Large deviations and metastability in zero-range condensation.

Paul Chleboun Stefan Grosskinsky

LAFNES 11 4/7/2011

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Motivation

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

• Jamming:

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Introduction

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• Use techniques from the theory of large deviations to derive the potential landscape for the maximum. » Gives rise to (Non-)Equivalence of ensembles.

» Behaviour of the maximum.

• Methods presented apply to systems that exhibit product stationary measure. Examples: » Zero range process

» Inclusion process [S. Grosskinsky, F. Redig, K. Vafayi (2011)]

» Misanthrope process

• Present methods in the context of a simple toy model.

Size dependent zero-range process

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Examples

• Independent particles:

• Decreasing rates (effective attraction):

eg:

» Condensation possible.

» Above some critical density a finite fraction of mass accumulates on a single site.

[Evans (2000)]

Stationary distributions

• Grand canonical product distribution

» Single site marginal:

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Stationary distributions

• Canonical (conditioned)

» The dynamics conserve the particle number.

» Restricted to

the dynamics are ergodic.

» Unique stationary distribution for fixed L and N:

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Equivalence of ensembles

• Do the Canonical measures converge to the Grand canonical measure at some appropriate chemical potential?

» Relative Entropy:

» Equivalence in terms of weak convergence,

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Entropy densities

• Define the entropies relative to the single-site weights:

» Grand canonical:

» Canonical

Equivalence of ensembles

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• Equivalence of ensembles holds at the level of measures whenever it holds at the level of thermodynamic functions [J. Lewis, C. Pfister, & W. Sullivan]

• Extends to restricted ensembles

Toy model

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[Grosskinsky, Schütz(2008)]

Toy model

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• Two immediate results:

» Grand canonical distributions exist for:

» GC distributions restricted to having maximum less than aL are still product measures and exist for:

Preliminary observations

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• Pressures

Grand canonical pressures

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Preliminary Results

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• Gärtner-Ellis Theorem:

»

Preliminary Results

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• Gärtner-Ellis Theorem:

»

Potential for Joint Density and Maximum

• What about above ρc?

• Break down of equivalence is often related to the appearance of a macroscopically occupied site

• is easier to calculate than but has many useful properties.

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Potential landscape of maximum

• Potential for the joint density and maximum:

• Canonical entropy:

• Canonical potential landscape for the max:

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Rate function for max

• ρ < ρc + a

»

» Unique minimum at m = 0

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Rate function for max

• ρc + a < ρ < ρtrans

»

» Global minimum at m = 0, second min at m = ρ - ρc

» Metastable condensed states.

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Rate function for max

• ρ > ρtrans

»

» Global minimum at m = ρ - ρc, second min at m = 0

» Metastable fluid states.

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Equivalence summary

Motion

• In the process we find the potential landscape for the two highest occupied sites.

» This allows us to understand the motion

» Two mechanisms

A)

B)

Rate function for two max

• ρ > ρtrans (not too much bigger)

» Motion via ‘fluid’

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Rate function for two max

• ρ >> ρtrans

» Motion via two macroscopically occupied sites

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Inclusion process

• Method applies to other Misanthrope processes .

» Example: The (size-dependent) inclusion process

» Definition:

Jump rate depends on number of particles on departure and target site:

Inclusion process

• Distribution of maximum and equivalence of ensembles:

» Case one: α tends to zero slower than 1/L:

» Fluid (all densities)

Inclusion process

• Distribution of maximum and equivalence of ensembles:

» Case two: α tends to faster than 1/L:

» Condensed (all densities)

Videos

• Motion of condensate via fluid phase in ZRP

• Fluid Inclusion process

• Condensed Inclusion process