Análisis de la demanda

3.4.3 Derivation of time scale

The mechanism of cluster interaction together with the time scale for motion⌧_Lmove

(3.19) and (3.22) determines the the time scale⌧L of coarsening and relaxation of the system, which we used in Figure 3.1. For the TASIP, condensates containing of order ⇢L particles have speed of order dL⇢L. Then the relative speed between any two condensates is also of this order, which leads to the average time between two encounters to be of orderL_·⌧_Lmove_⇠1/(⇢dL). Since every interaction leads to an unbiased exchange of orderp⇢Lparticles, order ⇢L exchanges are necessary to achieve a macroscopic change, and it leads to the time scale

⌧_La =L/dL (TASIP) , (3.28) which is independent of the particle density⇢.

Following the similar argument for the SIP, the average time between succes- sive encounters becomesL2·⌧_Lmove⇠L2/dL, since the condensates need to do order L2 jumps to meet as they perform symmetric random walks with rate dL. But as opposed to the TASIP, condensates can exchange a macroscopic number of particles so that we only need order 1 such encounters, leading to

⌧_Ls =L2/dL (SIP) , (3.29) which is again independent of⇢.

3.5

Coarsening and saturation

3.5.1 Dynamics in the coarsening regime

We use heuristic arguments to derive the coarsening dynamics, based on the dy- namics of a single ‘typical’ cluster and its interaction with others in a mean-field approximation.

Totally asymmetric dynamics

Let m(t) denote the typical size of a cluster in the coarsening regime, and n(t) the typical number of clusters per volume, so that we have n(t)m(t) = ⇢. We denote the speed of a typical cluster byv(t) =dLm(t) and the typical distance of two clusters is given bys(t) =m(t)/⇢. Then the rate at which two clusters meet is

3.5. Coarsening and saturation 49 v(t)/s(t) =⇢dL. As discussed in Section 3.4.2, when two clusters meet, they make an unbiased exchange of orderpm particles. So for one cluster to lose all its particles, it typically takes of order m(t) exchanges. Therefore, each cluster independently disappears with rate Ca⇢dL/m(t), where Ca is a proportionality constant which is hard to predict and we will just fit it from simulation data. These death events, which happen typically after time t =m(t)/(Ca⇢dLn(t)) per unit volume, drive the coarsening process. Each event e↵ectively increasesm(t) by m(t) =m(t)/n(t) per unit volume, which leads to

d

dtm(t) =

m(t)

t =Ca⇢dL. (3.30)

The initial condition is

m(0) = ⇢ n(0) =

⇢

r ,

where n(0) = r (cf. (3.3)) is the expected ratio of occupied sites after nucleation which we also fit from the data. The solution to (3.30) is then simply

m(t) =Ca⇢dLt+

⇢

r. (3.31)

Due to the clustered nature of configurations during the coarsening regime we have

2_{(t) =} m2(t) s(t) =⇢m(t) =Ca⇢ 2_d Lt+ ⇢2 r , which implies 2_(t) ⇢2 =CadLt+ 1 r . (3.32)

Note that there is no explicit system size dependence in the above analysis and this scaling law also holds on infinite lattices (given a fixed small parameter dL). On a finite lattice it only applies in a certain scaling window, after which the system saturates due to finite size e↵ects (see Figure 3.6(a)), reminiscent of the classical Family-Viscek scaling for coarsening dynamics in surface growth (see, e.g., Chapter 3.3 in [95]). The time scale⌧Lcharacterises this scaling window and the relaxation of the system, and is determined by the scaling solution reaching its maximal stationary value, i.e.

m(⌧_La) =Ca⇢dL⌧_La+⇢

r =⇥(N) .

This implies ⌧_La = ⇥(L/dL) and corresponds to the time when all clusters have merged to a single condensate. This agrees with our previous prediction for the

3.5. Coarsening and saturation 50 10−1 100 101 102 103 101 102 103 dLt σ 2/ ρ 2 L=256_ρ=2 L=256_ρ=4 L=512_ρ=2 L=512_ρ=4 L=1024_ρ=1 L=1024_ρ=2 1024 512 256 (a) 10−1 100 101 102 103 104 101 102 dLt σ 2/ ρ 2 L=64ρ=1 L=64ρ=2 L=128ρ=1 L=128ρ=2 L=256ρ=1 256 128 64 (b)

Figure 3.6: Power-law scaling of 2_(t)/⇢2 _{in the coarsening regime. (a) Data for}

TASIP compared to the prediction (3.32) shown as a full line with fitted constant Ca = 1.8961 and initial condition r = 0.3851. (b) Data for SIP compared to the prediction (3.34) shown as a full line with fitted constantCs= 5.7614. Data points are averaged over 200 realisations. Errors are bounded by the size of the symbols.

asymmetric time scale in (3.28).

Symmetric dynamics

We can apply the same argument to the SIP and get similar results. Since particles jump symmetrically, the velocity of clustersv(t) = dL is indeed the di↵usivity(see (3.22)) but we keepv(t) with a slight abuse of notation. Withs(t) being the typical distance between clusters, the interaction rate of clusters scales likev(t)/s(t)2_{in one}

dimension. Unlike the TASIP, a single interaction of two clusters in the SIP leads to a macroscopic exchange of order m(t) particles as was derived in Section 3.4.2. Then we have d dtm(t) =Cs v(t)m(t) s(t)2 =Cs dL⇢2 m(t) , (3.33)

where Cs is again a proportionality constant for cluster interaction. With initial conditionm(0), we have the solution

3.5. Coarsening and saturation 51 As before, the second moment can be written as

2_{(t) =⇢m(t) =⇢}2p_2Csd

Lt+ ( 2(0)/⇢2)2,

and for the initial condition we now have the exact result of the nucleation regime (3.15) where 2(0)/⇢2= 3 + 1/⇢. This leads to

2_(t) ⇢2 = s 2CsdLt+ ✓ 3 +1 ⇢ ◆2 , (3.34)

where we only have to fit the parameterCs. This scaling law is confirmed in Figure 3.6(b), and the scaling window and time scale can again be determined from

m(⌧_Ls) = r 2Cs⇢2dL⌧_Ls+ ⇢2 r2 =⇥(N) . This implies ⌧s

L=⇥(L2/dL) which agrees with our previous prediction in (3.29).

3.5.2 Exponential saturation and stationarity

Having identified the time scales ⌧L of the coarsening window for symmetric and asymmetric dynamics, we expect that the power-law behaviour turns into an ex- ponential saturation of the system to the stationary value 1 of our observable

2_(t)/(⇢2_{L), i.e.}

2_(t)

⇢2_L '1 e

C0_t/⌧L

ast_{! 1}. (3.35)

This is essentially equivalent to the assertion thatC0/⌧L is indeed the spectral gap of the generator of the system, which usually describes the exponential approach to stationarity in finite systems as shown previously in Figure 3.1.

For symmetric dynamics, we can provide a simple derivation which includes a rough estimate of the constant C_s0. The late stage of the dynamics is dominated by 2 remaining clusters competing for particles. On average, both of them have roughly sizem'N/2, and from the derivation of (3.33) in the previous sub-section we see that under that assumption they meet at rate

Cs v(t) s(t)2 =Cs 4dL L2 = 4Cs/⌧ s L since s=m/⇢=L/2 .

As mentioned in Section 3.4.3, at each encounter the clusters can merge with prob- ability 1/2, which would lead to a single condensate and remaining in a typical stationary configuration. Since merge attempts are independent, this leads to an

3.5. Coarsening and saturation 52 t/τLa 0.5 1 1.5 2 2.5 3 1 − σ 2/ ρ 2L 10-2 10-1 100 L=256ρ=2 L=256ρ=4 L=512ρ=2 L=512ρ=4 Exp Function 1−2t/τ_La t/τLs 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 1 − σ 2/ ρ 2L 10-2 10-1 100 L=64ρ=2 L=64ρ=4 L=128ρ=2 L=128ρ=4 Exp Function 1− ! Ct/τ_Ls

Figure 3.7: Exponential relaxation in the saturation regime for TASIP (a) and SIP (b). The predictions (3.35) are shown as a full lines with best fit constantsC_a0 = 2.00 and C_s0 = 10.51. In both cases we plot the coarsening scaling law (dashed line) for comparison, which is only valid for short times on the scale ⌧a

L or ⌧Ls. Data points are averaged over 200 realisations. Errors are of the order 10 4.

e↵ective rate to reach stationarity roughly given by 2Cs/⌧Ls, and we expect

1

2_(t)

⇢2_L 'e

C0_st/⌧s

L ast! 1 , (3.36)

withC_s0 '2Cs. This is confirmed in Figure 3.7(b), where we see a good data collapse with exponential decay with a best fit parameterC_s0 = 10.51, which is similar to 2Cs as fitted in Figure 3.6. Given the crude approximation of two equal sized clusters in our derivation we cannot expect a perfect match of those constants.

For totally asymmetric dynamics two macroscopic clusters cannot merge in a single interaction event, but exchange only of orderpL particles in an unbiased fashion. Still, we expect the approach to stationarity to be governed by an expo- nential law of the form (3.35), and we can derive the constant by direct comparison with the coarsening dynamics. Expanding (3.35) for timest_⌧⌧L we get

2_(t) ⇢2_L '1 e Ca0t/⌧La 'C0 a t dL L , where we used ⌧a

L = L/dL. This matches the scaling law solution (3.32) and we see that in fact C_a0 =Ca. Again this is confirmed in Figure 3.7(a), where the best fit parameter for C_a0 is very close to 2 as is Ca. We currently do not have a good theoretical explanation to predict this value, but our numerics strongly suggest that the constant in the asymmetric case seems to be simply 2.

3.6. Summary 53