The shape of the emerging condensate in effective models of - PowerPoint PPT Presentation

The shape of the emerging condensate in effective models of condensation Volker Betz TU Darmstadt Venice, 22 August 2017 Joint project with Steffen Dereich, Peter M¨ orters, Daniel Ueltschi

Effective models of condensation Particle models: ◮ Condensation in particle systems: a macroscopic fraction of the particles in a microscopic fraction of state space. ◮ This means: the value of a suitable ’observable’ is the same for a macroscopic fraction of the particles ◮ Example: BEC, the observable is energy; ◮ Example: Selection-mutation models, the observable is fitness. ◮ Proving existence of condensation is very hard. Effective models: ◮ Model the dynamics of the relevant quantity directly as a differential or integral equation. ◮ Condensation in the effective model means that smooth initial conditions converge weakly to measures with a dirac at the relevant place. ◮ Dynamical condensation is known for a few models. ◮ We will be interested in the shape of the function (on the right scale) as it approaches a Delta peak. V. Betz (Darmstadt) The shape of the emerging condensate

Kingmans model of selection and mutation p n (d x ) : fitness distribution of a population. Fitness x ∈ [0 , 1] . Effective equation: p n +1 (d x ) = (1 − β ) x p n (d x ) + βr (d x ) w n � 1 with w n := 0 xp n (d x ) (mean fitness) 0 < β < 1 (mutation rate) r (d x ) = mutant distribution. Abstract form: p n +1 = B [ p n ] p n + C � 1 r (d x ) [Kingman 1978]: if γ = 1 − β 1 − x > 0 , then condensation of 0 size γ occurs at x = 1 , as t → ∞ . orters 2013]: If r ( (1 − h, 1) ) ∼ h α , then [Dereich, M¨ � x γ y α − 1 e − y d y. n →∞ p n ( (1 − x/n, 1) ) = lim Γ( α ) 0 Gamma distribution V. Betz (Darmstadt) The shape of the emerging condensate

A model for Bosons in a bath of Fermions F t ( k ) = density of bosons at energy k > 0 . Effective equation: � ∞ � F t ( y )( k 2 + F t ( k )) e − k − F t ( k )( y 2 + F t ( y )) e − y � ∂ t F t ( k ) = b ( k, y ) d y 0 with b > 0 . Abstract form: ∂ t F t ( k ) = B [ F t ]( k ) F t ( k ) + C [ F t ]( k ) . [Escobedo, Mischler 99, 01]: If � ∞ � ∞ k 2 m := F 0 ( k ) d k > m 0 := e k − 1 d k, 0 0 then convergence of strength m − m 0 at k = 0 occurs as t → ∞ . [Escobedo, Mischler, Velazquez 03]: For b = 1 , the scale on which the condensate emerges is 1 /t , and the shape is a Gamma distribution. Parameter of the Gamma-Distribution may depend on initial condition, explicit representation formula for b = 1 , formal asymptotic expansions otherwise. V. Betz (Darmstadt) The shape of the emerging condensate

A model for Bosons in a heat bath p t ( k ) = energy distribution of Bosons, ˆ C = Fourier transform of C ( z )( e βz − 1) . the heat bath correlation function, A ( z ) = ˆ F ( x ) = cx 1 / 2 . Effective equation: � ∞ ∂ t p t ( x ) = A ( y − x ) p t ( x ) p t ( y ) d y 0 � ∞ � ∞ ˆ ˆ − C ( y − x ) F ( y ) p t ( x ) d y + C ( x − y ) p t ( y ) F ( x ) d y. 0 0 Abstract form: ∂ t p t ( x ) = B [ p t ]( x ) p t ( x ) + C [ p t ]( x ) . e 84]: If [Buffet, de Smedt, Pul´ � ∞ � ∞ F ( x ) m := p 0 ( x ) d x > m 0 := e βx − 1 , 0 0 then the condensation of strength m − m 0 occurs at x = 0 as t → ∞ . No previous result about the shape of the emerging condensate. V. Betz (Darmstadt) The shape of the emerging condensate

The Boltzmann-Nordheim equation f t ( k ) = energy distribution of weakly interacting bosons. Effective equation: complicated; but it has the Abstract form: ∂ t f t ( k ) = B [ f t ]( k ) f t ( k ) + C [ f t ] . [Escobedo, Velazquez 2015]: Equation blows up in finite time. Nothing about the shape of the condensate is known. This equation is much more singular that the previous ones! V. Betz (Darmstadt) The shape of the emerging condensate

An abstract point of view Let ( p t ) t � 0 , p t ∈ L 1 ([0 , ∞ )) , solve the equation ∂p t ( x ) = A [ p t ]( x ) = B [ p t ]( x ) p t ( x ) + C [ p t ]( x ) for t > 0 with initial condition p 0 ∈ L 1 ([0 , ∞ ]) . A , B , C : { κδ 0 + f d x : κ � 0 , f ∈ L 1 ( R + ) } → C ( R + ) . and we write p t instead of 0 δ 0 + p t d x . We say that ( p t ) exhibits condensation at x = 0 as t → ∞ if � ε ρ 0 := lim ε → 0 lim inf p t ( x ) d x > 0 . t →∞ 0 ρ 0 is then called the mass of the condensate. We say that convergence to condensation is regular, with bulk q ∈ L 1 , if � ∞ ∀ c > 0 : lim | p t ( x ) − q ( x ) | = 0 . t →∞ c V. Betz (Darmstadt) The shape of the emerging condensate

Main result: assumptions ∂p t ( x ) = A [ p t ]( x ) = B [ p t ]( x ) p t ( x ) + C [ p t ]( x ) Assumption A1 : Assume that B : { κδ 0 + f d x : κ � 0 , f ∈ L 1 ( R + ) } → C 1 ( R + ) , and that there is q ∈ L 1 , α > 0 , c > 0 , κ > 0 with B [ κδ 0 + q d x ](0) = 0 , ∂ x B [ κδ 0 + q d x ](0) < 0 , x → 0 x − α C [ κδ 0 + q d x ]( x ) = c. lim Assumption A2: Assume that for any sequence ( p n ) ⊂ L 1 with � ∞ p n d x → κδ 0 + q d x weakly, and lim | p n ( x ) − q ( x ) | d x = 0 , n →∞ c we have n →∞ � B [ p n ] − B [ q ] � C 1 ([0 ,δ ]) = 0 , lim n →∞ � C [ p n ] − C [ q ] � C ([0 ,δ ]) = 0 . lim V. Betz (Darmstadt) The shape of the emerging condensate

Main result: statements B [ κδ 0 + q d x ](0) = 0 , ∂ x B [ κδ 0 + q d x ](0) < 0 , x → 0 x − α C [ κδ 0 + q d x ]( x ) = c. lim Assume ( p t ) solves ∂ t p t = A [ p t ] and condensates regularly to κδ 0 + q d x . Assume further that p 0 ( x ) ∼ x α 0 near x = 0 , with α 0 > 0 . Then 1 t p t ( x/t ) = c 1 e − γ ∞ (0) x � � 1 { α � α 0 } x α c 2 + 1 { α 0 � α } x α 0 η (0) lim . t →∞ The values of c 1 , c 2 and γ ∞ (0) are known explicitly (see below). V. Betz (Darmstadt) The shape of the emerging condensate

Application to Kingmans model Kingmans Model (in continuous time): x p n +1 (d x ) = (1 − β ) w [ p n ] p n (d x ) + βr (d x ) replaced by x � � ∂ t p t (d x ) = (1 − β ) w [ p t ] − 1 p t (d x ) + βr (d x ) � 1 with w [ p ] = 0 xp (d x ) . Stationary solution: � 1 q (d x ) = β r (d x ) r (d x ) � � 1 − x + 1 − β δ 1 (d x ) . 1 − x 0 We have w [ q ] = 1 − β, so B [ q ]( x ) = x − 1 , C [ q ] = βr (d x ) . Putting y = 1 − x brings this into our form (condensation at zero), and our theorem applies! V. Betz (Darmstadt) The shape of the emerging condensate

Application to the EMV-model � ∞ � F t ( y )( k 2 + F t ( k )) e − k − F t ( k )( y 2 + F t ( y )) e − y � ∂ t F t ( k ) = b ( k, y ) d y 0 so � ∞ b ( k, y ) e − y � F ( y )( e k − y − 1) − y 2 � B [ F ]( k ) = d y, 0 and � ∞ C [ F ]( k ) = k 2 e − k b ( k, y ) F ( y ) d y. 0 k 2 With q ( k ) = e k − 1 d k + κδ 0 we find � ∞ y 2 ∂ k B [ q ](0) = − 2 b (0 , y ) e y − 1 d y − 2 κb (0 , 0) < 0 . 0 So the conditions apply, for (A2) it is enough that b ∈ C 1 b . V. Betz (Darmstadt) The shape of the emerging condensate

The BSP model ∂ t p t ( x ) = B [ p t ]( x ) p t ( x ) + C [ p t ]( x ) with � ∞ � ∞ ˆ B [ p ]( x ) = A ( y − x ) p ( y ) d y − C ( y − x ) F ( y ) d y, 0 0 � ∞ ˆ C ( x ) = F ( x ) C ( x − y ) p ( y ) d y. 0 F ( x ) With q (d x ) = e βx − 1 + κδ 0 we find ∂ x B [ q ](0) = − β ˆ C ( − y ) q ( y ) d y − κβ ˆ C (0) < 0 , showing (A1). For (A2), we need ˆ C ∈ C 1 . V. Betz (Darmstadt) The shape of the emerging condensate

Proof part 1: reformulation of assumptions ∂p t ( x ) = A [ p t ]( x ) = B [ p t ]( x ) p t ( x ) + C [ p t ]( x ) Assume that p t ∈ L 1 solves this equation, and condenses regularly to q + κδ 0 . Then with b t ( x ) = B [ p t ]( x ) , c t ( x ) = C [ p t ]( x ) we have (B1): lim t →∞ b t (0) = 0 . (B2): γ t ( x ) := − 1 x ( b t ( x ) − b t (0)) (with x > 0 ) is continuous at x = 0 , and and that there exists a continuos, strictly positive function γ ∞ : [0 , δ ] → R + such that t →∞ sup lim | γ t ( x ) − γ ∞ ( x ) | = 0 . x ∈ [0 ,δ ] (B3): There exists a continuos function c ∞ with c ∞ (0) > 0 and t →∞ sup lim | c t ( x ) − c ∞ ( x ) | = 0 . x ∈ [0 ,δ ] V. Betz (Darmstadt) The shape of the emerging condensate

Proof part 1: variation of constant p 0 ( x ) = x α 0 η ( x ) ∂ t p t = b t p t + c t , Variation of constants gives: � t W t γ s,t ( x ) d s + W t e − tx ¯ γ 0 ,t ( x ) p 0 ( x ) , x α c s ( x ) e − ( t − s ) x ¯ p t ( x ) = W s 0 where � t � 1 s γ r ( x ) d r if t > s � s 0 b u (0) du , t − s W s = e ¯ γ s,t ( x ) = γ t ( x ) if t = s. For fixed s , we have ¯ γ s,t ( x ) → γ ∞ ( x ) as t → ∞ . Let β = min { α, α 0 } . Assume that for Q t ( β ) := W − 1 ( t + 1) 1+ β , Q ∞ := lim t →∞ Q t ( β ) t exists and is finite. Then � ∞ t ) = e − γ ∞ (0) x 1 t p t ( x Q s ( β ) � � 1 { β = α } x α ( s + 1) β c s (0) d s +1 { β = α 0 } x α 0 η (0) lim Q ∞ t →∞ 0 V. Betz (Darmstadt) The shape of the emerging condensate