Negative velocity fluctuations of pulled reaction fronts Racah - - PowerPoint PPT Presentation
Negative velocity fluctuations of pulled reaction fronts Racah Institute of Physics, Baruch Meerson Hebrew University of Jerusalem Pavel V. Sasorov ITEP, Moscow arXiv:1106.3547 Large Fluctuations in Non-Equilibrium Systems, MPIPKS, Dresden,
Baruch Meerson
Racah Institute of Physics, Hebrew University of Jerusalem Large Fluctuations in Non-Equilibrium Systems, MPIPKS, Dresden, July 5 2011
Pavel V. Sasorov
ITEP, Moscow
arXiv:1106.3547
considerably slower than its deterministic counterpart? For example, can the noise arrest the front motion for some time, or even make it move in the wrong direction?
The FKPP equation
Fisher 1937, Kolmogorov, Petrovsky and Piscounov 1937
2 2
x t
A mean-field theory of invasion of an unstable phase, q(x → ∞, t) = 0, by a stable phase, q(x → −∞, t) = 1. A fundamental model in mathematical genetics, population biology and chemical
Traveling front solutions (TFSs) :
2 , , , , ,
c c c c c
Legitimate TFSs exist for For a steep initial condition, the solution approaches, at long time, a TFS Q0,2(ξ) with c=2.
This special solution is selected by the dynamics of the leading edge of the front, where Eq. (1) can be linearized around q = 0. The most celebrated example of pulled fronts.
The FKPP equation ignores discreteness of particles and the shot noise. Their impact is dramatic:
Mueller and Munier 2006)
Brunet, Derrida, Mueller and Munier 2006) N>>1 is the number of particles inside the front region Df only probes typical, relatively small fluctuations of the front position What is the probability P(c) that a noisy front moves, during a long time interval τ, with average velocity c that is considerably smaller than c = 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. But of most interest is the regime of 2-c<<1 when P(c) is not vanishingly small
Microscopic stochastic model: on-site reactions, say A->2A and 2A->A, and random walk on a 1d lattice Evolution of multivariate probability distribution P(n,t)=P(n1,n2,…,nN,t)
WKB ansatz
, / )], , ( exp[ ) , ( K t KS t P n q q n = − =
K>>1 is the on-site population size, S(q,t) is smooth
t q
In the leading order in 1/K one arrives at a Hamilton-Jacobi equation with Hamilton’s function
H0(q,p): on-site Hamilton’s function, h: lattice constant When random walk is fast this becomes a field theory: a continuous-in-space Hamilton’s mechanics:
x x x
2
Hamilton’s equations: two coupled PDEs
2 2 2
x x p p t x x x p p t
− −
q(x,t) - rescaled particle concentration – “coordinate”, p(x,t): “momentum”
In context of population extinction risk: Elgart and Kamenev 2004, Meerson and Sasorov 2011
The “relaxation trajectories”, q=q (x,t), p=0 are described by the FKPP equation
2 2 2 2
x x p p t x x x p p t
− −
x t 2 2
From now on: A->2A, 2A->A
Boundary conditions in time: kink-like profiles q(x,0)=q1(x), q(x,τ)=q2(x) Boundary conditions in space are determined by fixed points of the on-site Hamiltonian:
q(-∞,t)=1, p(-∞,t)=0, q(+∞,t)=0
in rescaled variables
2
− p p
Bernard Derrida observed that Eqs. (1) and (2) also describe the WKB limit of the stochastic FKPP equation
2 2 2 2 2 2
x t x t
2 2
x t
Calculations simplify in new variables
Boundary conditions: Q(-∞,t)=1; P(-∞,t)=0; Q(+∞,t)= either 0, or 1; P(+∞,t)=-1 so most of the following holds for the sFKPP equation too. Generating function density:
2 2
On-site Hamiltonian:
…, Pechenik and Levine (1999), Doering, Mueller and Smereka (2003), …
The model A->2A, 2A->A and random walk has two remarkable symmetries
Then Q2= P1(x,-t)+1 and P2=Q1(x,-t) -1 is also a solution with the same BCs.
In old variables this is invariance of q(x,t) with respect to time reversal: consequence of detailed balance of underlying microscopic model.
Then Q2= -P1(-x,-t) and P2=-Q1(-x,-t) is also a solution with the same BCs. Once the Hamilton’s equations are solved, we can calculate the action along the “activation trajectory” q(x,t), p(x,t) and evaluate P(c):
t t
∞ ∞ − ∞ ∞ − τ τ
N=Kld /h>>1 is the number of particles in the front region
Main assumption: at large τ the activation trajectory is described by
a Traveling Front Solution (TFS): Q=Q(x-ct), P=P(x-ct)
2 2 2 2
Find instanton: heteroclinic orbit of the 4th order dynamical system Conservation law:
reduces the order to 3rd. Still, for general c the problem appears non-integrable. Numerical solution, however, is straightforward For a TFS P(c) and the action become:
c c
− ∞ ∞ − ∞ ∞ − ∞ ∞ −
2 2
c ≥ -2
Follows from detailed balance of microscopic model
Example of numerical fluctuating TFS, c=3/2
Exact solution for c=0:
x x
−
2 x x
As a result,
c≤-2: equilibrium manifold Q=1, or p=ln q As a result, Here the activation trajectory is a time-reversed mean-field TFS with c≥2
2 2
x t
Results for the action and ln P(c)
ds/dt=c
Including exact results for
c=0 and c<-2
What happens when c is close to 2?
Perturbation theory for 2-c << 1
Strongly non-Gaussian, exponentially small in N, and rapidly falls off as c goes down Strong separation of Q and P profiles. Joint region where two asymptotes can be matched. Action comes from region where |P|~1.
C=1.96
Work in progress: Comparison with results of Brunet, Derrida, Mueller, and Munier (2006)
∞ = −
3 2 2
β
n n n X
∞ + ∞ − +
i i g X
σ σ β β
) (
We evaluated the integral at X<0 via saddle-point approximation. Two different regions: Gaussian dist. and left tail
cumulants
n 3 2 2
≥
X: front displacement relative to the mean-field front position
Thanks to Bernard Derrida for participating in comparison!
Asymptotes
(2011) MS 1 2 , 2 exp ... 0074 . (2006) Brunet ln , 2 ln ) ( exp ln 2 ~ ) ( ln (2006) Brunet ln , 4 ln ) ( 3
3 2 3 3 1 2 3 4 3 2
<< − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − << ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − << − −
− − − −
c c N al et c c N N c c N e c al et N c c N c c π π π τ π
γ
P 1
3 3 2 2 2
Put c=c0+(c-c0) in the WKB asymptote. After algebra:
Brunet et al (2006)
conjecture
Brunet and Derrida (1997)
− 2 3 3 2 /
2
π
a
Observation: Asymptote (1) holds, up to a c-independent factor, for a broad class of microscopic models and Langevin-type stochastic FKPP equations. For example, for all reactions belonging to the DP universality class close to transition.
ds/dt=c