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clusters move and exchange particles, following a power law scaling. We also de- scribe the exponential approach to stationarity in the saturation regime, and the initial nucleation dynamics where isolated particle clusters form on a fast time scale. The coarsening behaviour in condensing systems has already been studied heuristically in [11] and subsequent work for zero-range processes [9, 10, 12, 13, 37] and related models [38, 39, 40]. A rigorous description of the coarsening dynamics has also been studied recently in [92]. In contrast to zero-range models, in the inclu- sion process and related models condensates are mobile on the coarsening time scale and coarsening is in fact driven by condensate motion and interaction [31, 32, 41, 42]. Further developments in this direction include explosive condensation in a totally asymmetric model [31, 32] which exhibits a slinky motion of the condensate, also observed recently in [35] for non-Markovian zero-range dynamics. In this chapter we are able to give a detailed picture of this phenomenon in the totally asymmetric inclusion process by studying the interaction of two clusters. Further recent results on non-condensing inclusion processes include a hydrodynamic scaling limit for the symmetric system making use of self-duality of the model [93], which is not available for the totally asymmetric model we consider in this chapter and will be discussed later in Chapter 6.

3.2

Condensation and dynamical regimes

Recall the dynamics of the inclusion processes are defined by the generator acting on bounded test functionsf :XL!R,

LLf(⌘) = X x,y2⇤L

p(x, y)⌘x(dL+⌘y)(f(⌘x,y) f(⌘)). (2.27 revisited)

In this chapter we focus on TASIP and SIP on one-dimensional lattices as introduced in Section 2.3.2:

p(x, y) = 1

2( y,x+1+ y,x 1) (SIP)

p(x, y) = y,x+1 (TASIP) .

3.2.1 Condensation

For fixed Land dLthe range of densities is R([0,1)) = [0,1) and the process does not exhibit condensation in the usual sense of zero-range processes [9, 12] or related

3.2. Condensation and dynamical regimes 32 models [42], where this range is bounded. But it has been established in [74, 87] that in the thermodynamic limit (2.32) with vanishing di↵usion rate

L, N! 1, dL!0 such that N

L !⇢>0 and dLL!0 , the system exhibits complete condensation. In this case,

max

x2⇤⌘x/N !1 in distribution⇡L,N , (3.1) so if the di↵usion rate scales asdL⌧1/Lalmost all particles in the system condense on a single site. Furthermore, in [87] stationary large deviations for the maximum occupation number are computed in the limit (2.32), and for condensing systems the most likely value for the maximum scales as the total number of particles N in the system. We will assume dL ⌧ 1/L for the rest of the chapter and for all simulation results presented we usedL= _L12, but have checked the validity also for

other scalings ofdL.

In contrast to zero-range processes, the condensate and large clusters move on the same time scale as the system approaches stationarity. The motion and interaction of clusters dominates the coarsening process, as will be explained in detail in the following. This has been established rigorously in [41] for the simpler setting of symmetric systems on fixed lattices. This mechanism is very similar to recent results in [31, 32] on explosive condensation, where the jump rates are essentially ⌘x(✏+⌘y) with fixed ✏ > 0 and > 2. In this case domination of attractive e↵ects and condensation is caused by the non-linearity in the rates. For the inclusion process it is the scalingdL!0 that causes domination of the attractive interaction.

To describe the dynamics of condensation we consider the second moment

2_{(t) =E}⇥_⌘2

x(t) ⇤

for somex₂⇤L , (2.4 revisited) which does not depend on x since we will assume the initial distribution to be translation invariant. This is the simplest observable that captures the temporal evolution of the condensed phase, since the first moment is constant in time due to conservation of the number of particles. Due to spatial homogeneity, in simula- tions we measure 2(t) by spatial averageD1/LPL_x=1⌘_x2Eto have better statistics, where_h·i denotes averaging over a large number (typically 200 in our simulations) of realisations.

3.2. Condensation and dynamical regimes 33 formly and independently on the lattice, which leads to ⌘(0) having a symmetric multinomial distribution withN trials and success probability 1/L. ForL! 1and N/L_!⇢ the occupation numbers are asymptotically independent Poisson random variables ⌘x(0)⇠Poi(⇢), and have second moment 2(0) =⇢(1 +⇢). Furthermore, in stationarity as t! 1we know that up to fluctuations all particles condense on a single site, and we expect 2

L(t) ' L1(⇢L)2 = ⇢2L. So we consider the rescaled variable _L2(t)/⇢2L, which increases from very small values of order 1/Lto 1 during the formation of the condensate from homogeneous initial conditions. This process can be divided into four di↵erent regimes (see Figure 3.1):

(I). Nucleation Regime: Due to the inclusion rate ⌘x⌘y, neighbouring pairs of sites exchange particles with order 1 rates until the process reaches a state where all occupied sites are separated by at least one empty site. This happens simultaneously everywhere on the lattice and takes at most of order logLtime. After this regime, a fraction of at most 1/2 of all sites is occupied and particles can only jump to another site by the di↵usion part of the dynamics with slow ratedL. Details can be found in Section 3.3.

(II). Coarsening Regime: Particle clusters formed in regime (I) can move to empty neighbouring sites or exchange particles at rate⌘xdL, but typically do not split on this timescale. This drives a coarsening process with a decreasing number of clusters of increasing size, which grow to large clusters of orderN size. This coarsening process happens on a characteristic time scale 1/dL, as explained in detail in Section 3.5. As expected, 2(t) follows an approximate power law in this regime.

(III). Saturation Regime: The coarsening scaling law no longer holds since the system reaches its finite size limit, and the remaining clusters merge to form a single condensate. As expected close to stationarity, the observable 2_{(t) con-}

verges exponentially to its stationary value, as explained in detail in Section 3.5.2. The characteristic time scale for this regime is up to constant factors the relaxation time of the system, and turns out to be of order⌧L =L/dL for the TASIP andL2/dL for the SIP (see Section 3.4.3).

(IV). Stationary Regime: Once there is only a single condensate left on the lattice, it continues to move according to the same rules and time scales as in regimes II and III. The observable 2 does not detect this motion, but it

3.3. Nucleation regime 34