(Super)diffusive asymtotics for perturbed Lorentz or Lorentz-like - PowerPoint PPT Presentation

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz (Super)diffusive asymtotics for perturbed Lorentz or Lorentz-like processes Domokos Sz´ asz Budapest University of Technology joint w. P´ eter N´ andori and Tam´ as Varj´ u Hyperbolic Dynamical Systems in the Sciences INdAM, Corinaldo, June 1, 2010

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz A Lorentz orbit

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Notions and notations: Lorentz Process Lorentz process - billiard dynamics (uniform motion + specular reflection) (Ω , T , µ ) Q = R d \ ∪ ∞ ˆ i =1 O i is the configuration space of the Lorentz flow (the billiard table), where the closed sets O i are pairwise disjoint, strictly convex with C 3 − smooth boundaries Ω = Q × S + is its phase space for the billiard ball map (where Q = ∂ ˆ Q and S + is the hemisphere of outgoing unit velocities) T : Ω → Ω its discrete time billiard map (the so-called Poincar´ e section map) µ the T -invariant (infinite) Liouville-measure on Ω

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Notions and notations: Lorentz Process Lorentz process - billiard dynamics (uniform motion + specular reflection) (Ω , T , µ ) Q = R d \ ∪ ∞ ˆ i =1 O i is the configuration space of the Lorentz flow (the billiard table), where the closed sets O i are pairwise disjoint, strictly convex with C 3 − smooth boundaries Ω = Q × S + is its phase space for the billiard ball map (where Q = ∂ ˆ Q and S + is the hemisphere of outgoing unit velocities) T : Ω → Ω its discrete time billiard map (the so-called Poincar´ e section map) µ the T -invariant (infinite) Liouville-measure on Ω

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Notions and notations: Lorentz Process Lorentz process - billiard dynamics (uniform motion + specular reflection) (Ω , T , µ ) Q = R d \ ∪ ∞ ˆ i =1 O i is the configuration space of the Lorentz flow (the billiard table), where the closed sets O i are pairwise disjoint, strictly convex with C 3 − smooth boundaries Ω = Q × S + is its phase space for the billiard ball map (where Q = ∂ ˆ Q and S + is the hemisphere of outgoing unit velocities) T : Ω → Ω its discrete time billiard map (the so-called Poincar´ e section map) µ the T -invariant (infinite) Liouville-measure on Ω

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Notions and notations: Lorentz Process Lorentz process - billiard dynamics (uniform motion + specular reflection) (Ω , T , µ ) Q = R d \ ∪ ∞ ˆ i =1 O i is the configuration space of the Lorentz flow (the billiard table), where the closed sets O i are pairwise disjoint, strictly convex with C 3 − smooth boundaries Ω = Q × S + is its phase space for the billiard ball map (where Q = ∂ ˆ Q and S + is the hemisphere of outgoing unit velocities) T : Ω → Ω its discrete time billiard map (the so-called Poincar´ e section map) µ the T -invariant (infinite) Liouville-measure on Ω

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Notions and notations: Periodic Lorentz → Sinai Billiard If the scatterer configuration { O i } i is Z d -periodic , then the corresponding dynamical system will be denoted by (Ω per = Q per × S + , T per , µ per ). It makes sense then to factorize it by Z d to obtain a Sinai billiard (Ω 0 = Q 0 × S + , T 0 , µ 0 ). The natural projection Ω → Q (and analogously for Ω per and for Ω 0 ) will be denoted by π q . Finite horizon (FH) versus infinite horizon ( ∞ H )

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Notions and notations: Periodic Lorentz → Sinai Billiard If the scatterer configuration { O i } i is Z d -periodic , then the corresponding dynamical system will be denoted by (Ω per = Q per × S + , T per , µ per ). It makes sense then to factorize it by Z d to obtain a Sinai billiard (Ω 0 = Q 0 × S + , T 0 , µ 0 ). The natural projection Ω → Q (and analogously for Ω per and for Ω 0 ) will be denoted by π q . Finite horizon (FH) versus infinite horizon ( ∞ H )

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Why are local perturbations/ ∞ H interesting? Local perturbations Lorentz, 1905: described the transport of conduction electrons in metals (still in the pre-quantum era). Natural to consider models with local impurities; Non-periodic models (M. Lenci, ’96-, Sz., ’08: Penrose-Lorentz process). ∞ H Hard ball systems in the nonconfined regime have ∞ H Crystals Non-trivial asymptotic behavior and new kinetic equ. (Bourgain, Caglioti, Golse, Wennberg, ...; ’98-, Marklof-Str¨ ombergsson, ’08-).

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Stochastic properties: Correlation decay Let f , g M (= Ω 0 , billiard phase space ) → R d be piecewise H¨ older. Definition With a given a n : n ≥ 1 ( M , T , µ ) has { a n } n -correlation decay if ∃ C = C ( f , g ) such that ∀ f , g H¨ older and ∀ n ≥ 1 � � � � � � f ( g ◦ T n ) d µ − � fd µ gd µ � ≤ Ca n � � � M M M The correlation decay is exponential (EDC) if ∃ C 2 > 0 such that ∀ n ≥ 1 a n ≤ exp ( − C 2 n ) . The correlation decay is stretched exponential (SEDC) if ∃ α ∈ (0 , 1) , C 2 > 0 such that ∀ n ≥ 1 a n ≤ C 1 exp ( − C 2 n α ) .

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Diffusively scaled variant Definition Assume { q n ∈ R d | n ≥ 0 } is a random trajectory. Then its diffusively scaled variant ∈ C [0 , 1] ( or ∈ C [0 , ∞ ]) is defined as follows: for N ∈ Z + denote q j W N ( j N ) = (0 ≤ j ≤ N or j ∈ Z + ) and define otherwise √ N W N ( t )( t ∈ [0 , 1] or R + ) as its piecewise linear, continuous extension. E. g. κ ( x ) = π q ( Tx ) − π q ( x ) : M → R d , the free flight vector of a Lorentz process. From now on q n = q n ( x ) = � n − 1 k =0 κ ( T k x ) , n = 0 , 1 , 2 , . . . is the Lorentz trajectory.

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Diffusively scaled variant Definition Assume { q n ∈ R d | n ≥ 0 } is a random trajectory. Then its diffusively scaled variant ∈ C [0 , 1] ( or ∈ C [0 , ∞ ]) is defined as follows: for N ∈ Z + denote q j W N ( j N ) = (0 ≤ j ≤ N or j ∈ Z + ) and define otherwise √ N W N ( t )( t ∈ [0 , 1] or R + ) as its piecewise linear, continuous extension. E. g. κ ( x ) = π q ( Tx ) − π q ( x ) : M → R d , the free flight vector of a Lorentz process. From now on q n = q n ( x ) = � n − 1 k =0 κ ( T k x ) , n = 0 , 1 , 2 , . . . is the Lorentz trajectory.

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Stochastic properties: CLT & LCLT Definition CLT and Weak Invariance Principle W N ( t ) ⇒ W D 2 ( t ) , the Wiener process with a non-degenerate covariance matrix D 2 = µ 0 ( κ 0 ⊗ κ 0 ) + 2 � ∞ j =1 µ 0 ( κ 0 ⊗ κ n ). Local CLT Let x be distributed on Ω 0 according to µ 0 . Let the distribution of [ q n ( x )] be denoted by Υ n . There is a constant c such that n →∞ n Υ n → c − 1 l lim where l is the counting measure on the integer lattice Z 2 and → stands for vague convergence. In fact, c − 1 = 1 det D 2 . √ 2 π

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz 2D, Periodic case: Some Results SEDC EDC CLT LCLT B-S, ’81 M-partitions X X B-Ch-S, ’91 M-sieves X X Y, ’98 M-towers X X Sz-V, ’04 X SEDC - Stretched Exponential Decay of Correlations EDC - Exponential Decay of Correlations CLT - Central Limit Theorem LCLT - Local CLT

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Locally perturbed FH Lorentz Sinai’s problem, ’81: locally perturbed FH Lorentz Sz-Telcs, ’82: locally perturbed SSRW for d = 2 has the same diffusive limit as the unperturbed one Idea: local time is O (log n ) thus the √ n scaling eates perturbation up Method: there are log n time intervals spent at perturbation couple the intervals spent outside perturbations to SSRW

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz Locally perturbed FH Lorentz Dolgopyat-Sz-Varj´ u, 09: locally perturbed FH Lorentz has the same diffusive limit as the unperturbed one Method: Martingale method of Stroock-Varadhan Tools: Chernov-Dolgopyat, 05-09: standard pairs growth lemma Young-coupling Sz-Varj´ u, 04: local CLT for periodic FH Lorentz Dolgopyat-Sz-Varj´ u, 08: recurrence properties of FH Lorentz (extensions of Thm’s of Erd˝ os-Taylor and Darling-Kac from SSRW to FH Lorentz)

Introduction Planar FH Lorentz Process ∞ H Lorentz Martingale method ∞ H Lorentz ∞ H periodic Lorentz Reminder: κ ( x ) = π q ( Tx ) − π q ( x ) : M → R 2 , the free flight vector of a Lorentz process. q n = q n ( x ) = � n − 1 k =0 κ ( T k x ) is the Lorentz trajectory. Now: for N ∈ Z + denote � j � q j W N = √ N log N (0 ≤ j ≤ N or j ∈ Z + ) N and define otherwise W N ( t )( t ∈ [0 , 1] or R + ) as its piecewise linear, continuous extension.