1. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 1 Two-Component Symmetric Exclusion Process with Open Boundaries Andreas Brzank 1,2 and Gunter M. Schütz 1,3 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. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 4 Where does it happen? Biology: ion channels (e.g. pumps: symporter, antiporter) Randomness: - Diffusion - Thermal activation

5. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 5 Where does it happen? Colloidal systems: etched channels or optical lattices Randomness: - Thermal activation

6. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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 also light HC components Fibrous zeolites: - quasi-one-dimensional channel network - channel length up to 100 cross sections - pronounced single-file effect MFI-type zeolite

7. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 7 Where does it happen? zeolite pore wall (quasi 1-D open system) Gas Gas Phase Phase Heavy HC molecules (toluene)) Light molecules (propane) 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).

8. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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 ~ t 1/2 (infinite system, rigorous for SEP) Longest relaxation time ~ L 3 (finite system, scaling and numerics) More ?? Use lattice gas model to study generic large-scale behaviour!

9. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 9 Questions (2) 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! Stochastic particle systems as models for hydrodynamic behaviour: One-component systems (identical particles): Well-understood

10. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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 D A, D B, colour) non-equilibrium steady state (open boundaries)

11. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 11 I. Two-component Symmetric Simple Exclusion Process (2) boundary chemical potentials - A,B = A,B / A,B, A,B = A,B / A,B boundary densities = /(1+ ) (exclusion) boundary processes = coupling to infinite reservoirs Physical interpretation of boundary processes:

12. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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 = ta 2 coarse-grained deterministic density A,B (x,t) (law of large numbers) local stationarity (large microscopic time)

14. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 14 Ansatz (ignore boundary, rigorous for D A =D B [Quastel, 1992]): Conservation law macroscopic continuity equation current t S (x,t) = - x [ - x D self (x,t) S (x,t) + b(x,t) S (x,t) ] diffusive background Diffusive motion of tracer particle, interacting with background Relaxation of background II. Hydrodynamic Limit for Open Boundaries (2)

15. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 15 II. Hydrodynamic Limit for Open Boundaries (3) Background relaxation: Introduce weighted density field = A /D A + B /D B Exact linear equation d/dt = x 2 Plug into ansatz b = 1/ x (D self - )

16. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 16 II. Hydrodynamic Limit for Open Boundaries (4) Self-diffusion coefficient: Vanishes in infinite system (subdiffusion) Finite system: D self = 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. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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 n i bond-symmetric ZRP with bimodal quenched disorder w(n i ) = D A, D B 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. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 18 II. Hydrodynamic Limit for Open Boundaries (7) Step 2) Einstein relation and locally driven ZRP: Introduce hopping bias e E/2 of tagged particle (external driving field) stationary velocity v(E) Define (for E=0) lim t 1 h X 2 (t) i/t = 2 D s 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. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 19 II. Hydrodynamic Limit for Open Boundaries (8) Step 3) Stationary current in locally driven ZRP: Invariant measure: (inhomogeneous) product measure [Benjamini et al (1996)] with marginals Prob[n i = n] = z i n (1-z i ) with local fugacity z i, Here for finite lattice with L sites and periodic boundary conditions: j(E) = D i+1 (z i – z i+1 ) i -1 = D 0 (e E/2 z -1 – e -E/2 z 0 ) p.b.c. 0=N j(E,z 0 ), z 0 given by in 2c-SEP proves D self = L -1 (1- )/ Corollary: z k = z 0 + i=1 k D i -1 linear on large scales (LLN) with jump at 0

20. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 20 II. Hydrodynamic Limit for Open Boundaries (9) Nonlinear diffusion equation t = x (D x : Diffusion matrix A A B /D B - A /D A D = 1/ + D self B B - B /D A A /D A Boundary conditions: Left boundary: A,B (0,t) = A,B - Right boundary: A,B (L,t) = A,B +

21. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 21 II. Hydrodynamic Limit for Open Boundaries (10) Standard procedure for boundary conditions, BUT Vanishing self-diffusion coefficient Overdetermined boundary-value problem Conjecture: Keep D self as regularization

22. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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 = x 2 Linear density profile (x) = - + ( + - - ) x / L Non-Fickian weighted current J = - ( + - - ) / L

23. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 23 III. Steady State (2) Transformation h = A / linear ode h(x) = h - + (h + - h - ) [1 - (1-( + - - )/(1- - ) x/L) L ] / [1 - ((1- + )/(1- - )) L ] (x) = [ - + ( + - - ) x/L] h(x) / [D B + (1-D B /D A ) h(x)] j A = - ( + - - )/L [h + (1- - ) L - h - (1- + ) L ] / [(1- - ) L - (1- + ) L ] Profile of light particles (A-component) Nonlinear equation: 1/L x [ A (1- )/ ] + (1+1/L) A / x = - j A A-current (integration constant)

24. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 24 III. Steady State (4) Simulation results for tagged-particle problem L=200, - A = - B =0.3, + A ¼ 0.68, + B ¼ 0.09 ( + > - ) Left boundary layer of finite width Non-monotone A-profile (pumping: current flows against gradient)

25. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 25 III. Steady State (3) Profile of light particles (cont.) Vanishing reservoir gradient + = - = : j A = (1- ) / L 2 j = j A + j B 0 (for D A D B ) Current of order 1/L 2 rather than 1/L Total current vanishes only if hopping rates are equal

26. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 26 IV. Boundary-Induced Non- Equilibrium Phase Transition (1) Thermodynamic Limit L 1 Non-analytic behaviour at vanishing reservoir gradient + = - - h + ( + - - )/L for + > - j A = 0 for + = - - h - ( + - - )/L for + < - Positive (negative) gradient: current determined by right (left) boundary Mean total density h + ( + + - )/2 for + > - = ( + + - )/2 for + = - h - ( + + - )/2 for + < - Discontinuous non-equilibrium phase transition

27. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 27 IV. Boundary-Induced Non- Equilibrium Phase Transition (2) Phase diagram 1 = h + av = h - av - Larger boundary density determines bulk density - Current is maximized R L

28. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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/ ] / [D B + (1-D B /D A ) (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 D B /D A Scaled variable r = x/L: Jump discontinuity at r=0 for L 1

29. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 29 IV. Boundary-Induced Non- Equilibrium Phase Transition (4) L-Phase (negative reservoir gradient + > - ) Reflection symmetry interchange (+, –) and (x, L-x) boundary layer jumps to right boundary at discontinuous transition Phase transition line diverges Dependence of bulk profile on D B /D A

30. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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. Two-Component Single-File Diffusion with Open Boundaries Fritzfest, Technical University of Budapest, 27-29 March 2008 Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn 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