The Fisher KPP equation: an exactly soluble version Eric Brunet and - - PowerPoint PPT Presentation

The Fisher KPP equation: an exactly soluble version

Eric Brunet and Bernard Derrida

LPS ENS Paris, Collège de France Analytical Results in Statistical Physics in memory of Bernard Jancovici

Paris November 2015

Bernard Jancovici

Outline Introduction to F-KPP equations An exactly soluble case Branching Brownian motion Glass transition and replicas

The Fisher KPP equation

• R. Fisher Annals of Eugenics 1937

The wave of advance of advantageous gene

• A. Kolmogorov, I. Petrovsky, N. Piscounov, Bull. Univ. État Moscou 1937

Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique

du dt = d2u dx2 + u − u2

The Fisher KPP equation

• R. Fisher Annals of Eugenics 1937

The wave of advance of advantageous gene

• A. Kolmogorov, I. Petrovsky, N. Piscounov, Bull. Univ. État Moscou 1937

Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique

du dt = d2u dx2 + u − u2

u = n(x, t) n(x, t) + n(x, t)

The Fisher KPP equation

• R. Fisher Annals of Eugenics 1937

The wave of advance of advantageous gene

• A. Kolmogorov, I. Petrovsky, N. Piscounov, Bull. Univ. État Moscou 1937

Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique

du dt = d2u dx2 + u − u2

u = n(x, t) n(x, t) + n(x, t)

F-KPP equation

du dt = d 2u dx2 +u−u2

1 u(x,0) u(x,t)

◮ Reaction diffusion model: A → 2A and 2A → A

u(x, t) is the density

◮ Branching Brownian motion

u(x, t) is the distribution of the rightmost particle Mc Kean 1975

◮ Directed polymers on a tree

u(x, t): generating function of the partition function

• D. Spohn 1988

◮ extreme value statistics, particle physics, evolution in biology,

etc..

F-KPP equation

du dt = d 2u dx2 +u−u2

1 u(x,0) u(x,t)

◮ A one parameter family of traveling waves

u(x, t) = Wv(x − vt) where v is the velocity

F-KPP equation

du dt = d 2u dx2 +u−u2

1 u(x,0) u(x,t)

◮ A one parameter family of traveling waves

u(x, t) = Wv(x − vt) where v is the velocity

◮ If u(x, 0) ∼ e−γx with γ < γc

(where γc satisfies v ′(γc) = 0) u(x, t) ≃ Wv(γ)(x − v(γ)t) with v(γ) = γ + 1 γ

F-KPP equation

du dt = d 2u dx2 +u−u2

1 u(x,0) u(x,t)

◮ A one parameter family of traveling waves

u(x, t) = Wv(x − vt) where v is the velocity

◮ If u(x, 0) ∼ e−γx with γ < γc

(where γc satisfies v ′(γc) = 0) u(x, t) ≃ Wv(γ)(x − v(γ)t) with v(γ) = γ + 1 γ

◮ If u(x, 0) ∼ e−γx with γ > γc the minimum velocity is selected

u(x, t) ≃ Wv(γc)(x − v(γc)t + o(t))

The position of the front

Bramson 78,83 Ebert, van Saarloos 98 Steep enough initial condition

(i.e. u(x, 0) = θ(−x)

• r u(x, 0) ∼ e−γx with γ > γc)

u(x, t) ≃ Wv(γc)(x − Xt) with Xt = v(γc)t− 3 2γc log t + Const − 3

• 2π

γ5

c v ′′(γc)t−1/2 + · · ·

The F-KPP class u = 1 stable ; u = 0 unstable

◮ F-KPP

du dt = d2u dx2 + u − u2

◮ other non-linearities

du dt = d2u dx2 + f (u)

◮ other diffusion terms

du(x, t) dt =

• ρ(ǫ)u(x − ǫ, t) dǫ + f (u(x, t))

◮ discrete time or/and space ◮ etc..

Our exactly soluble version Brunet D. 2015

a > 0 du(n, t) dt =    a u(n − 1, t) + u(n, t) if 0 ≤ u(n, t) < 1

• therwise

Our exactly soluble version Brunet D. 2015

a > 0 du(n, t) dt =    a u(n − 1, t) + u(n, t) if 0 ≤ u(n, t) < 1

• therwise

n u(n,0), u(n,t)

tn= first time that u(n, t) = 1

Our exactly soluble version Brunet D. 2015

a > 0 du(n, t) dt =    a u(n − 1, t) + u(n, t) if 0 ≤ u(n, t) < 1

• therwise

Time tn when u(n, t) = 1 for the first time

∞

• n=1

u(n, 0) λn = − aλ 1 + aλ + a + 1 1 + aλ

∞

• n=1

e−(1+aλ)tn λn

Our exactly soluble version Brunet D. 2015

a > 0 du(n, t) dt =    a u(n − 1, t) + u(n, t) if 0 ≤ u(n, t) < 1

• therwise

Time tn when u(n, t) = 1 for the first time

∞

• n=1

u(n, 0) λn = − aλ 1 + aλ + a + 1 1 + aλ

∞

• n=1

e−(1+aλ)tn λn

Step initial condition u1(0) = u2(0) = · · · = 0 ⇒

tn = n v(γc)+ 3 2 γc v(γc) log n + Const + 3

• 2π

γ7

c v ′′(γc) v(γc)n−1/2 + · · ·

Branching Brownian motion

Branching Brownian motion Pro(Xmax(t))

The rightmost particle and the Fisher-KPP equation

Branching Brownian Motion

◮ Particles diffuse

(∆x)2 = 2dt

◮ They split at rate 1

The rightmost particle and the Fisher-KPP equation

Branching Brownian Motion

◮ Particles diffuse

(∆x)2 = 2dt

◮ They split at rate 1

Distribution of the rightmost particle Q(x, t) = Proba[Xmax(t) < x]

The rightmost particle and the Fisher-KPP equation

Branching Brownian Motion

◮ Particles diffuse

(∆x)2 = 2dt

◮ They split at rate 1

Distribution of the rightmost particle Q(x, t) = Proba[Xmax(t) < x] Q(x, 0) =

• 1

The rightmost particle and the Fisher-KPP equation

Branching Brownian Motion

◮ Particles diffuse

(∆x)2 = 2dt

◮ They split at rate 1

Distribution of the rightmost particle Q(x, t) = Proba[Xmax(t) < x] Q(x, 0) =

• 1

Q(x, t + dt) = (1 − dt)Q(x + ∆x) + dt Q(x, t)2

Distribution of the rightmost particle McKean 1975

Q(x, t) = Proba[Xmax(t) < x] Q(x, 0) =

• 1

Q(x, t + dt) = (1 − dt)Q(x + ∆x) + dt Q(x, t)2

Distribution of the rightmost particle McKean 1975

Q(x, t) = Proba[Xmax(t) < x] Q(x, 0) =

• 1

Q(x, t + dt) = (1 − dt)Q(x + ∆x) + dt Q(x, t)2

Taking dt ≪ 1 (and as (∆x)2 = 2dt) one gets

The Fisher-KPP equation ∂tQ = ∂2

xQ − Q + Q2

Distribution of the rightmost particle McKean 1975

Q(x, t) = Proba[Xmax(t) < x] Q(x, 0) =

• 1

Q(x, t + dt) = (1 − dt)Q(x + ∆x) + dt Q(x, t)2

Taking dt ≪ 1 (and as (∆x)2 = 2dt) one gets

The Fisher-KPP equation ∂tQ = ∂2

xQ − Q + Q2

u = 1 − Q

Mean field theory of directed polymers

• D. Spohn 88

Zt =

N(t)

• i=1

eβXi(t)

Evolution Zt+dt =

• Zt eη

√ dt

with prob. 1 − dt Z (1)

t

+ Z (2)

t

dt Generating function Q(x, t) =

• exp
• −e−βxZt
• ∂tQ = ∂2

xQ − Q + Q2

Mean field theory of directed polymers

• D. Spohn 88

Zt =

N(t)

• i=1

eβXi(t)

Evolution Zt+dt =

• Zt eη

√ dt

with prob. 1 − dt Z (1)

t

+ Z (2)

t

dt Generating function Q(x, t) =

• exp
• −e−βxZt
• ∂tQ = ∂2

xQ − Q + Q2

u(x, 0) = 1−Q(x, 0) = 1−exp

• −e−βx

Mean field theory of directed polymers

Q(x, t) =

• exp
• −e−βxZt
• ∂tQ = ∂2

xQ − Q + Q2

v(γ) = γ + 1

γ and v ′(γc) = 0

◮ β < γc (high temperature phase)

log Zt ≃ tβv(β)

◮ β > γc (glassy phase)

log Zt ≃ tβv(γc) + corrections

Replica approach

Zt =

N(t)

• i=1

eβXi(t)

log Zt = lim

n→0

logZ n

t

n

Replica approach

Zt =

N(t)

• i=1

eβXi(t)

log Zt = lim

n→0

logZ n

t

n

1 step of broken replica symmetry

Parisi Look for a saddle point of the form

n replicas in n

µ groups of µ replicas

Replica approach

Zt =

N(t)

• i=1

eβXi(t)

log Zt = lim

n→0

logZ n

t

n

1 step of broken replica symmetry

Parisi Look for a saddle point of the form

n replicas in n

µ groups of µ replicas

Saddle point: log Z n

t = extremum0<µ<1

• t

n µ + nµβ2

Replica approach

Zt =

N(t)

• i=1

eβXi(t)

log Zt = lim

n→0

logZ n

t

n

1 step of broken replica symmetry

Parisi Look for a saddle point of the form

n replicas in n

µ groups of µ replicas

Saddle point: log Z n

t = extremum0<µ<1

• t

n µ + nµβ2

• with P. Mottishaw

Fluctuations around the saddle point → the finite size corrections

Conclusion An "exactly soluble" F-KPP equation → a problem of complex analysis Replica for this problem Noisy version

• E. Brunet, B. Derrida

An Exactly Solvable Travelling Wave Equation in the Fisher KPP Class

• J. Stat. Phys. 2015
• B. Derrida, P. Mottishaw

Finite size corrections in the random energy model and the replica approach Journal of Statistical Mechanics: Theory and Experiment, 2015