Intrinsic fluctuations on Fisher fronts Svetozar Nesic Departamento de Matem´ aticas, Universidad Carlos III de Madrid February 8, 2012 GISC workshop, Universidad Carlos III de Madrid
Intrinsic fluctuations on Fisher frontsSvetozar Nesic
February 8, 2012
Fisher waves in a Reaction-Diffusion process
Kinetic reaction:
ρ is the concentration of particles
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Fisher waves in a Reaction-Diffusion process
Kinetic reaction:
ρ is the concentration of particles
Fisher-Kolmogorov equation
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Fisher waves in a Reaction-Diffusion process
Kinetic reaction:
ρ is the concentration of particles
Fisher-Kolmogorov equation
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Fisher waves in a Reaction-Diffusion process
Kinetic reaction:
ρ is the concentration of particles
Fisher-Kolmogorov equation
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Fisher waves in a Reaction-Diffusion process
Kinetic reaction:
ρ is the concentration of particles
Fisher-Kolmogorov equation
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Fisher waves in a Reaction-Diffusion process
Kinetic reaction:
ρ is the concentration of particles
Fisher-Kolmogorov equation
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Fisher waves in a Reaction-Diffusion process
Kinetic reaction:
ρ is the concentration of particles
Fisher-Kolmogorov equation
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Fisher waves in a Reaction-Diffusion process
N - number of particles
ρ is the concentration of particles
Fisher-Kolmogorov equation (what if ρ is not a continuous
function)
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Fisher waves in a Reaction-Diffusion process
N - number of particles
ρ is the concentration of particles
Fisher-Kolmogorov-Petrovsky-Piscunov equation
∂ρ(~r, t) ∂t
= D4ρ(~r, t) + kρ(~r, t)− λρ2(~r, t) + √ ρ(~r, t)/Nη(~r, t)
R. A. Fisher, Ann. Eugenics 7, 355 (1937), A. Kolmogorov, I.
Petrovsky, N. Piscounov, Moscou Univ. Bull. Math. A 1, 1 (1937)
2/19
Master equation approximated by a diffusion process
∂Pn(t) ∂t
= k V
nPn − λ
∂P(ρ, t) ∂t
= − (k1 − k2)N V
∂ρ2 ρ(1− ρ)P(ρ, t)
where γ = (k1−k2)N V , in the thermodynamic limit V ∼ N and σ2 =
k1+k2
V ' 1 N
3/19
∂ρ
x
ρ
4/19
Velocity of the front:
4/19
+ √ ρ/Nη
vN = 2 √
D− C log2 N
E. Brunet and B. Derrida, Phys. Rev. E 56, 2597 (1997)
4/19
+ √ ρ/Nη
x
How does the surface interface scale?
A. L. Barabasi, H. E. Stanley, Fractal Concepts in Surface Growth,
CUP, 1995
4/19
x
S(q) = hqh−q
A. L. Barabasi, H. E. Stanley, Fractal Concepts in Surface Growth,
CUP, 1995 6/19
Kinetic roughening
Lz
t
α = zβ
A. L. Barabasi, H. E. Stanley, Fractal Concepts in Surface Growth,
CUP, 1995 6/19
Two different results using particle model for A→ 2A and 2A→ A
process
Conjecture: In general, 2 dimensional Fisher waves belong to the
KPZ universality class in 2 dimensions, even though the front is
described by a 1 dimensional interface! G. Tripathy and W van
Saarloos, Phys. Rev. Lett. 85, 3556 (2000)
2 dimensional Fisher waves belong to the KPZ universality class in
the 1 dimension. E. Moro, Phys. Rev. Lett. 87, 238303 (2001)
7/19
Conjecture:
∂φ
where V ′(φ) = f (φ) = φ− φ3
φ
V
and g(φ) = σ(φ− φ3)α, belong to the KPZ universality class of the
same dimensionality!
Leading edge transformation φ ∼ 0
φ = e−λξψ and α = 1 give
∂ψ
Only works when α = 1.
G. Tripathy and W van Saarloos, Phys. Rev. Lett. 86, 5215 (2001) U.
Ebert and W van Saarloos, Phys. Rev. Lett. 80, 1650 (1998)
8/19
When N is really large, simulations are costly!
G. Tripathy, W. van Saarloos, Phys. Rev. Lett. 85, 3556 (2000) C.
R. Doering, M. A. Burschka, W. Horsthemke, J. Stat. Phys. 65, 953
(1991)
VS
∂ρ∗(x,t) ∂t = D∂2ρ∗(x,t)
∂x2 + ρ∗(x, t)− ρ∗2(x, t)
ρ∗ → ρ0ρ→ ρ∗
I. Dornic, H. Chate, M. A. Munoz, Phys. Rev. Lett. 94, 100601
(2005) E. Moro, Phys. Rev. E, 70, 045102 (2004)
9/19
10 2
10 3
10 4
10 -3
10 -2
10 -1
10 0
10 -6
10 -3
10 0
10 3
10 2
10 3
10 4
α = 1 2
Both macroscopic front line and microscopic edge line belong to 1d
KPZ universality class!
11/19
SN(q/B(N))
we scale curves to the curve for which N = 106
A(σ) = a + a1 log N B(σ) = b + b1 log N 8 16log N
1
1.5
2
2.5
10-2 100 102q10-12
N=106
N=1010
N=1014
N=1018
N=1022
N=1026
When q→ 0 we have S(q) = Teff q2 we get that effective
temperature
scales as:
No physical significance of ρ < 1 N
δ - distance ahead of the edge The position of the front
changes
Xt+δt = {
Xt + vcutoff t + R(δ) with prob. P(δ)
Xt + vcutoff t with prob. 1 − P(δ)
where P(δ) is the probability of a relevant fluctuation during the
time t
The effect of noise is so weak that, most of time, it can be
ignored and the cutoff theory describes accurately the evolution of
the front!
E. Brunet, B. Derrida, A. H. Mueller, and S. Munier, Phys. Rev. E
73, 056126 (2006)
13/19
No physical significance of ρ < 1 N
δ - distance ahead of the edge The position of the front
changes
Xt+δt = {
Xt + vcutoff t + R(δ) with prob. P(δ)
Xt + vcutoff t with prob. 1 − P(δ)
where P(δ) is the probability of a relevant fluctuation during the
time t
The effect of noise is so weak that, most of time, it can be
ignored and the cutoff theory describes accurately the evolution of
the front!
E. Brunet, B. Derrida, A. H. Mueller, and S. Munier, Phys. Rev. E
73, 056126 (2006)
13/19
hedge(x, t) = 1 N
hfront(x, t) = 1 2
15/19
7800 8200 8600
!
Cross-Correlation between Wfront(t) and Wedge(t)
Strong fluctuations at the tip will happen at the front after
∼ ln2 N Ly
Microscopic tip fluctuations drive the morphology of macroscopic
Fisher fronts
17/19
Conclusions:
One dimensional equipotential lines of stochastic Fisher fronts
belong to the one dimensional KPZ universality class contrary to
the conjecture!
Even for a large number of particles the front still feels the
noise!
Fluctuations propagate from the edge to the front providing the
same universality class for all equipotential lines.
18/19
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