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Multiple Condensates in Zero Range Processes David Mukamel. Seoul, July 1 st 2008. Zero Range Process A model which has been used to probe condensation phenomena In non-equilibrium driven systems. condensation. Condensed state. Macroscopically homogeneous. - PowerPoint PPT Presentation
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Multiple Condensates in Zero Range Processes
David Mukamel
Seoul, July 1st 2008
Zero Range Process
A model which has been used to probecondensation phenomena In non-equilibriumdriven systems.
Macroscopically homogeneous Condensed state
Under what conditions does one expect condensationdo take place?
condensation
A B
AB BA1
q
dynamics
Minimal model: Asymmetric Simple Exclusion Process (ASEP)
Steady State:q=1 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).
A B
AB BA1
q
dynamics
Extensions of the model:
more species (e.g. ABC model)
transition rate (q) depends on local configuration (e.g.KLS model Katz, Lebowitz, Spohn)
Ordering and condensation in such models (?)
Given a 1d driven process, does it exhibit phase separation?
Criterion for condensation
1nj 2nj 3nj 4nj
1n 2n 3n 4n
)(njdomains exchange particles via their current
if decreases with n: coarsening may be expected to take place.
)(nj
quantitative criterion?
Phase separation takes place in one of two cases:
n)exp( n
0)( njCase A: as
e.g.
Case B: 0j and
01/)( jnj slower than 2/n
) 1s /1or 2 / ( snbnb
Zero Range Processes
pu(n)(1-p)u(n)
Particles in boxes with the following dynamics:
Steady state distribution of ZRP processes:
product measure
n
k ku
znp
1 )()( where z is a fugacity
For example if u(n)=u (independent of n)
/)( nn
eu
znp
n
nnpn )(with (determines )
An interesting choice of u(n) provides a condensationtransition at high densities.
b
n
b
n
n
e
n
znp
/
)(
)(
)(
)/1()(
1
n
k ku
znp
nbnu
nbk
bku
n
k
ln)1ln()(ln1
The correlation length is determined by
ndnnnp )( b
n
n
enp
/
)(
For b<2 there is always a solution with a finitefor any density. Thus no condensation.
2for 1
: 1
bdnnb
-average number of particles in a box n
2for 1
: 1
bcdnnb
ndnnnp )( b
n
n
enp
/
)(
For b>2 there is a maximal possible density, andhence a transition of the Bose Einstein Condensationtype.
For densities larger than c a condensate is formed whichcontains a macroscopically large number of particles.
Phase diagram for b>2
2
1
b0
p(n)
n
b
n
n
e /
bn
1 ))((1
Lnan cb
L-1000, N=3000, b=3
fluid
condensate
Use ZRP to probe possible types of ordering.
Multiple condensates?
Can the critical phase exist over a whole regionrather than at a point in parameter space?
u
n
k
L
nc
n
bnu
1)( L sites
Non-monotonic hopping rate – multiple condensatesY. Schwatrzkopf, M. R. Evans, D. Mukamel. J. Phys. A, 41, 205001 (2008)
Typical occupation configuration obtained from simulation(b=3):
L=1000, N=2000, b=3
L=1000 b=4 k=1 4
)(
)(1
n
k ku
znp
k
L
nc
n
bnu
1)(
)c/(kaL
na
n
znp
k
k
b
n
1 exp)(1
LNnnpn
/)( the fugacity z (L) is determined by the density:
Analysis of the occupation distribution
p(n)
*n
minn *n n
bn
1
)1/(**
)1/(2*
)1/(*
)1/(*
)(
)(
kk
kk
kk
kk
Lnnpw
Lnp
Ln
Ln
Up to logarithmic corrections the peak parametersscale with the system size as:
The peak is broad
*n
minn *n n
(“condensate” weight)
Number of particles in a condensate: )1/(* kkLn
Number of condensates:)1/(1**)( kLnnLpLw
The condensed phase is composed of a largenumber (sub-extensive) of meso-condensateseach contains a sub-extensive number of particlessuch that the total occupation of all meso-condensates isextensive.
)exp(1)( aLnn
bnu
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 afinite number of condensates.
L=1000, N= 2300, a=1, b=4 critical density = 0.5
Lnnpw
Lnp
Ln
LaLn
/1)(
)(
ln
**
2/3*
2/1*
*
Results of scaling analysis:
number of condensates is Lw = O(1)
Dynamics
temporal evolution of the occupation of a single site
p(n)
*n
minn *n n
)1/()1(
min
)1/(
min
)(
)(
1
kkbe
kbkc
Lnp
w
Lnp
creation time
evaporation time
Arrhenius law approach:
b=4, k=5, density=3
Can one have a critical phase for a whole rangeof densities (like self organized criticality)?
A ZRP model with non-conserving processes
Connection to network dynamics
k
s
L
nna
Lc
n
bnu
)(
1
1)(hopping rate:
creation rate:
annihilation rate:
pu(n)(1-p)u(n) c
a(n)
Non conserving ZRPA. Angel, M.R. Evans, E. Levine, D. Mukamel, PRE 72, 046132 (2005);JSTAT P08017 (2007)
)1()()()()(
)()()1()1()1()(
npcnpnanu
npcnpnanut
np
Evolution equation (fully connected)
1
)()(n
npnuhopping current
1
)()(n
npnu
0
1)(n
np
1
)()(n
cnpna
Sum rules:
normalization
hopping current
steady state density
)0()()(
)()(
1
pmuma
cnp
nm
n
k
eff L
n
n
bnanunu
1)()()(
Steady state distribution
Like a conserving ZRP with an effective hopping rate
and cz
effu
n
k
eff L
n
n
bnu
1)(
)(1
npL
nL
k
n
s
)(
1
npnLn
ksk
)0()()(
)()(
1
pmuma
cnp
nm
n
Steady state distribution
Is determined by the sum rule
1
)()(n
npnac
(s,k) phase diagram
large s - small creation rate (1/Ls) - low density
small s – large creation rate – high density
intermediate s ?
b=2.6, L=10,000
conserving model )66.1( 4 c
non-conserving s=1.96 k=3
strong high density
weak high density
critical B
critical Alow density
Phase diagram
)0()()(
)()(
1
pmuma
cnp
nm
n
k
L
n
n
bnanu
1)()(
nLk
n
bce
nnP
k
k
)(1
)( )1(
1
)(1
npnLn
ksk
Low density phase
nksLnp )()(
ks
skLp )1(
The density vanishes in the thermodynamic limit.The lattice is basically empty.
Critical phase A bk/(k+1)<s<k
yL
ng
be
nnP
1)(
1
bk
sky
critical phase with a cutoff at Ly which divergesin the thermodynamic limit
Critical phase B 2k/(k+1)<s<bk/(k+1)
skk
s
kk
Lnnpw
Lnp
Lnn
)1/(**
*
)1/(**
)(
)(
p(n)
n* n
*n Weak peak
0)1/(2* skkwp Lwnn
c
Most of the sites form a fluid with typically lowoccupation. In addition a sub-extensive numberof sites Lw are highly occupied with n=n*
(meso-condensates)
)1/(*
)1/(1
kk
kks
Ln
LLwno. of condensates
condensate occupation
Weak high density k/(k+1)<s<2k/(k+1)
skk
s
kk
Lnnpw
Lnp
Lnn
)1/(**
*
)1/(**
)(
)(
p(n)
n* n
*n Weak peak
skkwp Lwnn )1/(2*
Strong high density phase s<k/(k+1)
p(n)
n* n
1
/1*
w
Ln ks
no fluid, all site are highly occupied by n* particles
L=5000 b=2.6 k=3
S=2, 1.7, 1.2, 0.4
fully connected
+ 1d lattice
strong high density
weak high density
critical B
critical Alow density
Phase diagram
Network dynamics
correspondence between networks and ZRP
node --- site link to a node --- particle
Rewiring dynamics
1
1
jj
ii
nn
nni
j
)( inurate:
Networks with multiple and self links (tadpoles).
i
j
u(ni)a(ni)
c
rewiring – creation – evaporation processes
one obtains the same results as for thenon-conserving ZRP
)1()2(
)()()()()()2(
)1()1()1()1()(
npc
npnnanunpc
npnnanut
np
1
1
)()(
)()()(
l
l
lplu
N
nclpla
N
nn
Here is the evaporation current resultingfrom other nodes
)(n
Hence
N
nc
L
n
n
bnnanuu
k
eff
1)()()(
b=2.6 k=3 s=2 L=1000, 2000, 4000
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
A B C
AB BA1
q
BC CB1
q
CA AC1
q
Random sequential update
ABC Model
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