Multiple%20Condensates%20in%20Zero%20Range%20Processes

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Multiple Condensates in Zero Range
Processes David Mukamel
Seoul, July 1st 2008
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Zero Range Process A model
which has been used to probe condensation
phenomena In non-equilibrium driven systems.
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condensation
Condensed state
Macroscopically homogeneous
Under what conditions does one expect
condensation do take place?
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Minimal model Asymmetric Simple Exclusion
Process (ASEP)
dynamics
Steady State q1 corresponds to an Ising model
at T All microscopic states are equally
probable. Density is macroscopically
homogeneous. No liquid-gas transition (for any
density and q).
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Extensions of the model more species (e.g. ABC
model) transition rate (q) depends on local
configuration (e.g. KLS model Katz, Lebowitz,
Spohn)
dynamics
Ordering and condensation in such models (?)
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Criterion for condensation
domains exchange particles via their current
quantitative criterion?
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Phase separation takes place in one of two cases
Case A
as
e.g.
and
Case B
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Zero Range Processes
Particles in boxes with the following dynamics
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Steady state distribution of ZRP processes
product measure
where z is a fugacity
For example if u(n)u (independent of n)
with
(determines )
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An interesting choice of u(n) provides a
condensation transition at high densities.
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The correlation length is determined by
-average number of particles in a box
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For bgt2 there is a maximal possible density,
and hence a transition of the Bose Einstein
Condensation type.
For densities larger than c a condensate is
formed which contains a macroscopically large
number of particles.
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Phase diagram for bgt2
0
p(n)
n
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fluid
condensate
L-1000, N3000, b3
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Use ZRP to probe possible types of
ordering. Multiple condensates? Can the
critical phase exist over a whole region rather
than at a point in parameter space?
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Non-monotonic hopping rate multiple
condensates Y. Schwatrzkopf, M. R. Evans, D.
Mukamel. J. Phys. A, 41, 205001 (2008)
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Typical occupation configuration obtained from
simulation (b3)
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L1000, N2000, b3
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Analysis of the occupation distribution
the fugacity z (L) is determined by the density
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Up to logarithmic corrections the peak
parameters scale with the system size as
(condensate weight)
The peak is broad
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Number of particles in a condensate
Number of condensates
The condensed phase is composed of a large number
(sub-extensive) of meso-condensates each contains
a sub-extensive number of particles such that the
total occupation of all meso-condensates
is extensive.
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Another choice of u(n) a sharp
(exponential) cutoff at large densities
Here we expect condensates to contain up to aL
particles (extensive occupation), and hence
a finite number of condensates.
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L1000, N 2300, a1, b4 critical density 0.5
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Results of scaling analysis
number of condensates is Lw O(1)
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Dynamics
temporal evolution of the occupation of a single
site
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Arrhenius law approach
creation time
evaporation time
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b4, k5, density3
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Can one have a critical phase for a whole
range of densities (like self organized
criticality)?
A ZRP model with non-conserving
processes Connection to network dynamics
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Non conserving ZRP
A. Angel, M.R. Evans, E. Levine, D. Mukamel, PRE
72, 046132 (2005) JSTAT P08017 (2007)
hopping rate creation rate annihilation
rate
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Evolution equation (fully connected)
hopping current
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Sum rules
normalization
hopping current
steady state density
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Steady state distribution
Like a conserving ZRP with an effective hopping
rate
and
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n
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Steady state distribution
Is determined by the sum rule
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(s,k) phase diagram
large s - small creation rate (1/Ls) - low
density small s large creation rate high
density intermediate s ?
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conserving model
non-conserving s1.96 k3
b2.6, L10,000
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Phase diagram
critical A
low density
critical B
weak high density
strong high density
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Low density phase
The density vanishes in the thermodynamic
limit. The lattice is basically empty.
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Critical phase A
bk/(k1)ltsltk
critical phase with a cutoff at Ly which
diverges in the thermodynamic limit
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Critical phase B 2k/(k1)ltsltbk/(k1)
Weak peak
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Most of the sites form a fluid with typically
low occupation. In addition a sub-extensive
number of sites Lw are highly occupied with
nn (meso-condensates)
no. of condensates condensate occupation
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Weak high density k/(k1)ltslt2k/(k1)
Weak peak
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Strong high density phase sltk/(k1)
no fluid, all site are highly occupied by n
particles
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L5000 b2.6 k3
S2, 1.7, 1.2, 0.4
fully connected
1d lattice
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Phase diagram
critical A
low density
critical B
weak high density
strong high density
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Network dynamics
correspondence between networks and ZRP
node --- site link to a node --- particle
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Rewiring dynamics
Networks with multiple and self links (tadpoles).
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rewiring creation evaporation processes
c
i
a(ni)
u(ni)
j
one obtains the same results as for
the non-conserving ZRP
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Here is the evaporation current
resulting from other nodes
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Hence
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b2.6 k3 s2 L1000, 2000, 4000
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summary
The Zero-Range-Process may be used to probe
ordering phenomena in driven systems. Multiple
meso-condensates may result in certain
cases Generic critical phase (like self
organized criticality) Network dynamics may
exhibit similar phenomena
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ABC Model
Random sequential update