Classification of phase transitions in reaction-diffusion models

Elgart, Vlad; Kamenev, Alex

doi:10.1103/physreve.74.041101

Cited by 70 publications

(147 citation statements)

References 82 publications

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“…This trajectory exits, at t = −∞, the "endemic" point A along its two-dimensional unstable manifold and enters, at t = ∞, the fluctuational disease extinction point B, along its two-dimensional stable manifold. As in one-dimensional birth-death systems [4,12], one can show that there is no trajectory going directly from A to C. Therefore, the fluctuational extinction point B, not present in the mean-field dynamics, plays a crucial role in the disease extinction.…”

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confidence: 99%

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Extinction of an infectious disease: A large fluctuation in a nonequilibrium system

2008

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We develop a theory of first passage processes in stochastic non-equilibrium systems of birthdeath type using two closely related epidemiological models as examples. Our method employs the probability generating function technique in conjunction with the eikonal approximation. In this way the problem is reduced to finding the optimal path to extinction: a heteroclinic trajectory of an effective multi-dimensional classical Hamiltonian system. We compute this trajectory and mean extinction time of the disease numerically and uncover a non-monotone, spiral path to extinction of a disease. We also obtain analytical results close to a bifurcation point, where the problem is described by a Hamiltonian previously identified in one-species population models.PACS numbers: 87.23.Cc, 02.50.Ga Statistics of large fluctuations in stochastic nonequilibrium systems has received much recent attention [1]. While the equilibrium fluctuation probability is determined by the Boltzmann distribution, there is no similar general principle away from equilibrium. The underlying reason is the absence of time reversal symmetry between the relaxation and excitation dynamics in out-of-equilibrium systems. As a consequence, the most probable fluctuation path is not determined by the relaxation trajectory of the underlying deterministic system.An important class of stochastic non-equilibrium systems is reaction kinetics, or birth-death, systems [2]. Rather than being caused by external factors, the noise in these systems is intrinsic, as it originates from discreteness of the reacting agents and random character of their interactions. When the typical number of agents is large, the Fokker-Planck (FP) approximation to the master equation, see e.g. Ref.[2], can accurately describe small deviations from the probability distribution maxima. It fails, however, in determining the probability of large fluctuations [3,4,5]. Therefore, developing adequate theoretical tools for dealing with large fluctuations is an important task.One of the areas where the birth-death models have been very successful is mathematical epidemiology, see Refs. [6]. In this Letter we consider two closely related models of spread of disease in a population. Although they have served as standard multi-population epidemiological models, the analysis of large fluctuations in each of these models has not been satisfactory. We will use the two models as prototypical examples of multidimensional stochastic non-equilibrium systems.Observing the dynamics of a disease in a finite population, one notices the remarkable phenomenon of extinction of the disease in a finite time. The expected time to extinction (and the possibility to affect it) is of great practical interest. Here we develop an efficient theoretical approach capable of computing, among other things, this quantity. The approach employs the probability generating function formalism in conjunction with the eikonal approximation. In this way the problem is reduced to the dynamics of an effective classical Hamiltonian sys...

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“…This Hamiltonian appears in the theory of a class of singlespecies models in the vicinity of a bifurcation point [12]. As H r (q 2 , p 2 ) is independent of time, it is an integral of motion.…”

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Extinction of an infectious disease: A large fluctuation in a nonequilibrium system

2008

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“…The calculations greatly simplify in the new variables Q = qe −p and P = e p − 1 [21,22]. The generating function of this canonical transformation is h −1 dxF (q, Q), where…”

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“…It also holds, close to the transition, for all sets of reactions belonging to the Directed Percolation universality class. The properly rescaled on-site Hamiltonian for this class of models is H 0 (Q, P ) = QP (P − Q + 1) [21].…”

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Negative velocity fluctuations of pulled reaction fronts

Meerson

Sasorov

2011

Phys. Rev. E

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The position of a reaction front, propagating into an unstable state, fluctuates because of the shot noise. What is the probability that the fluctuating front moves considerably slower than its deterministic counterpart? Can the noise arrest the front motion for some time, or even make it move in the wrong direction? We present a WKB theory that assumes many particles in the front region and answers these questions for the microscopic model A ⇄ 2A and random walk.PACS numbers: 02.50. Ga, 87.23.Cc, 05.10.Gg, 87.18.Tt The Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation [1],describes invasion of an unstable phase, q(x → ∞, t) = 0, by a stable phase, q(x → −∞, t) = 1. It is one of the most fundamental models in mathematical genetics and population biology [1,2], but similar equations appear in chemical kinetics [3], extreme value statistics [4], disordered systems [5] and even particle physics [6]. The invasion fronts are described by traveling-front solutions (TFSs) of Eq. (1): q(x, t) = Q 0,c (ξ), where ξ = x − ct. Q 0,c (ξ) solves the equationwhere the prime denotes the ξ-derivative. It is well known [7] that, for a steep enough initial condition, the invasion front of Eq. (1) converges at long times to the limiting TFS, Q 0,2 (ξ), of Eq. (2) with the velocity c 0 = 2. This special velocity is determined by the dynamics of the leading edge of the front, where Eq. (1) can be linearized around q = 0. One can say that the nonlinear front, described by Eq. (1), is "pulled" by its leading edge, hence the term "pulled fronts" [7], of which the FKPP equation (1) is the celebrated example. Being a mean-field equation, Eq. (1) ignores discreteness of particles and the resulting shot noise. Their impact on the front propagation is dramatic, and it has attracted a great deal of interest [8][9][10][11][12][13][14][15][16]. The noise leads to fluctuations of the front shape [8,12]. The particle discreteness and noise also cause deviations of the front position with time that include a systematic part -the front velocity shift -and a fluctuating part. If N ≫ 1 is the number of particles in the front region, the front velocity shift scales as ln −2 N [9-11], whereas the front diffusion coefficient scales as ln −3 N [11,13,15]. These properties of fluctuating pulled fronts are markedly different from those of fluctuating fronts propagating into metastable states, where the front velocity shift and the front diffusion coefficient scale as inverse powers of N and are therefore much smaller [14,17,18].The front diffusion coefficient probes typical, relatively small fluctuations of the front position. Here we ask the following question that has not been addressed before. What is the probability P(c) that a fluctuating front moves, during a long time interval τ , with average velocity c that is considerably smaller than the mean-field value c 0 = 2? This includes the extreme case of c = 0, when the front is standing on average, and even c < 0, when it moves "in the wrong direction." We will also obtain new results in the reg...

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“…The tube is centered at the most probable switching path (MPSP). For low fluctuation intensity, the MPSP is obtained from a variational problem, which also determines the switching activation barrier [13][14][15][16][17][18][19][20][21][22]. Despite its fundamental role, the concept of the narrow tube of switching paths has not been tested experimentally nor has this tube been characterized quantitatively.…”

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Paths of Fluctuation Induced Switching

2008

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We demonstrate that the paths followed by a system in fluctuation-activated switching form a narrow tube in phase space. A theory of the path distribution is developed and its direct measurement is performed in a micromechanical oscillator. The experimental and theoretical results are in excellent agreement, with no adjustable parameters. We also demonstrate the lack of time-reversal symmetry in switching of systems far from thermal equilibrium. [6,7] in modulated traps and rf-driven Josephson junctions [8,9] and nano-and micromechanical resonators [10 -12]. Nonequilibrium systems generally lack detailed balance, and the switching rates may not be found by a simple extension of the Kramers approach.The analysis of switching in both nonequilibrium systems and complex equilibrium systems, including biomolecules, relies on the idea that, even though the system motion is random, the switching paths form narrow tubes in phase space. The tube is centered at the most probable switching path (MPSP). For low fluctuation intensity, the MPSP is obtained from a variational problem, which also determines the switching activation barrier [13][14][15][16][17][18][19][20][21][22]. Despite its fundamental role, the concept of the narrow tube of switching paths has not been tested experimentally nor has this tube been characterized quantitatively. Prior experimental studies [23] and simulations [24,25] focused on the distribution in space and time of fluctuational paths to a certain space-time point [26]. The methods [23][24][25][26] do not apply to the switching paths distribution for systems with more than one dynamical variable because of the motion slowing down near the stable states and the related loss of path synchronization.In this Letter, we present a theory of the switching paths distribution and report its direct observation in a wellcharacterized system, a micromechanical oscillator driven into parametric resonance. The observed distribution is shown in Figs. 1(a) and 1(b). Our experimental and theoretical results are in excellent agreement, with no adjustable parameters. In addition to proving the concept of the narrow tube of switching paths and putting it on a quantitative basis, the results provide the first demonstration of the lack of time-reversal symmetry in switching of systems far from thermal equilibrium. The results also open the possibility of efficient control of the switching probability based on the measured narrow path distribution.We consider a bistable system with several dynamical variables q q 1 ; . . . ; q N . The stable states A 1 and A 2 are located at q A 1 and q A 2 , respectively. For low fluctuation intensity, the physical picture of switching is as follows. The system prepared initially near state A 1 , for example,

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Classification of phase transitions in reaction-diffusion models

Cited by 70 publications

References 82 publications

Extinction of an infectious disease: A large fluctuation in a nonequilibrium system

Extinction of an infectious disease: A large fluctuation in a nonequilibrium system

Negative velocity fluctuations of pulled reaction fronts

Paths of Fluctuation Induced Switching

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