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 1

Motivation • Granular clustering [van der Meer, van der Weele, Lohse, Mikkelsen, Versluis (2001-02)] • Jamming: 2

Introduction • 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. 3

Size dependent zero-range process 4

Examples • Independent particles: • Decreasing rates (effective attraction): eg: [Evans (2000)] » Condensation possible. » Above some critical density a finite fraction of mass accumulates on a single site.

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 • 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 10

Toy model [Grosskinsky, Schütz(2008)] 11

Toy model 12

Preliminary observations • Two immediate results: » Grand canonical distributions exist for: » GC distributions restricted to having maximum less than aL are still product measures and exist for: 13

Grand canonical pressures • Pressures 14

Preliminary Results • Gärtner-Ellis Theorem: » 16

Preliminary Results • Gärtner-Ellis Theorem: » 17

Potential for Joint Density and Maximum • What about above ρ c ? • 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. 18

Potential landscape of maximum • Potential for the joint density and maximum: • Canonical entropy: • Canonical potential landscape for the max: 19

Rate function for max • ρ < ρ c + a » » Unique minimum at m = 0 20

Rate function for max • ρ c + a < ρ < ρ trans » » Global minimum at m = 0, second min at m = ρ - ρ c » Metastable condensed states. 21

Rate function for max • ρ > ρ trans » » Global minimum at m = ρ - ρ c , second min at m = 0 » Metastable fluid states. 22

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 > ρ trans (not too much bigger) » Motion via ‘fluid’ 25

Rate function for two max • ρ >> ρ trans » Motion via two macroscopically occupied sites 26

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