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 dt = d 2 u du dx 2 + u − u 2

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 dt = d 2 u du dx 2 + u − u 2 n ( x , t ) u = 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 dt = d 2 u du dx 2 + u − u 2 n ( x , t ) u = n ( x , t ) + n ( x , t )

F-KPP equation 1 dt = d 2 u du dx 2 + u − u 2 u(x,t) u(x,0) 0 ◮ Reaction diffusion model: A → 2 A and 2 A → 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 1 dt = d 2 u du dx 2 + u − u 2 u(x,t) u(x,0) 0 ◮ A one parameter family of traveling waves u ( x , t ) = W v ( x − vt ) where v is the velocity

F-KPP equation 1 dt = d 2 u du dx 2 + u − u 2 u(x,t) u(x,0) 0 ◮ A one parameter family of traveling waves u ( x , t ) = W v ( x − vt ) where v is the velocity ◮ If u ( x , 0 ) ∼ e − γ x with γ < γ c (where γ c satisfies v ′ ( γ c ) = 0) with v ( γ ) = γ + 1 u ( x , t ) ≃ W v ( γ ) ( x − v ( γ ) t ) γ

F-KPP equation 1 dt = d 2 u du dx 2 + u − u 2 u(x,t) u(x,0) 0 ◮ A one parameter family of traveling waves u ( x , t ) = W v ( x − vt ) where v is the velocity ◮ If u ( x , 0 ) ∼ e − γ x with γ < γ c (where γ c satisfies v ′ ( γ c ) = 0) with v ( γ ) = γ + 1 u ( x , t ) ≃ W v ( γ ) ( x − v ( γ ) t ) γ ◮ If u ( x , 0 ) ∼ e − γ x with γ > γ c the minimum velocity is selected u ( x , t ) ≃ W v ( γ c ) ( x − v ( γ c ) 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 ) or u ( x , 0 ) ∼ e − γ x with γ > γ c ) u ( x , t ) ≃ W v ( γ c ) ( x − X t ) with � X t = v ( γ c ) t − 3 2 π c v ′′ ( γ c ) t − 1 / 2 + · · · log t + Const − 3 2 γ c γ 5

The F-KPP class u = 1 stable ; u = 0 unstable ◮ F-KPP dt = d 2 u du dx 2 + u − u 2 ◮ other non-linearities dt = d 2 u du dx 2 + f ( u ) ◮ other diffusion terms du ( x , t ) � = ρ ( ǫ ) u ( x − ǫ, t ) d ǫ + f ( u ( x , t )) dt ◮ discrete time or/and space ◮ etc..

Our exactly soluble version Brunet D. 2015 a > 0  a u ( n − 1 , t ) + u ( n , t ) if 0 ≤ u ( n , t ) < 1 du ( n , t )  = dt 0 otherwise 

Our exactly soluble version Brunet D. 2015 a > 0  a u ( n − 1 , t ) + u ( n , t ) if 0 ≤ u ( n , t ) < 1 du ( n , t )  = dt 0 otherwise  u(n,0), u(n,t) t n = first time that u ( n , t ) = 1 n

Our exactly soluble version Brunet D. 2015 a > 0  a u ( n − 1 , t ) + u ( n , t ) if 0 ≤ u ( n , t ) < 1 du ( n , t )  = dt 0 otherwise  Time t n when u ( n , t ) = 1 for the first time ∞ ∞ 1 + a λ + a + 1 a λ u ( n , 0 ) λ n = − � � e − ( 1 + a λ ) t n λ n 1 + a λ n = 1 n = 1

Our exactly soluble version Brunet D. 2015 a > 0  a u ( n − 1 , t ) + u ( n , t ) if 0 ≤ u ( n , t ) < 1 du ( n , t )  = dt 0 otherwise  Time t n when u ( n , t ) = 1 for the first time ∞ ∞ 1 + a λ + a + 1 a λ u ( n , 0 ) λ n = − � � e − ( 1 + a λ ) t n λ n 1 + a λ n = 1 n = 1 Step initial condition u 1 ( 0 ) = u 2 ( 0 ) = · · · = 0 ⇒ � 3 2 π n c v ′′ ( γ c ) v ( γ c ) n − 1 / 2 + · · · 2 γ c v ( γ c ) log n + Const + 3 t n = v ( γ c )+ γ 7

Branching Brownian motion

Branching Brownian motion Pro ( X max ( t ))

The rightmost particle and the Fisher-KPP equation Branching Brownian Motion ◮ Particles diffuse � (∆ x ) 2 � = 2 dt ◮ They split at rate 1

The rightmost particle and the Fisher-KPP equation Branching Brownian Motion ◮ Particles diffuse � (∆ x ) 2 � = 2 dt ◮ They split at rate 1 Distribution of the rightmost particle Q ( x , t ) = Proba [ X max ( t ) < x ]

The rightmost particle and the Fisher-KPP equation Branching Brownian Motion ◮ Particles diffuse � (∆ x ) 2 � = 2 dt ◮ They split at rate 1 Distribution of the rightmost particle � 1 � Q ( x , t ) = Proba [ X max ( t ) < x ] Q ( x , 0 ) = 0 0

The rightmost particle and the Fisher-KPP equation Branching Brownian Motion ◮ Particles diffuse � (∆ x ) 2 � = 2 dt ◮ They split at rate 1 Distribution of the rightmost particle � 1 � Q ( x , t ) = Proba [ X max ( t ) < x ] Q ( x , 0 ) = 0 0 Q ( x , t + dt ) = ( 1 − dt ) � Q ( x + ∆ x ) � + dt Q ( x , t ) 2

Distribution of the rightmost particle McKean 1975 � 1 � Q ( x , t ) = Proba [ X max ( t ) < x ] Q ( x , 0 ) = 0 0 Q ( x , t + dt ) = ( 1 − dt ) � Q ( x + ∆ x ) � + dt Q ( x , t ) 2

Distribution of the rightmost particle McKean 1975 � 1 � Q ( x , t ) = Proba [ X max ( t ) < x ] Q ( x , 0 ) = 0 0 Q ( x , t + dt ) = ( 1 − dt ) � Q ( x + ∆ x ) � + dt Q ( x , t ) 2 Taking dt ≪ 1 (and as � (∆ x ) 2 � = 2 dt ) one gets The Fisher-KPP equation ∂ t Q = ∂ 2 x Q − Q + Q 2

Distribution of the rightmost particle McKean 1975 � 1 � Q ( x , t ) = Proba [ X max ( t ) < x ] Q ( x , 0 ) = 0 0 Q ( x , t + dt ) = ( 1 − dt ) � Q ( x + ∆ x ) � + dt Q ( x , t ) 2 Taking dt ≪ 1 (and as � (∆ x ) 2 � = 2 dt ) one gets The Fisher-KPP equation ∂ t Q = ∂ 2 x Q − Q + Q 2 u = 1 − Q

Mean field theory of directed polymers D. Spohn 88 N ( t ) � e β X i ( t ) Z t = i = 1 √ � Z t e η dt with prob. 1 − dt Evolution Z t + dt = Z ( 1 ) + Z ( 2 ) dt t t − e − β x Z t � � �� Generating function Q ( x , t ) = exp ∂ t Q = ∂ 2 x Q − Q + Q 2

Mean field theory of directed polymers D. Spohn 88 N ( t ) � e β X i ( t ) Z t = i = 1 √ � Z t e η dt with prob. 1 − dt Evolution Z t + dt = Z ( 1 ) + Z ( 2 ) dt t t − e − β x Z t � � �� Generating function Q ( x , t ) = exp ∂ t Q = ∂ 2 x Q − Q + Q 2 − e − β x � u ( x , 0 ) = 1 − Q ( x , 0 ) = 1 − exp �

Mean field theory of directed polymers ∂ t Q = ∂ 2 x Q − Q + Q 2 − e − β x Z t � � �� Q ( x , t ) = exp v ( γ ) = γ + 1 γ and v ′ ( γ c ) = 0 ◮ β < γ c (high temperature phase) log Z t ≃ t β v ( β ) ◮ β > γ c (glassy phase) log Z t ≃ t β v ( γ c ) + corrections

Replica approach N ( t ) log � Z n � t � e β X i ( t ) Z t = � log Z t � = lim n n → 0 i = 1

Replica approach N ( t ) log � Z n � t � e β X i ( t ) Z t = � log Z t � = lim n n → 0 i = 1 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 N ( t ) log � Z n � t � e β X i ( t ) Z t = � log Z t � = lim n n → 0 i = 1 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: � � n �� � log Z n µ + n µβ 2 t � = extremum 0 <µ< 1 t

Replica approach N ( t ) log � Z n � t � e β X i ( t ) Z t = � log Z t � = lim n n → 0 i = 1 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: � � n �� � log Z n µ + n µβ 2 t � = extremum 0 <µ< 1 t 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