Large deviations in models of growing clusters with symmetry-breaking transitions

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Large deviations in models of growing clusters with symmetry-breaking transitions

Robert L. Jack

Phys. Rev. E 100, 012140 – Published 25 July 2019

Abstract

We analyze large deviations of the magnetization in two models of growing clusters. The models have symmetry-breaking transitions, so the typical magnetization of a growing cluster may be either positive or negative, with equal probability. For large clusters, the magnetization obeys a large-deviation principle. We show that the corresponding rate function is zero for values of the magnetization that are intermediate between the two steady state values, which means that fluctuations with these values of the magnetization are much less unlikely than previously thought. We show that their probabilities decay as power laws in the cluster size, instead of the exponential scaling that would be expected from the large-deviation principle. We discuss how this observation is related to dynamical phase coexistence phenomena. We also comment on the typical size of magnetization fluctuations.

Received 17 April 2019

DOI:https://doi.org/10.1103/PhysRevE.100.012140

Physics Subject Headings (PhySH)

Dynamical phase transitions Growth processes

Nonequilibrium systems

Large deviation & rare event statistics

Statistical Physics & Thermodynamics

Authors & Affiliations

Robert L. Jack

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom and Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

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Vol. 100, Iss. 1 — July 2019

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Images

Figure 1
Upper bounds on the rate function for the LDP (6) in the irreversible model, for the representative case $J = 1.4$ . We show the bound of KGGW, which is obtained using control forces that are constant (independent of $k$ ). This is compared with our new bound for $| m | < m_{S}$ , which uses control forces that depend on $k$ . Our new result (with $k$ -dependent forces) implies that $I (m) = 0$ whenever $| m | < m_{S}$ .
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Figure 2
Results for the irreversible growth model in the representative case $J = 1.4$ , for which $m_{S} \approx 0.82$ . (a) Trajectories of the controlled dynamics for $K = 10^{8}$ , which all achieve $m_{K} \approx 0.4$ . For the $k$ -dependent force then $(b, k^{*}) = (0.10, 2.0 \times 10^{6})$ . (b) Results for $H_{K}$ using the same controlled dynamics, as a function of $K$ . (For the time-dependent forces, the values of $b, k^{*}$ are different for each $K$ .) Error bars are comparable with symbol sizes. The dashed line is the prediction of (24), without any fitting parameters. (c) The behavior of $H_{K}$ as a function of $m$ . The data in (b), (c) and the agreement with (24) indicate that ${lim}_{K \to \infty} H_{K} (m, ε) = 0$ throughout the range $| m | < m_{S} - ε$ , so ${lim}_{K \to \infty} I_{K} (m, ε) = 0$ within this range, by (14).
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Figure 3
(a) Estimated probability density for $m_{K}$ obtained by direct sampling of the irreversible growth model at $J = 1.3$ . Note that $K$ varies over more than two decades. The binning parameter is $ε = 10^{- 3}$ for values of $m$ near the peaks and $ε = 10^{- 2}$ for intermediate values. (Results depend weakly on $ε$ , which is chosen to reduce statistical uncertainties while maintaining adequate resolution.) (b) The logarithms of the same probability densities, scaled by $ln K$ . The prediction (28) is that for every $| m | < m_{S} - ε$ one should observe convergence of this quantity to some negative (nonzero) limit, as $K \to \infty$ . The data are consistent with this prediction.
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Figure 4
Irreversible model with $J = 1.3$ . (a) Probability distribution of $m_{K}$ under the controlled dynamics, with $k^{*}$ chosen as in (29). The form of the distribution depends weakly on $K$ and the probability density is of order unity across a wide range of $m$ . (b) The bound (31) for $K = 10^{5}$ , compared with the (numerically) exact distribution $ρ_{K}$ for the original model, obtained by direct sampling. (c) The negative of the asymptotic bound ${\tilde{H}}_{K}^{\infty}$ that appears in (33), for various $K$ . As an illustrative comparison, this is compared with the (numerically) exact distribution at $K = 10^{5}$ from panel (b), shown as a thin black line. We emphasize that the asymptotic bound is not a bound on this finite- $K$ distribution.
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Figure 5
Probability distribution $ρ_{K} (m, ε)$ for $J = 1.6$ with $K = 10^{5}$ and $ε = 8 \times 10^{- 4}$ , showing the behavior close to the peak at $m^{*} = m_{S} \approx 0.8906$ . In this case $Λ \approx 0.67$ . The probability density is shown on a logarithmic scale; it has been divided by $z_{0} = ρ_{K} (m^{*}, ε)$ so that the maximal value is unity. Numerical data were obtained by direct sampling and are compared with the predictions obtained using constant control forces [Eq. (44)] and $k$ -dependent control forces [Eq. (47)]. There are no fitting parameters. The predictions are accurate if their respective controlled processes capture the typical mechanism for fluctuations of $m_{K}$ ; they also require that $(m - m^{*})$ be small. The controlled process with constant forces does not fully capture the fluctuation mechanism so it underestimates the relevant probability. The process with $k$ -dependent forces captures the mechanism and gives accurate predictions close to the peak, but it does not capture the (mild) skewness of the exact distribution, because the Taylor expansion in (43) is truncated at second order.
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Figure 6
Reversible growth model with $(c, J) = (5.0, 4.0)$ , which is representative of the regime in which the model has three steady states. (a) Bounds on the rate function in (9), obtained using control forces that are independent of $k$ , and with $k$ -dependent forces. Compare with Fig. 1. The data with $I (m) > 0$ are obtained by numerical minimization of (56) but do not involve simulations of the growth process itself. (b) Trajectories of the controlled dynamics for $K = 10^{9}$ , which achieve either $m_{K} \approx 0.2$ or $m_{K} \approx 0.65$ . (c) Results for $H_{K}$ using these choices for the controlled dynamics. Data are shown for increasing $K$ , analogously to Fig. 2. For the time-dependent control force, this bound decays as $K^{- 1 / 2}$ , consistently with (60). (d) The behavior of $H_{K}$ is shown as a function of $m$ , analogously to Fig. 2. The decrease of $H_{K}$ with $K$ occurs for all $m$ within this range.
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