite, rescaled system size L’ = aL • Colourblind profile • Stationary equation of motion for weighted density : • 0 = x2 • Linear density profile (x) = - + (+--) x / L • Non-Fickian weighted current J = - (+ - -) / L

  23. III. Steady State (2) Profile of light particles (A-component) Nonlinear equation: 1/L x [A(1-)/] + (1+1/L) A/x = - jA A-current (integration constant) Transformation h = A/  linear ode h(x) = h- + (h+ - h-) [1 - (1-(+--)/(1--) x/L)L] / [1 - ((1-+)/(1--))L] A(x) = [- + (+ - -) x/L] h(x) / [DB + (1-DB/DA) h(x)] jA = - (+ - -)/L [h+(1--)L - h-(1-+)L] / [(1--)L - (1-+)L]

  24. III. Steady State (4) • Simulation results for tagged-particle problem L=200, -A=-B=0.3, +A¼ 0.68, +B¼ 0.09 (+ > -) • Left boundary layer of finite width • Non-monotone A-profile (pumping: current flows against gradient)

  25. III. Steady State (3) • Profile of light particles (cont.) Vanishing reservoir gradient + = - =  : jA = (1-) / L2 j = jA + jB 0 (for DA DB) • Current of order 1/L2 rather than 1/L • Total current vanishes only if hopping rates are equal

  26. IV. Boundary-Induced Non-Equilibrium Phase Transition (1) • Thermodynamic Limit L 1 • Non-analytic behaviour at vanishing reservoir gradient + = - - h+ (+ - -)/L for + > - jA = 0 for + = - - h- (+ - -)/L for + < - • Positive (negative) gradient: current determined by right (left) boundary Mean total density h+ (+ + -)/2 for + > - A = (+ + -)/2 for + = - h- (+ + -)/2 for + < - • Discontinuous non-equilibrium phase transition

  27. Phase diagram 1 + A = h+av A = h-av 0 - 1 IV. Boundary-Induced Non-Equilibrium Phase Transition (2) - Larger boundary density determines bulk density - Current is „maximized“ R L

  28. IV. Boundary-Induced Non-Equilibrium Phase Transition (3) • Density profiles • Consider R-phase (positive reservoir gradient + > -) • A(x) = [- + (+ - -) x/L]£ • [h+ - (h+ - h-) e-x/] / [DB + (1-DB/DA) (h+ - (h+ - h-) e-x/)] • Left boundary layer with localization length  = [ ln (+ - -)/(1--) ]-1 • Far from boundary (x À): A(x) = h+(x) •  no dependence on DB/DA • Scaled variable r = x/L: Jump discontinuity at r=0 for L 1

  29. IV. Boundary-Induced Non-Equilibrium Phase Transition (4) • Phase transition line •  diverges • Dependence of bulk profile on DB/DA • L-Phase (negative reservoir gradient + > -) • Reflection symmetry  interchange (+, –) and (x, L-x) • boundary layer jumps to right boundary at discontinuous transition

  30. V. Conclusions • Exact hydrodynamic description of microscopic two-component SEP with open boundaries self-diffusion regularization of diffusion matrix for single-file systems • Discontinuous boundary-induced non-equilibrium phase transition  caused by boundary layers • Current is ,,maximal`` (high density boundary), boundary layer is at other edge • Current may flow against density gradient (pumping)  strong correlations in boundary layer Boundary and finite-size effects?

  31. Acknowledgments • Thanks to: • R. Harris (London), D. Karevski (Nancy), J. Kärger (Leipzig), • H. van Beijeren (Utrecht) • Isaac Newton Institute for Mathematical Sciences (Cambridge) • Deutsche Forschungsgemeinschaft, SPP1155 “Molekulare • Modellierung und Simulation in der Verfahrenstechnik“

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