The Zero Range Process Condensation in non-equilibrium systems

The Zero Range Process 

Condensation in non-equilibrium systems 

Yoni Schwarzkopf 

Department of Physics of Complex Systems 

Weizmann Institute of Science 

February 10, 2006 

The Zero Range Process

Yoni Schwarzkopf February 10, 2006 

Introduction 

• Phase transitions in equilibrium systems are well studied: 

BEC (Bose-Einstein Condensates) 

Liquid Gas Transitions 

. 

• 1d equilibrium systems have no phase transitions (Only for strong long range 

interactions) 

The Zero Range Process 1


Introduction (2) 

• Non equilibrium systems offer rich phenomena even in 1d: 

TAESP totally asymmetrical exclusion process 

• Need a simple coarse grained model to offer universal insights 

TAESP → Zero Range Process 



Non-Equilibrium: General properties 

• Let C be a configuration of a system 

• Define P(C,t): the probability to be in a configuration of a system at time t. 

• W(C → C ′ ): transition rate between configurations 

• Master Equation 1 : 

∂P(C,t) 

∂t 

= 

W(C ′ → C)P(C ′ , t) − 

W(C → C ′ )P(C,t) (1) 

C ′ 

1 Phase Transitions in Nonequilibrium Systems. D. Mukamel (2000) 

The Zero Range Process 3 

C


Non-Equilibrium: General properties (2) 

• In thermal equilibrium we have hamiltonian H 

• P(C) ∼ e −E(C)/K BT 

• Equilibrium ⇒ detailed balance 

W(C ′ → C)P(C ′ ,t) = W(C → C ′ )P(C,t) ∀C, C ′ 

• Non-Equilibrium ⇒ NO detailed balance 


(2)


Zero Range Process (ZRP) 

The ZRP is a non-linear stochastic model of interacting particles on a Lattice, 

hopping from site to site with a rate that depends solely on the occupation of the 

departure site. 



Zero Range Process (ZRP) (2) 

• The hopping rate from a lattice site ℓ with nℓ particles occupying it is denoted 

as uℓ(nℓ). 




• Homogenous case - u(nℓ) independent of position . 

• Ring Topology: 1D lattice with periodic boundaries of length L and N particles. 




• A configuration of the lattice is defined by the occupation of each site 

C ≡ {n1, n2, ...,nL} 

• Define the probability to find the lattice in configuration C at time t 

P(C,t) ≡ P(n1,n2,...,nL,t) 



Zero Range Process (5) 

• The master equation 

∂P({nℓ},t) 

∂t 

= 

L 

[u(nℓ−1)P(...,nℓ−1 + 1,nℓ − 1,...) − u(nℓ)P({nℓ})]Θ(nℓ) 

ℓ=1 

• examine steady state configurations 

∂P({nℓ},t) 

∂t 


= 0


Steady State in the Canonical Ensemble 

• In the canonical ensemble the probability distribution can be written using 

steady state single site weights-f 

P({nℓ}) = Z −1 

L,N 

L 

f(nℓ)δ( 

ℓ=1 

L 

nℓ = N) (3) 

ℓ=1 

• The canonical partition function is defined as follows 

ZL,N ≡ 

{n ℓ} 

L 

f(nℓ)δ( 

ℓ=1 

L 

nℓ = N) (4) 


ℓ=1


Steady State in the Canonical Ensemble (2) 

• From the master equation 

f(n) = 

• The probability factorizes as follows 

P({nℓ}) = 

L 

m=1 

1 

u(m) 

(5) 

L 

p(nℓ) (6) 

• We will mainly be interested in the single site distribution function 

ℓ=1 

p(n) = f(n) ZL−1,N−n 

ZL,N 


(7)


Grand Canonical Ensemble 

• The grand canonical partition function is defined as follows 

with F(λ) and f(m) defined as 

ZL(λ) = [F(λ)] L 

F(λ) = 

f(m) = 

∞ 

m=0 

λ m f(m) 

m 

u(i) −1 


i=1 

(8)


Grand Canonical Ensemble (2) 

• The fugacity λ corresponds to the steady state current < u(n) >= λ and is 

determined through the density 

ρ = λ F ′ (λ) 

F(λ) 


(9)


Grand Canonical Ensemble (3) 

•The single site probability 

p(n) = λ n f(n) ZL−1(λ) 

ZL(λ) 

plugging in the definition of ZL yields 

p(n) = λn f(n) 

F(λ) 

(10) 



TASEP ⇒ ZRP 

a) 

b) 

Site: 

Particle: 

u(3) 

u(1) 

1 2 3 4 

u(2) 

u(3) u(1) 

u(2) 

1 2 3 4 

• In the TASEP hopping rates depend on inter-particle distance 

5 

• Phase separation in the driven model (TASEP) corresponds to a macroscopic 

occupation in the ZRP model. 

2 Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models. Evans et al (2005). 


5 

2


Condensation in the ZRP 

• The density (given in eq. 9) is clearly an increasing function of the fugacity z. 

ρ = z F ′ (z) 

F(z) = 

∞ 

m=0 m zm f(m) 

∞ 

m=0 zm f(m) 

• The maximum density is given by taking z → 1 

ρmax = F ′ (1) 

F(1) = 

∞ 

m=0 mf(m) 

∞ 

m=0 f(m) 

(11) 

(12) 



Condensation in the ZRP (2) 

• A system with a finite maximum density (critical density) ρc has three phases: 

1. ρ < ρc low density phase: 

A fluid obeying the steady state distribution. 

2. ρ = ρc critical phase: 

A critical fluid with diverging correlation lengths . 

3. ρ > ρc condensed phase: 

A condensate in coexistence with the critical fluid. 



Analytically solvable case 

• TASEP: the particle current out of a domain of length n decays as 3 

• Consider ZRP with hopping rate 

j(n) ∼ β(1 + b/n) 

u(n) = 1 + b/n (13) 

• Interested in condensation ⇒ investigate large n behavior 

3 Novel phase-separation transition in one-dimensianal driven models. 

Y.Kafri, E.Levine, D.mukamel, G.M.Schutz and R.D.Willmann. (2002) 



Analytically solvable case (2) 

• The steady state probability can be calculated numerically using a recursion 

relation for the partition function 

ZL,N = 

N 

n=0 

• The single site probability distribution is 

f(n)ZL−1,N−n 

p(n) = f(n) ZL−1,N−n 

ZL,N 

(14) 

(15) 




ln p(n) 

0.001 

1e-06 

1e-09 

1e-12 

1 10 100 1000 10000 

ln n 

Figure 1: Probability distribution calculated using the recursion relation. For 

L = 1000, b = 5 and ◦, ⋄ for ρ = 1/4, 4 respectively 

4 





• large n analysis: 

f(n) = 

ln (f(n)) = − 

≈ − 

n 

m=1 

the single site weight is approximately 

[1 + b/m] −1 

n 

m=1 

n 

m=1 

ln (1 + b/m) 

b 

m 

f(n) ≈ n −b 

≈ −b ln(n) 




• The large n probability 

• The density is given by 

ρ = 

p(n) ≈ λn 

n bp(0) 

N 

np(n) ≈ 

n=1 

• The maximum density (λ → 1) behaves as 

ρmax ∼ 

N 

n=1 

N 

n=1 

λ n 

n b−1p(0) 

1 

n b−1 

(16) 

(17) 




• The condensation criterion (in the L → ∞ thermodynamic limit) 

• The critical density is given by 5 

b ≤ 2 ρmax → ∞ 

b > 2 ρmax → ρc 

ρc = 1 

b − 2 


(18) 




For b > 2 the system has three phases 

1. low density phase: 

p(n) ∼ 1 

n 

where the correlation length ξ is defined as 

b exp−n/ξ 

ξ = 1 

|ln(λ)| 

(19) 

(20) 

For b < 2 one can easily see that ρ ≡< n > diverges hence no condensation 

will appear. 




 

 

 

 

 

 

 

 

 

 

Figure 2: Probability distribution for L = 1000, b = 5 and ρ = 0.25. 




 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3: Lattice occupation for L = 1000, b = 5 and ρ = 0.25. 




2 critical phase: 

p(n) ∼ 1 

nb (21) 

Since the correlation length ξ diverges for λ → ∞ the critical fluid obeys a 

power distribution. 

Such a power distribution is called a scale free distribution 




 

 

 

 

 

 

 

 

 

Figure 4: Probability distribution for L = 1000, b = 5 and ρ = ρc = 1/3. 




 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5: Lattice occupation for L = 1000, b = 5 and ρ = ρc = 1/3. 




3 condensed phase: 

p(n) ∼ 1 

nb (22) 

A scale free distribution with a single peak in the distribution centered 

around n = L(ρ − ρc) corresponding to a single condensate in the L → ∞ 

thermodynamic limit. 




 

 

 

 

 

 

 

 

 

Figure 6: Probability distribution for L = 1000 , b = 5 and ρ = 2/3. The 

condensate is expected to be centered at ∆ = L(ρ − ρc) = 333. 




 

 

 

 

 

 

 

 

 

 

Figure 7: Lattice occupation for L = 1000, b = 5 and ρ = 2/3. The condensate 

is expected to be centered at ∆ = L(ρ − ρc) = 333. 




 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 8: The effect of the probability distribution for L = 1000 , b = 5 for 

different densities . 




 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 9: The effect of the probability distribution for L = 1000, b = 2.6 for 

different densities . 



Dynamics Of The Condensate 

• We have studied the steady state distribution for these processes 

 

 

 

 

 

 

 

 

 

 

 

Figure 10: The probability distribution for L = 1000, b = 2.6 and ρ = 4. 



Dynamics Of The Condensate (2) 

• These states are not static but dynamic 

 

 

 

 

 

 

 

 

 

 

 

• Investigate the dynamical behavior of the condensate 




• The population of the condensate is constant on average. 

while the fluctuations 

can be separated to two regimes 

< n >= 

np(n) ∼ 

n 1−b 

n 

< n 2 >= 

n 2 p(n) ∼ 

n 2−b 

2 

3 

n 


n 

n


• These states are not static but dynamic 

• Let us look at the behavior of the condensate 




 

 

 

 

 

 

 

 

 

 

 

 

Figure 11: The occupation and position of the condensate as a function of time. 

Simulated for L = 100, b = 4 and ρ = 2 




 

 

 

 

 

 

 

 

 

 

Figure 12: The occupation and position of the condensate as a function of time. 

Simulated for L = 100, b = 2.6 and ρ = 4 




• Model the condensate occupation to a random walk 6 on the segment 

[0,∆ = L(ρ − ρc)] usinf MF dynamics 

- The walker steps to the right with a rate 

λ =< u(n) >= 

u(n)p(n) 

In the supercritical phase the fugacity approaches 1 

- To the left with a rate of u(n) 

6 C. Godrèche and J.M.Luck Dynamics of the condensate in zero-range processes cond-mat/0505640 


n



• Evaporation and Creation times of the condensate are mapped to walking back 

and forth the interval [0,∆]. 

• The evaporation time scales with the system size as follows 

τ ∼ (ρ − ρc) b+1 L b ∼ ∆b+1 

L 



Condensation into more than one site 

• We would like to consider the possibility to condense into more than one site 

• Restrict the accumulation of particles in a site 

• Such a macroscopic restriction can be achieved by adding the term 

 

n 

k L 



Condensation into more than one site (2) 

• The hopping rate is no longer monotonically decreasing but has a minimum at 

n ∼ L k/(k+1) 

 

 

 

 

 

 

 

 

 

 

 

 




• The new hopping rate is approximately unchanged for small n. 

 

 

 

u(n) = 1 + b/n + c (n/L) k 

 

 

 

 




• The fluid will not feel the added term and all the properties are un changed, 

including the critical density ρc. 

• This is true in the L → ∞ limit and there are finite size effects such as a 

critical fluid for b = 1.9 < 2 and L = 1000. 

 

 

 

 

 

 

 

 

 

 

 




• In the super critical phase The fluid corresponds to a scale free distribution 

with a peak corresponding to the condensates. 

• The peak is weaker than the single condensate case and is algebraic. 

As can be seen in the figure for L = 1000, b = 2.6, c = 1, k = 5 and ρ = 4. 

 

 

 

 

 

 

 

 

 

 

 




• In order to understand the dynamics we wish to analyze the scaling of the 

probability distribution. 

• The leading behavior in the system size L is as follows 

n ∗ ∼ L k/(k+1) ln(L) 1/(k+1) 

p(n ∗ ) ∼ L −2k/(k+1) 

∆n ∗ ∼ L k/(k+1) ln(L) 1/(k+1) 

w ∼ 

1/(k+1) ln(L) 

L k 

(23) 




• The dip in the probability scales as follows 

nmin ∼ 

p(nmin) ∼ 

k/(k+1) L 

ln(L) 

− bk 

L k+1 

ln(L) 

(24) 




• One conclusion is that these are not condensates but meso-condensates sice 

they scale sub-linearly with the system size. 

• Another conclusion the number of meso-condensates Ncon ∼ w L ∼ L 1/(k+1) 

• The number of meso-condensates diverges with the system size 



Conclusions 

• We discussed the general properties of the ZRP and condensation. 

• We solved for the case of condensation for an analytical hopping rate 

• We discussed the possibility of condensation into more than one site 

• Thank you for your time....

The Zero Range Process Condensation in non-equilibrium systems

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