Giant Amplification of Noise in Fluctuation-Induced Pattern Formation

Biancalani, Tommaso; Jafarpour, Farshid; Goldenfeld, Nigel

doi:10.1103/physrevlett.118.018101

Cited by 66 publications

(54 citation statements)

References 32 publications

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“…In particular, the DLEs derived here suggest that the average steady-state concentration of particles in each domain is independent of the arrangement and shape of domains. While we have focused here on the deterministic parts of the lattice Langevin equations associated with diffusion in inhomogeneous media, the formalism employed here can be extended [24,33,34,[38][39][40][41][42] to carry out a systematic analysis of the fluctuations induced by the random hopping of particles in inhomogeneous media, and to connect the DLEs derived here to generalized diffusion equations with spatially-varying diffusion coefficients [16][17][18][19][20][21]25].…”

Section: Discussionmentioning

confidence: 99%

Distribution of randomly diffusing particles in inhomogeneous media

Kahraman

Haselwandter

2017

Phys. Rev. E

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Diffusion can be conceptualized, at microscopic scales, as the random hopping of particles between neighboring lattice sites. In the case of diffusion in inhomogeneous media, distinct spatial domains in the system may yield distinct particle hopping rates. Starting from the master equations (MEs) governing diffusion in inhomogeneous media we derive here, for arbitrary spatial dimensions, the deterministic lattice equations (DLEs) specifying the average particle number at each lattice site for randomly diffusing particles in inhomogeneous media. We consider the case of free (Fickian) diffusion with no steric constraints on the maximum particle number per lattice site as well as the case of diffusion under steric constraints imposing a maximum particle concentration. We find, for both transient and asymptotic regimes, excellent agreement between the DLEs and kinetic Monte Carlo simulations of the MEs. The DLEs provide a computationally efficient method for predicting the (average) distribution of randomly diffusing particles in inhomogeneous media, with the number of DLEs associated with a given system being independent of the number of particles in the system. From the DLEs we obtain general analytic expressions for the steady-state particle distributions for free diffusion and, in special cases, diffusion under steric constraints in inhomogeneous media. We find that, in the steady state of the system, the average fraction of particles in a given domain is independent of most system properties, such as the arrangement and shape of domains, and only depends on the number of lattice sites in each domain, the particle hopping rates, the number of distinct particle species in the system, and the total number of particles of each particle species in the system. Our results provide general insights into the role of spatially inhomogeneous particle hopping rates in setting the particle distributions in inhomogeneous media.

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Section: Discussionmentioning

confidence: 99%

Distribution of randomly diffusing particles in inhomogeneous media

Kahraman

Haselwandter

2017

Phys. Rev. E

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“…Although the second part T † + T X is also zero when averaged over Eq. (8), it does not need to be the case.…”

Section: Generators Of Transient Behaviourmentioning

confidence: 99%

“…This shortcoming in physics literature can be traced back to the work of Orr [1] in the hydrodynamical context. Since then, similar ideas were revived in the context of fluid dynamics [2,3], plasma physics [4,5], diffusion in porous media [6] or pattern formation [7,8]. Further motivation for this work is rooted in the ecological literature on biological networks [9][10][11].…”

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confidence: 99%

What drives transient behavior in complex systems?

Grela

2017

Phys. Rev. E

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We study transient behaviour in the dynamics of complex systems described by a set of nonlinear ODE's. Destabilizing nature of transient trajectories is discussed and its connection with the eigenvalue-based linearization procedure. The complexity is realized as a random matrix drawn from a modified May-Wigner model. Based on the initial response of the system, we identify a novel stable-transient regime. We calculate exact abundances of typical and extreme transient trajectories finding both Gaussian and Tracy-Widom distributions known in extreme value statistics. We identify degrees of freedom driving transient behaviour as connected to the eigenvectors and encoded in a non-orthogonality matrix T0. We accordingly extend the May-Wigner model to contain a phase with typical transient trajectories present. An exact norm of the trajectory is obtained in the vanishing T0 limit where it describes a normal matrix.One of the key problems in studying complex systems is answering the question of stability. The standard linearization approach relies heavily on the large time asymptotics and can be misleading for intermediate times. This is especially pronounced when systems develop transient behaviour -the analysis based on eigenvalues loses its significance and different approach is needed. This shortcoming in physics literature can be traced back to the work of Orr [1] in the hydrodynamical context. Since then, similar ideas were revived in the context of fluid dynamics [2,3], plasma physics [4,5], diffusion in porous media [6] or pattern formation [7,8]. Further motivation for this work is rooted in the ecological literature on biological networks [9][10][11].The physical mechanism of transient behaviour is both relatively simple and quite general. It needs the system's components to interact asymmetrically and be stabilized by an effective dissipation mechanism. Asymmetry is indispensable since only then inter-eigenmodes fluxes of "energy" can be formed. Such an unbalanced flow renders particular eigenmodes overpopulated or amplified. Crucially, this mechanism does not break the overall stability -the dissipation eventually wins over and the amplification effect is only temporary or transient.In the paper we focus on such transient trajectories for a system of non-linear ODE's. On an elementary example we show how eigenvalue-based linearization technique becomes misleading and how simultaneously the transient property is developed. We utilize the May-Wigner model to include this mechanism and inspect its generic features. We find a new transient regime where transient trajectories are present although uncommon. Based on these findings, we identify relevant degrees of freedom driving the transient behaviour and propose a natural extension of the May-Wigner model. Such a modification leads to a generic transient behaviour arising as a robust transient phase. * jacek.grela@lptms.u-psud.fr A. Stability of systemsWe focus on a typical complex dynamical system of a set of N first-order non-linear ordinary differential equa-

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“…Systems undergoing chemical reactions offer a promising and fertile field where selection effects due to external noise can be tested, observed and refined for specific purposes in mind. The realm of complex chemical phenomena that could be manipulated and controlled in this way with external noise includes sustained chemical oscillations [6][7][8][9], pattern formation [10,11], excitable dynamics and front propagation [12][13][14][15], or any non-linear chemical systems where the number of constituents is sufficiently large so as to allow smooth concentrations to be defined [16].…”

Section: Introductionmentioning

confidence: 99%

Selection and control of pathways by using externally adjustable noise on a stochastic cubic autocatalytic chemical system

2018

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We investigate the effect of noisy feed rates on the behavior of a cubic autocatalytic chemical reaction model. By combining the renormalization group and stoichiometric network analysis, we demonstrate how externally adjustable random perturbations (extrinsic noise) can be used to select reaction pathways and therefore control reaction yields. This method is general and provides the means to explore the impact that changing statistical parameters in a noisy external environment (such as noisy feed rates, fluctuating reaction rates induced by noisy light, etc) has on chemical fluxes and pathways, thus demonstrating how external noise may be used to control, promote, direct and optimize chemical progress through a given reaction pathway. *

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Giant Amplification of Noise in Fluctuation-Induced Pattern Formation

Cited by 66 publications

References 32 publications

Distribution of randomly diffusing particles in inhomogeneous media

Distribution of randomly diffusing particles in inhomogeneous media

What drives transient behavior in complex systems?

Selection and control of pathways by using externally adjustable noise on a stochastic cubic autocatalytic chemical system

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