Large deviations and metastability in zero-range condensation. Paul - - PowerPoint PPT Presentation

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:

2

Introduction

3

• 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

4

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:

6

Stationary distributions

• Canonical (conditioned)

» The dynamics conserve the particle number. » Restricted to the dynamics are ergodic. » Unique stationary distribution for fixed L and N:

7

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,

8

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