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The zero-range process (ZRP) is a stochastic particle system with no restriction on the number of particles per site, and with jump rates depending only on the number of particles occupying on the departure site. It was originally introduced by Spitzer [4], and the stationary measure has a simple product structure [4, 46], as covered in Theorem 2.3. This model has recently drawn great research interests since a particular class of this model exhibits condensation transitions, which was established in a series of papers [9, 10, 11, 12]. And more variants of models in this class have been studied recently, including a non-Markovian version with slinky condensate motion [35, 79]. Recent reviews of the literature can be found in [14, 42]. If the particle density⇢in the system exceeds a critical value⇢c, the system separates into a fluid phase at density⇢cand a condensate, as explained in the previous section. The dynamics and time scaling of this condensation have been studied heuristically in [37]. For a large but finite system, the location of the condensate changes on a slow timescale and converges to a random walk on the lattice in the limit of diverging density [80, 81]. Recent extensions include a non-equilibrium version [82, 83] and models with size-dependent transition rates [84]. The zero range process so far has been the most studied model in the family of interacting particle systems without restrictions on local occupation numbers, and provides inspiring ideas to work on other models including the inclusion process. It is also well known that the zero range process can be mapped to simple exclusion process if sites are considered as particles and masses as hole clusters (see, e.g., [6, 85]). In this thesis we do not study the zero range process directly, but only extract ideas from the relevant work and extend them to the inclusion process, and in some cases compare these two models. In this section we give basic definitions and properties of the zero-range process. For further details see [4, 73].

2.3. Models 23

x y

p(x, y)g(⌘x)

Figure 2.1: Illustration of the dynamics of zero-range process. Particles perform random walks with rate p(x, y)g(⌘x) , which is independent of particles on target sitey.

Definitions

The local state space of zero-range process is E = N, and we focus on finite translation invariant lattices with periodic boundary conditions. Denote the one- dimensional torus byTn=Z/nZ={1,2,3, ..., n}. We consider zero-range processes defined on ˆd-dimensional torus, ⇤L = (Tn)dˆ, of L = ndˆ sites. In one-dimensional case this is⇤L={1,2,3, ..., L}with periodic boundary conditions. The state space is then

XL={⌘= (⌘x)x2⇤L : ⌘x2N}=N

⇤L_,

where⌘ is the full configuration and⌘x is the local configuration on sitex.

Particles on the lattice jump to other sites with rates depending only on the number of particles residing on departure sites (zero-range). This is a sharp contrast to the inclusion process and macroscopically makes a significant di↵erence to the system’s behaviour. The dynamics is described by the generator acting on bounded test functionsf 2Cb(X), choosing u(n) =g(n) and v(n)⌘1 as given in (2.11)

LLf(⌘) = X x,y2⇤

p(x, y)g(⌘x) (f(⌘x,y) f(⌘)), (2.23)

where p(x, y) 0 are transition rates of an arbitrary, irreducible random walk on ⇤L. In this thesis we restrict jumps to be spatially homogeneous,

p(x, y) =q(y x), for allx, y₂⇤L, (2.24) andq(x) is further assumed to be normalised and of finite range,

X x2⇤L

2.3. Models 24 φ 0 0.2 0.4 0.6 0.8 1 R_(φ) 0 0.2 0.4 0.6 0.8 1 φc ρc b_{= 1} b_{= 4} (a) Density ρ 0 0.5 1 1.5 2 j(ρ) 0 0.2 0.4 0.6 0.8 1 ρc ρc ρc ρc ρc ρc L= 128 L= 256 L= 512 Thermodynamic Limit (b)

Figure 2.2: Density and fundamental diagram of zero-range process (2.23) withg(n) given in (2.26). (a): DensityR( ) (2.15) with = 1,b = 1 and = 1, b= 4. For b= 4, ⇢c = 1/(b 2) = 1/2 and c = 1. (b): Fundamental diagram. The canonical current with = 1 andb= 4 for various (finite) systems are plotted (see Algorithms in Appendix C). The black line is the thermodynamic current as a function of the system density⇢, given by the inverse of functionR( ) for⇢_⇢c and byv c(g) = 1

for⇢>⇢c using Theorem 2.4.

where B is a bound independent of L. The jump rates g(⌘x) are assumed to be strictly positive on positive integers and have bounded variation,

sup

k2N|g(k+ 1) g(k)|<1 and g(k) = 0,k= 0.

The process can also be defined on infinite lattices under certain constraints, see [46, 86] for details.

Stationary measure

As discussed in previous section, the stationary product measures of zero-range processes are given by (2.12), where the weights can be specified as

w(n) = n Y k=1

g(k) 1, n >0.

Since v _⌘ 1 in this model, the grand canonical current (2.21) can be written as jgc = . Recall the average particle density (2.15) can be computed as R( ) =

2.3. Models 25

x y

p(x, y)⌘x(dL+⌘y)

Figure 2.3: Illustration of the dynamics of one-dimensional inclusion process. Parti- cles perform independent random walks with rate p(x, y)dL and attract each other with rate p(x, y)⌘x⌘y, which is called the inclusion part of the dynamics.

ofR( ). A standard example of zero-range process is given by g(n) = 1 + b

n , for all n 1, and g(0) = 0, (2.26) which was first studied in [9]. Condensation as defined in Section 2.2 occurs for 2(0,1), b >0 or = 1 andb >2, and the weights show a stretched exponential or power law decay, respectively (see [9, 12] for more details). Rigorous results on this transition have been published in a series of papers [12, 26, 54, 55, 56], and heuristic results on the dynamics have also been obtained in the areas like equilibration and coarsening [10, 12] and stationary dynamics of the condensate [37]. Figure 2.2 illustrates the density R( ) (2.15) and the fundamental diagram of the zero-range process with (2.26) with numerics and the thermodynamic limit. In Figure 2.2(b) we observe that the canonical currents are converging tojgc(⇢), a consequence of weak convergence in Theorem 2.4.