Modeling of fractional dynamics using L evy walks - recent advances - PowerPoint PPT Presentation

Modeling of fractional dynamics using L´ evy walks - recent advances Marcin Magdziarz Hugo Steinhaus Center Faculty of Pure and Applied Mathematics Wrocław University of Science and Technology, Poland Fractional PDEs: Theory, Algorithms and Applications ICERM, Brown University, June 2018 Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 1 / 45

Outline Examples of applications of L´ evy walks Basic definitions of L´ evy walks Asymptotic (diffusion) limits of multidimensional L´ evy walks Corresponding fractional diffusion equations Explicit densities in multidimensional case Other results Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 2 / 45

Examples of applications of L´ evy walks Light transport in optical materials P. Barthelemy, P.J. Bertolotti, D.S. Wiersma, Nature 453, 495 (2008). Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 3 / 45

Examples of applications of L´ evy walks Foraging patterns of animals M. Buchanan, Nature 453, 714 (2008). Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 4 / 45

Examples of applications of L´ evy walks Migration of swarming bacteria G. Ariel et al., Nature Communications (2015). Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 5 / 45

Examples of applications of L´ evy walks Blinking nanocrystals G. Margolin, E. Barkai, Phys. Rev. Lett. 94, 080601 (2005) F.D. Stefani, J.P. Hoogenboom, E. Barkai, Physics Today, 62 (2009) Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 6 / 45

Examples of applications of L´ evy walks Human travel D. Brockmann, L. Hufnagel, and T. Geisel, Nature 439, 462 (2006). Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 7 / 45

Basic definitions – 1D case Waiting times: T i , i = 1 , 2 , ... – sequence of iid positive random 1 variables with power-law distribution P ( T i > t ) ∝ t α , α ∈ ( 0 , 1 ) . Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 8 / 45

Basic definitions – 1D case Waiting times: T i , i = 1 , 2 , ... – sequence of iid positive random 1 variables with power-law distribution P ( T i > t ) ∝ t α , α ∈ ( 0 , 1 ) . Jumps: J i = Λ i T i where Λ i are iid random variables with P (Λ i = 1 ) = p , P (Λ i = − 1 ) = 1 − p . They govern the direction of the jumps (velocity v = 1). | T i | = | J i | . Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 8 / 45

Basic definitions – 1D case Waiting times: T i , i = 1 , 2 , ... – sequence of iid positive random 1 variables with power-law distribution P ( T i > t ) ∝ t α , α ∈ ( 0 , 1 ) . Jumps: J i = Λ i T i where Λ i are iid random variables with P (Λ i = 1 ) = p , P (Λ i = − 1 ) = 1 − p . They govern the direction of the jumps (velocity v = 1). | T i | = | J i | . Number of jumps up to time t: N t = max { n ≥ 0 : T 1 + ... + T n ≤ t } . Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 8 / 45

Definition: Wait-First L´ evy Walk – 1D case N t � R WF ( t ) = J i i = 1 Note that | R WF ( t ) | ≤ t . T 6 T 2 J 6 J 1 T 1 J 2 0 J 5 T 3 R WF (t) T 5 J 3 J 4 T 4 0 t Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 9 / 45

Definition: Jump-First L´ evy Walk – 1D case N t + 1 � R JF ( t ) = J i i = 1 T 5 T 1 J 6 J 1 J 2 J 5 0 T 6 R JF (t) T 2 T 4 J 3 J 4 T 3 0 t Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 10 / 45

Definition: Standard L´ evy Walk – 1D case N t � R ( t ) = J i + ( t − T ( N t ))Λ N t + 1 , i = 1 where T ( n ) = � n i = 1 T i . Note that | R ( t ) | ≤ t . T 6 T 2 J 6 J 1 J 5 J 2 0 T 1 T 3 R(t) T 5 J 3 J 4 T 4 0 t Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 11 / 45

Basic definitions – d -dimensional case Waiting times: T i , i = 1 , 2 , ... – sequence of iid positive random 1 variables with power-law distribution P ( T i > t ) ∝ t α , α ∈ ( 0 , 1 ) . Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 12 / 45

Basic definitions – d -dimensional case Waiting times: T i , i = 1 , 2 , ... – sequence of iid positive random 1 variables with power-law distribution P ( T i > t ) ∝ t α , α ∈ ( 0 , 1 ) . Jumps in R d : J i = Λ i T i where Λ i are iid unit random vectors in R d with the distribution Λ( dx ) on d − dimensional sphere S d . They govern the direction of the jumps in R d (velocity v = 1). We have | T i | = � J i � . Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 12 / 45

Basic definitions – d -dimensional case Distribution Λ on S 2 Trajectory 5 2 x 10 90 o 112.5 o 67.5 o a) d) 135 o 45 o 157.5 o 22.5 o 0 180 o 0 o −2 202.5 o 337.5 o 225 o 315 o −4 247.5 o 292.5 o 270 o −4 −2 0 5 x 10 5 x 10 90 o 112.5 o 67.5 o b) e) 6 135 o 45 o 157.5 o 22.5 o 4 180 o 0 o 2 202.5 o 337.5 o 225 o 315 o 247.5 o 292.5 o 0 270 o 0 2 4 6 5 x 10 5 2 x 10 90 o 112.5 o 67.5 o c) f) 135 o 45 o 157.5 o 22.5 o 0 180 o 0 o −2 202.5 o 337.5 o 225 o 315 o −4 247.5 o 292.5 o 270 o −2 0 2 5 x 10 Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 13 / 45

Basic definitions – d -dimensional case evy Walk in R d Wait-First L´ N t � R WF ( t ) = J i i = 1 Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 14 / 45

Basic definitions – d -dimensional case evy Walk in R d Wait-First L´ N t � R WF ( t ) = J i i = 1 evy Walk in R d Jump-First L´ N t + 1 � R JF ( t ) = J i i = 1 Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 14 / 45

Basic definitions – d -dimensional case evy Walk in R d Wait-First L´ N t � R WF ( t ) = J i i = 1 evy Walk in R d Jump-First L´ N t + 1 � R JF ( t ) = J i i = 1 evy Walk in R d Standard L´ N t � R ( t ) = J i + ( t − T ( N t ))Λ N t + 1 , i = 1 where T ( n ) = � n i = 1 T i . Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 14 / 45

Diffusion limits of L´ evy walks – d -dimensional case Theorem (Diffusion limit of Wait-First L´ evy walk) The following convergence in distribution holds as n → ∞ R WF ( nt ) d α ( S − 1 → L − − α ( t )) . n Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 15 / 45

Diffusion limits of L´ evy walks – d -dimensional case Theorem (Diffusion limit of Wait-First L´ evy walk) The following convergence in distribution holds as n → ∞ R WF ( nt ) d α ( S − 1 → L − − α ( t )) . n Here: L α (t) – d -dimensional α -stable L´ evy motion (limit of jumps) with Fourier transform � � � S d |� k , s �| α ( i sgn ( Φ L α ( t ) ( k ) = exp t � k , s � ) tan ( πα/ 2 ) − 1 )Λ( d s ) Λ( ds ) - distribution of jump direction Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 15 / 45

Diffusion limits of L´ evy walks – d -dimensional case Theorem (Diffusion limit of Wait-First L´ evy walk) The following convergence in distribution holds as n → ∞ R WF ( nt ) d α ( S − 1 → L − − α ( t )) . n Here: L α (t) – d -dimensional α -stable L´ evy motion (limit of jumps) with Fourier transform � � � S d |� k , s �| α ( i sgn ( Φ L α ( t ) ( k ) = exp t � k , s � ) tan ( πα/ 2 ) − 1 )Λ( d s ) Λ( ds ) - distribution of jump direction S α (t) – α -stable subordinator (limit of waiting times) S − 1 α ( t ) = inf { τ ≥ 0 : S α ( τ ) > t } Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 15 / 45

Diffusion limits of L´ evy walks – d -dimensional case Theorem (Diffusion limit of Wait-First L´ evy walk) The following convergence in distribution holds as n → ∞ R WF ( nt ) d α ( S − 1 → L − − α ( t )) . n Here: L α (t) – d -dimensional α -stable L´ evy motion (limit of jumps) with Fourier transform � � � S d |� k , s �| α ( i sgn ( Φ L α ( t ) ( k ) = exp t � k , s � ) tan ( πα/ 2 ) − 1 )Λ( d s ) Λ( ds ) - distribution of jump direction S α (t) – α -stable subordinator (limit of waiting times) S − 1 α ( t ) = inf { τ ≥ 0 : S α ( τ ) > t } Coupling! | ∆ L α ( t ) | = ∆ S α ( t ) Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 15 / 45

Diffusion limits of L´ evy walks – 1D case 0 ... 0 t Figure: Trajectory of the diffusion limit of Wait-First L´ evy walk. It can have infinitely many jumps on finite interval. Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 16 / 45

Diffusion limits of L´ evy walks – d -dimensional case Theorem (Diffusion limit of Jump-First L´ evy walk) The following convergence in distribution holds as n → ∞ R JF ( nt ) d → L α ( S − 1 − α ( t )) . n Marcin Magdziarz (Wrocław) Modelling of fract. dynamics - L´ evy walks ICERM 2018 17 / 45