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