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 Two-Component Symmetric Exclusion Process with Open Boundaries Andreas Brzank 1,2 and Gunter M. Schütz 1,3 ) Institut für Festkörperforschung, Forschungszentrum Jülich 2) Institut für Experimentalphysik, Universität Leipzig nterdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn ffusion Fundamentals 4, 7.1-7.12 (2006) Stat. Mech: Theory and Experiment (2007)
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1 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 Leipzig3) Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
Diffusion Fundamentals 4, 7.1-7.12 (2006)
J. Stat. Mech: Theory and Experiment (2007)
gschuetz
2 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
Where does it happen?
• Biology: ion channels (e.g. pumps: symporter, antiporter)
Randomness:
- Diffusion- Thermal activation
5 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
Where does it happen?
• Colloidal systems: etched channels or optical lattices
Randomness:
- Thermal activation
6 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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
II. Hydrodynamic Limit for Open Boundaries (2)
15 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
• 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 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 DB/DA
30 Two-Component Single-File Diffusion with Open BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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 BoundariesFritzfest, Technical University of Budapest, 27-29 March 2008
Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
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“