Two -Component Symmetric Exclusion Process with Open Boundaries

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



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



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)



Where does it happen?

• Biology: ion channels (e.g. pumps: symporter, antiporter)

Randomness:

- Diffusion- Thermal activation




• Colloidal systems: etched channels or optical lattices

Randomness:

- Thermal activation




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




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).



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!



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



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)




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:




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



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)



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





Background relaxation:

• Introduce weighted density field = A/DA + B/DB

• Exact linear equation

d/dt = x2

Plug into ansatz

b = 1/ x (Dself - )




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)




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)




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

• ZRP: hopping asymmetry across bond (-1,0) (local driving field)

• Velocity v(E) stationary particle current j(E)




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




Nonlinear diffusion equation t = x (D x:

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+




Standard procedure for boundary conditions,

BUT

Vanishing self-diffusion coefficient Overdetermined boundary-value problem

Conjecture:

Keep Dself as regularization



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




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) / [DB + (1-DB/DA) h(x)]

jA = - (+ - -)/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 = - jA

A-current (integration constant)




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)




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



IV. Boundary-Induced Non-Equilibrium Phase Transition (1)

Thermodynamic Limit L 1Non-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 + > -

= (+ + -)/2 for + = -

h- (+ + -)/2 for + < -

Discontinuous non-equilibrium phase transition




Phase diagram

1

= h+ av

= h- av

- Larger boundary density determines bulk density

- Current is „maximized“RL




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




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



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?



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“

Two -Component Symmetric Exclusion Process with Open Boundaries

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