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 Leipzig
3) Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn
Diffusion Fundamentals 4, 7.1-7.12 (2006)
J. Stat. Mech: Theory and Experiment (2007)
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
Outline:
1) Single-File Diffusion: Definition, Examples and Questions
2) Symmetric Exclusion Process with Open Boundaries
2
•
Two-Component Symmetric Simple Exclusion Process
•
Hydrodynamic Limit for Open Boundaries
•
Steady State Behaviour
•
Conclusions
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
Definition
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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)
3
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
Where does it happen?
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
• Biology: ion channels (e.g. pumps: symporter, antiporter)
Randomness:
- Diffusion
- Thermal activation
4
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
Where does it happen?
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
• Colloidal systems: etched channels or optical lattices
Randomness:
- Thermal activation
5
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
Where does it happen?
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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
6
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
Where does it happen?
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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).
zeolite pore wall (quasi 1-D open system)
Gas
Phase
Gas
Phase
Heavy HC molecules (toluene))
7
Light molecules (propane)
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
Questions (1)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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!
8
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
Questions (2)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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!
9
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
I. Two-component Symmetric Simple
Exclusion Process (1)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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)
10
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
I. Two-component Symmetric Simple
Exclusion Process (2)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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
11
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
I. Two-component Symmetric Simple
Exclusion Process (3)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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):
•
12
No exact results
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (1)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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)
13
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (2)
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
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background
•
Diffusive motion of tracer particle, interacting with background
•
Relaxation of background
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (3)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Background relaxation:
•
Introduce weighted density field  = A/DA + B/DB
•
Exact linear equation
d/dt  = x2
Plug into ansatz
b = 1/ x (Dself - )
15
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (4)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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):
16
•
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)
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (6)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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)
17
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (7)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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  1 h X2(t) i/t = 2 Ds
•
Einstein relation (E=0):
d/dE v(E) = D_s
18
•
ZRP: hopping asymmetry across bond (-1,0) (local driving field)
•
Velocity v(E)  stationary particle current j(E)
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (8)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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)
= D0 (eE/2 z-1 – e-E/2 z0)
i  -1
p.b.c. 0=N
 j(E,z0), z0 given by  in 2c-SEP
 proves Dself = L-1 (1-)/
•
19
Corollary: zk = z0 + i=1k Di-1 linear on large scales (LLN) with jump at 0
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (9)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Nonlinear diffusion equation t y = x (D xy):
Diffusion matrix
A A
D = 1/
B B
B/DB - A/DA
+ Dself
- B/DA
A/DA
Boundary conditions:
Left boundary: A,B(0,t) = A,BRight boundary: A,B(L,t) = A,B+
20
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
II. Hydrodynamic Limit for Open
Boundaries (10)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Standard procedure for boundary conditions,
BUT
Vanishing self-diffusion coefficient 
Overdetermined boundary-value problem
Conjecture:
Keep Dself as regularization
21
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
III. Steady State (1)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Steady State properties
•
Stationary density profiles in finite, rescaled system size L’ = aL
Colourblind profile
Stationary equation of motion for weighted density :
0 = x2 
22

Linear density profile
(x) = - + (+--) x / L

Non-Fickian weighted current
J = - (+ - -) / L
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
III. Steady State (2)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
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]
23
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
III. Steady State (4)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Simulation results for tagged-particle problem
L=200, -A=-B=0.3, +A ¼ 0.68, +B ¼ 0.09 (+ > -)


24
Left boundary layer of finite width
Non-monotone A-profile (pumping: current flows against gradient)
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
III. Steady State (3)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Profile of light particles (cont.)
Vanishing reservoir gradient + = - =  :
jA = (1-) / L2
j = jA + jB  0
25
(for DA  DB)

Current of order 1/L2 rather than 1/L

Total current vanishes only if hopping rates are equal
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
IV. Boundary-Induced NonEquilibrium Phase Transition (1)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Thermodynamic Limit L  1
Non-analytic behaviour at vanishing reservoir gradient + = -
- h+ (+ - -)/L
0
- h- (+ - -)/L
jA =

for + > for + = for + < -
Positive (negative) gradient: current determined by right (left) boundary
Mean total density
A =

26
h+ (+ + -)/2
(+ + -)/2
h- (+ + -)/2
for + > for + = for + < -
Discontinuous non-equilibrium phase transition
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
IV. Boundary-Induced NonEquilibrium Phase Transition (2)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Phase diagram
1
- Larger boundary density
determines bulk density
A = h+ av
+
- Current is „maximized“
R
L
A = h- av
0
27
-
1
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
IV. Boundary-Induced NonEquilibrium Phase Transition (3)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Density profiles
Consider R-phase (positive reservoir gradient + > -)
•
28
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
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
IV. Boundary-Induced NonEquilibrium Phase Transition (4)
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Phase transition line
•
 diverges
•
Dependence of bulk profile on DB/DA
L-Phase (negative reservoir gradient + > -)
29
•
Reflection symmetry  interchange (+, –) and (x, L-x)
•
boundary layer jumps to right boundary at discontinuous transition
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
V. Conclusions
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn

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?
30
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008
Acknowledgments
Interdisziplinäres Zentrum für
Komplexe Systeme, Universität Bonn
Thanks to:
31
•
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“
Two-Component Single-File Diffusion with Open Boundaries
Fritzfest, Technical University of Budapest, 27-29 March 2008