Large deviations and metastability in zero-range...
Transcript of Large deviations and metastability in zero-range...
![Page 1: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/1.jpg)
Large deviations and metastability in zero-range condensation.
Paul Chleboun Stefan Grosskinsky
LAFNES 11 4/7/2011
1
![Page 2: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/2.jpg)
Motivation
• Granular clustering [van der Meer, van der Weele, Lohse, Mikkelsen, Versluis (2001-02)]
• Jamming:
2
![Page 3: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/3.jpg)
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.
![Page 4: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/4.jpg)
Size dependent zero-range process
4
![Page 5: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/5.jpg)
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)]
![Page 6: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/6.jpg)
Stationary distributions
• Grand canonical product distribution
» Single site marginal:
6
![Page 7: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/7.jpg)
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
![Page 8: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/8.jpg)
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
![Page 9: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/9.jpg)
Entropy densities
• Define the entropies relative to the single-site weights:
» Grand canonical:
» Canonical
![Page 10: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/10.jpg)
Equivalence of ensembles
10
• 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
![Page 11: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/11.jpg)
Toy model
11
[Grosskinsky, Schütz(2008)]
![Page 12: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/12.jpg)
Toy model
12
![Page 13: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/13.jpg)
• 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
13
![Page 14: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/14.jpg)
• Pressures
Grand canonical pressures
14
![Page 15: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/15.jpg)
Preliminary Results
16
• Gärtner-Ellis Theorem:
»
![Page 16: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/16.jpg)
Preliminary Results
17
• Gärtner-Ellis Theorem:
»
![Page 17: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/17.jpg)
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.
18
![Page 18: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/18.jpg)
Potential landscape of maximum
• Potential for the joint density and maximum:
• Canonical entropy:
• Canonical potential landscape for the max:
19
![Page 19: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/19.jpg)
Rate function for max
• ρ < ρc + a
»
» Unique minimum at m = 0
20
![Page 20: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/20.jpg)
Rate function for max
• ρc + a < ρ < ρtrans
»
» Global minimum at m = 0, second min at m = ρ - ρc
» Metastable condensed states.
21
![Page 21: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/21.jpg)
Rate function for max
• ρ > ρtrans
»
» Global minimum at m = ρ - ρc, second min at m = 0
» Metastable fluid states.
22
![Page 22: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/22.jpg)
Equivalence summary
![Page 23: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/23.jpg)
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)
![Page 24: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/24.jpg)
Rate function for two max
• ρ > ρtrans (not too much bigger)
» Motion via ‘fluid’
25
![Page 25: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/25.jpg)
Rate function for two max
• ρ >> ρtrans
» Motion via two macroscopically occupied sites
26
![Page 26: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/26.jpg)
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:
![Page 27: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/27.jpg)
Inclusion process
• Distribution of maximum and equivalence of ensembles:
» Case one: α tends to zero slower than 1/L:
» Fluid (all densities)
![Page 28: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/28.jpg)
Inclusion process
• Distribution of maximum and equivalence of ensembles:
» Case two: α tends to faster than 1/L:
» Condensed (all densities)
![Page 29: Large deviations and metastability in zero-range condensation.lafnes11/Slides/Paul_Chleboun_LAFNES11… · Large deviations and metastability in zero-range condensation. Paul Chleboun](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc399caac3a0243ba2cdb96/html5/thumbnails/29.jpg)
Videos
• Motion of condensate via fluid phase in ZRP
• Fluid Inclusion process
• Condensed Inclusion process