multiple condensates in zero range processes david mukamel
DESCRIPTION
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 PresentationTRANSCRIPT
![Page 1: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/1.jpg)
Multiple Condensates in Zero Range Processes
David Mukamel
Seoul, July 1st 2008
![Page 2: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/2.jpg)
Zero Range Process
A model which has been used to probecondensation phenomena In non-equilibriumdriven systems.
![Page 3: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/3.jpg)
Macroscopically homogeneous Condensed state
Under what conditions does one expect condensationdo take place?
condensation
![Page 4: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/4.jpg)
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).
![Page 5: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/5.jpg)
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 (?)
![Page 6: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/6.jpg)
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?
![Page 7: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/7.jpg)
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
![Page 8: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/8.jpg)
Zero Range Processes
pu(n)(1-p)u(n)
Particles in boxes with the following dynamics:
![Page 9: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/9.jpg)
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 )
![Page 10: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/10.jpg)
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
![Page 11: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/11.jpg)
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
![Page 12: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/12.jpg)
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.
![Page 13: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/13.jpg)
Phase diagram for b>2
2
1
b0
p(n)
n
b
n
n
e /
bn
1 ))((1
Lnan cb
![Page 14: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/14.jpg)
L-1000, N=3000, b=3
fluid
condensate
![Page 15: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/15.jpg)
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?
![Page 16: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/16.jpg)
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)
![Page 17: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/17.jpg)
Typical occupation configuration obtained from simulation(b=3):
![Page 18: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/18.jpg)
L=1000, N=2000, b=3
![Page 19: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/19.jpg)
L=1000 b=4 k=1 4
![Page 20: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/20.jpg)
)(
)(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
![Page 21: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/21.jpg)
p(n)
*n
minn *n n
bn
1
![Page 22: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/22.jpg)
)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)
![Page 23: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/23.jpg)
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.
![Page 24: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/24.jpg)
)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.
![Page 25: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/25.jpg)
L=1000, N= 2300, a=1, b=4 critical density = 0.5
![Page 26: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/26.jpg)
Lnnpw
Lnp
Ln
LaLn
/1)(
)(
ln
**
2/3*
2/1*
*
Results of scaling analysis:
number of condensates is Lw = O(1)
![Page 27: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/27.jpg)
Dynamics
temporal evolution of the occupation of a single site
![Page 28: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/28.jpg)
p(n)
*n
minn *n n
)1/()1(
min
)1/(
min
)(
)(
1
kkbe
kbkc
Lnp
w
Lnp
creation time
evaporation time
Arrhenius law approach:
![Page 29: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/29.jpg)
b=4, k=5, density=3
![Page 30: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/30.jpg)
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
![Page 31: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/31.jpg)
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)
![Page 32: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/32.jpg)
)1()()()()(
)()()1()1()1()(
npcnpnanu
npcnpnanut
np
Evolution equation (fully connected)
1
)()(n
npnuhopping current
![Page 33: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/33.jpg)
1
)()(n
npnu
0
1)(n
np
1
)()(n
cnpna
Sum rules:
normalization
hopping current
steady state density
![Page 34: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/34.jpg)
)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
![Page 35: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/35.jpg)
effu
n
k
eff L
n
n
bnu
1)(
![Page 36: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/36.jpg)
)(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
![Page 37: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/37.jpg)
(s,k) phase diagram
large s - small creation rate (1/Ls) - low density
small s – large creation rate – high density
intermediate s ?
![Page 38: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/38.jpg)
b=2.6, L=10,000
conserving model )66.1( 4 c
non-conserving s=1.96 k=3
![Page 39: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/39.jpg)
strong high density
weak high density
critical B
critical Alow density
Phase diagram
![Page 40: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/40.jpg)
)0()()(
)()(
1
pmuma
cnp
nm
n
k
L
n
n
bnanu
1)()(
nLk
n
bce
nnP
k
k
)(1
)( )1(
1
)(1
npnLn
ksk
![Page 41: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/41.jpg)
Low density phase
nksLnp )()(
ks
skLp )1(
The density vanishes in the thermodynamic limit.The lattice is basically empty.
![Page 42: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/42.jpg)
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
![Page 43: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/43.jpg)
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
![Page 44: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/44.jpg)
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
![Page 45: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/45.jpg)
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*
![Page 46: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/46.jpg)
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
![Page 47: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/47.jpg)
L=5000 b=2.6 k=3
S=2, 1.7, 1.2, 0.4
fully connected
+ 1d lattice
![Page 48: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/48.jpg)
strong high density
weak high density
critical B
critical Alow density
Phase diagram
![Page 49: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/49.jpg)
Network dynamics
correspondence between networks and ZRP
node --- site link to a node --- particle
![Page 50: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/50.jpg)
Rewiring dynamics
1
1
jj
ii
nn
nni
j
)( inurate:
Networks with multiple and self links (tadpoles).
![Page 51: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/51.jpg)
i
j
u(ni)a(ni)
c
rewiring – creation – evaporation processes
one obtains the same results as for thenon-conserving ZRP
![Page 52: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/52.jpg)
)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
![Page 53: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/53.jpg)
Hence
N
nc
L
n
n
bnnanuu
k
eff
1)()()(
![Page 54: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/54.jpg)
b=2.6 k=3 s=2 L=1000, 2000, 4000
![Page 55: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/55.jpg)
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
![Page 56: Multiple Condensates in Zero Range Processes David Mukamel](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813e35550346895da81bc6/html5/thumbnails/56.jpg)
A B C
AB BA1
q
BC CB1
q
CA AC1
q
Random sequential update
ABC Model