Negative velocity fluctuations of pulled reaction fronts

Abstract

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...

Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts

“…Furthermore, the first line coincides (up to the numerical pre-factor which is apparently non-universal) with prediction from the phenomenological theory of Derrida et al [11]. The validity of MFT demands that δc ≫ 2π 2 ln −3 N [17]. Needless to say, N must be huge to have these pronounced asymptotic regimes.…”

“…Similarly to the model studied in Ref. [17], there is a symmetry relation among q and p profiles of the TFS:…”

“…The (instanton-type) solutions of these equations were found analytically in Ref. [17] for some values of c [most importantly for c close, but not too close, to c * ] and numerically otherwise. The resulting ln P(c) turns out to be proportional to T , strongly non-Gaussian with respect to c and rapidly falling with an increase of δc ≡ c * − c. The N -dependence of the quantity −T −1 ln P(c) undergoes a major change depending on δc:…”

“…The fluctuating traveling front conjecture [15,17] assumes that, at T → ∞, S(X, T ) has the following scaling form: S(X, T ) = T F (X/T ), so that…”

“…[17] that the aforementioned properties of ln P(c) also hold for all sets of on-site reactions which belong to the directed percolation universality class: under condition that the system is sufficiently close to the characteristic (transcritical) bifurcation of the mean-field theory. This is the setting we address here.…”