Finite size effects in a stochastic condensation model measures: Grand canonical ensemble There...

Finite size effects in a stochasticcondensation model

Paul ChlebounStefan Grosskinsky

Complexity Science Doctoral Training CentreUniversity of Warwick

28th January 2010

Finite size effects in a stochastic condensation model

Model: The zero-range process

A continuous-time Markov Chain. Driven diffusive system.

1-p p

[Spitzer (1970), Andjel (1982)]

Lattice: ΛL = {1, . . . ,L} with pbc

Configuration: η = (ηx)x∈ΛL with ηx ∈ N0

State space: XL = NΛL0

Jump rates: g : N0 → [0,∞)g(n) = 0 ⇐⇒ n = 0

Introduction Observations Results Conclusions Model: ZRP Motivation Previous results


Motivation

g(k)↘ ⇒ effective attraction, condensation possible

g(k) ' 1 +bkγ, b > 0, γ ∈ (0, 1)

[Evans (2000)]

Many applications. [Evans and Hanney (2005)]

Including to granular media and traffic flow.

[van der Meer, van der Weele, Lohse, Mikkelsen,Versluis (2001-02)]



Motivation: Granular media

[van der Meer, van der Weele, Lohse, Mikkelsen, Versluis (2001-02)]stilton.tnw.utwente.nl/people/rene/clustering.html

Typical sizes L ∼ 10− 100, N ∼ 1000.



Motivation: Traffic

Mapping ZRP to ASEP:

1 2

L+N

1 2

7 8 9 10 11 12

ASEP can be considered as a simple traffic model[Kaupuzs, Mahnke, Harris (2008)]

Typical sizes L ∼ 1000, N ∼ 200.



Outline

1 IntroductionModel: The zero-range processHeuristic MotivationPrevious Results: Equivalence of ensembles

2 ObservationsNumericsMC simulations

3 ResultsFluid and Condensed approximationsRate functionCurrent matchingScaling limit

4 Conclusions



Stationary measures: Grand canonical ensemble

There exist stationary product measures.

Stationary weights: w(n) =n∏

k=1

g(k)−1

Grand canonical ensemble

νLφ(η) =

1z(φ)L

∏x∈ΛL

w(ηx)φηx where z(φ) =∞∑

n=0

w(n)φn

Fugacity: φ ∈ [0, φc)

(↗) Density: 〈ηx〉νφ = R(φ) = φ∂φ log z(φ), ρc = limφ→φc

R(φ) ∈ (0,∞]

Current: 〈g(ηx)〉νφ = φ



Stationary measures: Canonical ensemble

Dynamics conserve the total particle number:

ΣL(η) :=∑x∈ΛL

ηx = N

Fixed L and N: system is irreducible and finite state so ergodic.

Unique stationary measure:

Canonical Ensemble

πL,N(η) :=1

Z(L,N)

∏x∈ΛL

w(ηx)δ(∑

x

ηx − N)

Density: 〈ηx〉πL,N = ρ = N/L



Thermodynamic limit

Equivalence of ensembles:

Previous results [Großkinsky, Schütz, Spohn (2003)]

In the thermodynamic limit L,N →∞ , N/L→ ρ

πL,Nw−→ νΦ(ρ) where Φ(ρ) =

{R−1(ρ) if ρ < ρc

φc if ρ ≥ ρc

g(k) = 1 +bkγ, b > 0, γ ∈ (0, 1) =⇒ ρc <∞



Thermodynamic limit

Thermodynamic entropy:

s(ρ) = supφ∈[0,1)

(log z(φ)− ρ logφ)

Convergence of canonical entropy:

limL→∞

1L

log Z(L, [ρL]) = log z(Φ(ρ))− ρ log Φ(ρ) = s(ρ)

ρc

Density ρ

s(ρc)

0

s(ρ)

Φ(ρ) = e−∂ρs(ρ)



Numerics in the Canonical Ensemble

Recursion relation

Z(L,N) =N∑

k=0w(k) Z(L−1,N−k)

Canonical current

j := 〈g(ηx)〉πL,N =Z(L,N−1)

Z(L,N)

Large fluid current overshoot.Sharp transition to putative condensed phase.

Introduction Observations Results Conclusions Numerics MC simulations


Monte Carlo simulations

Left: Metastable switching between fluid and condensedcurrents.Right: Exponential distribution of waiting time in thecondensed and fluid ‘phases’.

Introduction Observations Results Conclusions Numerics MC simulations


Fluid and Condensed approximations

ρc 0.4 0.5 0.6 0.7 0.8 0.9 1

Density ρ1

1.1

1.2

1.3

1.4

1.5

1.6

curr

ent

φc

NumericsFluid approx.Current matchingSimulation

Current OvershootL = 1000, γ = 0.5, b = 4

5e+06 5.5e+06 6e+06 6.5e+06 7e+06Time

1

1.1

1.2

1.3

1.4

1.5

1.6

Describe where the approximations come fromHow they can be used to find the leading order effect

Introduction Observations Results Conclusions Approximations Rate function Current matching Scaling limit


Rate function

Aim:Find the scaling of the overshoot region.Understand the apparent metastability close to themaximum current .

Approach:Find an effective rate function which describes themetastability in some scaling limit in terms of thebackground density.

πL,N

(N −maxx∈Λ

ηx

L− 1� ρbg

)∼ exp

(−LβIρ(ρbg)

)



Rate function

Notation:

πL,N(ρbg)

: = πL,N

(N −max

x∈ΛLηx = ρbg (L− 1)

)= πL,N

(maxx∈ΛL

ηx = ηm

)where ρbg(L− 1) = N − ηm.

We have:

πL,N(ρbg)

=1

Z(L,N)Lw(ηm)

∑η∈X̃

L−1∏x=1

w(ηx)

where X̃ = {η :L−1∑x=1

ηx = ρbg(L−1), η1, . . . , ηL−1 ≤ N−ρbg(L−1)}.



Rate function

X̃ = {η :L−1∑x=1

ηx = ρbg(L−1), η1, . . . , ηL−1 ≤ N−ρbg(L−1)}

πL,N(ρbg)

=1

Z(L,N)Lw(ηm)

∑η∈X̃

L−1∏x=1

w(ηx)

=1

Z(L,N)Lw(ηm)φ−ρbg(L−1)zL−1

ηm(φ)νL−1

φ,ηm(ΣL−1 = ρbg(L−1))

where,

νL−1φ,ηm

(ΣL−1 = ρbg(L−1)) = νL−1φ (ΣL−1 = ρbg(L−1)|ηx ≤ ηm)



Rate function

logπN,L(ρbg)

=(L− 1)(log zηm(φ)− ρbg logφ

)+

+ log w(ηm) + log νL−1φ,ηm

(ΣL−1 = ρbg(L−1))−− log Z(L,N) + log L.

Holds for all φ ∈ [0,∞). Choose φ so that,

Rηm(φ) :=1

zηm(φ)

ηm∑k=1

kw(k)φk= ρbg

Define: Φηm = R−1ηm

.



Rate function

logπN,L(ρbg)

=(L− 1)(log zηm(Φηm(ρbg))− ρbg log Φηm(ρbg)

)+

+ log w(ηm) + log νL−1Φηm (ρbg),ηm

(ΣL−1 = ρbg(L−1))−

− log Z(L,N) + log L.

Holds for all φ ∈ [0,∞). Choose φ so that,

Rηm(φ) :=1

zηm(φ)

ηm∑k=1

kw(k)φk= ρbg

Define: Φηm = R−1ηm

.



Rate function

logπN,L(ρbg)


)+


(ΣL−1 = ρbg(L−1))−


Looks like the thermodynamic entropy, except withtruncation at ηm so exists for all ρbg.

Contribution due to the condensate.Can be approximated by Gaussian density at 0.The final terms are constant for fixed N and L.



Rate function

logπN,L(ρbg)


)+


(ΣL−1 = ρbg(L−1))−


Looks like the thermodynamic entropy, except withtruncation at ηm so exists for all ρbg.Contribution due to the condensate.

Can be approximated by Gaussian density at 0.The final terms are constant for fixed N and L.



Rate function

logπN,L(ρbg)


)+


(ΣL−1 = ρbg(L−1))−


Looks like the thermodynamic entropy, except withtruncation at ηm so exists for all ρbg.Contribution due to the condensate.Can be approximated by Gaussian density at 0.

The final terms are constant for fixed N and L.



Rate function

logπN,L(ρbg)


)+


(ΣL−1 = ρbg(L−1))−


Looks like the thermodynamic entropy, except withtruncation at ηm so exists for all ρbg.Contribution due to the condensate.Can be approximated by Gaussian density at 0.The final terms are constant for fixed N and L.



Rate function in the thermodynamic limit

ILρ(ρbg) = −1

Llogπ[ρL],L

(Σbg

L = [ρbg(L− 1)])

TheoremAs L→∞ with N/L = ρ and ρbg ∈ (0, ρ),

Iρ(ρbg) := limL→∞

ILρ(ρbg) = s(ρ)− s(ρbg) (1)

and so Iρ(ρbg) = 0 for each ρbg ≥ ρc.

0 0.2 0.4 0.6 0.8 1ρ

bg

0

0.1

0.2

I ρ(ρbg

)

ρc ρ

L = 8000L = 16000L = 32000L = 64000



Rate function (finite lattices)

Ignoring the final two normalising terms we can approximate− logπN,L

(ρbg)

very quickly. Still informative.

0.4 0.5 0.6Background desnity ρ

bg

0

0.005

0.01

0.015

0.02

I ρ(ρbg

)

ρc

L = 1000, N = 660, γ = 0.5, b = 4

ρ below sharp transition point.





(ρbg)



bg

-0.002

0

0.002

0.004

0.006

0.008

0.01

I ρ(ρbg

)

ρc

ρ

L = 1000, N = 700, γ = 0.5, b = 4

ρ near sharp transition point.





(ρbg)



bg

-0.002

0

0.002

0.004

0.006

0.008

0.01

I ρ(ρbg

)

ρc

L = 1000, N = 730, γ = 0.5, b = 4

ρ above sharp transition point.



Max and min of distribution

Under controllable approximations (to leading order in L):

logπN,L(ρbg)− logπN,L

(ρbg − 1/L

)' log g(ηm)− log Φηm

(ρbg)

First minimum and maximum are given by currentmatching.The final minimum is effectively a boundary minimum.



Current matching


bg

1

1.2

1.4

1.6

Current

ρc

ρ

Φηmax

(ρbg

)

g(ηmax

)

L = 1000, N = 700, γ = 0.5, b = 4


bg

-0.002

0

0.002

0.004

0.006

0.008

0.01

I ρ(ρbg)

ρc

ρ

L = 1000, N = 700, γ = 0.5, b = 4



Lifetime of fluid and condensed ‘phases’


bg

1

1.2

1.4

1.6

Current

ρc

ρ

Φηmax

(ρbg

)

g(ηmax

)

L = 1000, N = 700, γ = 0.5, b = 4


bg

-0.002

0

0.002

0.004

0.006

0.008

0.01

I ρ(ρbg)

ρc

ρ

L = 1000, N = 700, γ = 0.5, b = 4

Fixed Density N/L

Ratio of the areas will alwaysgive proportion of time spent ineach ‘phase’

For totally asymmetric processrandom walk argument very wellto predict the life time.



Approximations

ρc 0.4 0.5 0.6 0.7 0.8 0.9 1

Density ρ1

1.1

1.2

1.3

1.4

1.5

1.6

curr

ent

φc

NumericsFluid approx.Current matchingSimulation

Current OvershootL = 1000, γ = 0.5, b = 4

Fluid current: From the average current in a cut-off grandcanonical ensemble ΦN(ρ).Condensed current: From current matching, solve

Φηm(ρbg) = g(ηm)



Scaling limit (Currents)

Examine behaviour on the scale of the current overshootρ = ρc + δρL−α.

TheoremCondensate and background current are asymptotically:

g(ηLm) = g

((δρ− δρbg)L1−α) = 1 +

b(δρ− δρbg)γ

Lγ(α−1)

log ΦηLm(ρL

bg) = log ΦηLm(ρc + δρbgL−α) =

1σ2

cδρbgL−α + o(L−α).

From this we can derive the scaling so that the twocurrents collapse α = γ

1+γ .At this scale they cross at unique δρbg for fixed δρ



Scaling limit (Currents)

0 1 2 3Rescaled Density - ρ

c

0

2

4

6

8

Res

cale

d C

urre

nt -

1

L=10000L=40000L=80000L=160000L=320000

Blue: Linear scaling limit for fluid current in background.Red: Current out of condensate (Collapsed in scaling limit).



Scaling limit (Rate function)

Examine behaviour of the system at the critical scaling.

I(2)δρ (δρbg) := lim

L→∞L1−βIL

ρc+δρL−α(ρc + δρbgL−α)

Theorem

If α = γ1+γ , β = 1−γ

1+γ the limit exists and is given by

I(2)δρ (δρbg) =

δρ2bg

2σ2c

+b

1− γ(δρ− δρbg)1−γ + min{ δρ

2

2σ2c, f (ρbg)}

Poof: Bounds on remainder term in Taylor series of ΦηLm(ρL

bg)and local limit theorem for triangular arrays.



Scaling limit (Rate function)

0 1 2 3 4δρ

bg

0

0.5

1

1.5

2

2.5

δρ

L = 160000L = 320000L = 640000L = 1280000I

(2)

δρI

(2)δρ (δρbg)

I(2)δρ (δρbg) := lim

L→∞

1Lβ

logπL,[(ρc+δρL−α)L]

(ρbg � ρc + δρbgL−α

)Critical scaling and the position of global minima correspondswith recent results [Armendariz, Grosskinsky, Loulakis]



Data collapses

c(γ)

δρ0

2

4

6

8

δφ

0 1 2 3 4 5 6 7 8

L = 1000L = 2000L = 10000L = 20000Scaling limit

Also for each vertical slice (fixed rescaled density) we havethe scaling rate function.



Data collapses for γ = 1

0 1 2 3 4 5 6δ ρ

0

0.2

0.4

0.6

0.8

δ φ

L = 500L = 1000L = 2000L=10000Scaling limit



Conclusions and Summary

Large finite size effects observed for some parametervalues.Observed phenomena on a finite system such asmetastable switching are also observed in real worldclustering.May have implications for understanding traffic flowpatterns.

Results:Estimate the fluid current overshoot and condensedcurrent on large finite systems.Estimate the density at which an observed change in‘phase’ will occur on a large finite system.Predict asymptotic scaling.Describe metastable behaviour and predict switching timesheuristically for simple systems.

Introduction Observations Results Conclusions


Acknowledgements

Thanks toSecond supervisor Ellak Somfai.Discussions with Robin Ball.

Warwick Complexity Science DTCFunding from the EPSRC

Introduction Observations Results Conclusions

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