What is common between the two systems below? (Wikipedia / Kinetic - PowerPoint PPT Presentation

Intro Billiard Waves Summary What is common between the two systems below? (Wikipedia / Kinetic theory of gases) (Wikipedia / Shallow water equations) Jani Lukkarinen

Intro Billiard Waves Summary (Free) transport by particles and waves Jani Lukkarinen 19 Jun 2018 Jani Lukkarinen

Intro Billiard Waves Summary Classical Hamiltonian system N identical particles (hard spheres, radius r 0 ) Reflection on the boundary In the figure, dimension d = 2 Dynamics away from the boundary determined by the Hamiltonian function H : ( R d ) N × ( R d ) N → R , � N � N p 2 2 + 1 n H ( q , p ) = V ( q n − q n ′ ) 2 n =1 n , n ′ =1; n ′ � = n Pair interaction potential V : R d → R for hard spheres satisfies V ( r ) = g ( | r | / (2 r 0 )) , g ( r ) = 0 , for r > 1 Jani Lukkarinen

Intro Billiard Waves Summary Suppose that at time t 0 none of the spheres is touching each other, nor the boundary of the box: for all n ′ , n , | q n ( t 0 ) − q n ′ ( t 0 ) | > 2 r 0 , dist( q n ( t 0 ) , ∂ B ) > r 0 Then V ( q n ( t 0 ) − q n ′ ( t 0 )) = 0, and by continuity of the trajectories there must be a time-interval ε > 0 during which none of the spheres interact with each other nor with the boundary If t 0 < t < t 0 + ε , the Hamiltonian evolution equations read � d � d t q n ( t ) = ∇ p n H ( q , p ) q = q ( t ) , p = p ( t ) = p n ( t ) � � d � d t p n ( t ) = −∇ q n H ( q , p ) q = q ( t ) , p = p ( t ) = − ∇ V ( q n ( t ) − q n ′ ( t )) = 0 n ′ � = n ⇒ free motion = movement with constant velocity , q n ( t ) = q n ( t 0 ) + ( t − t 0 ) p n ( t 0 ) , p n ( t ) = p n ( t 0 ) Jani Lukkarinen

Intro Billiard Waves Summary In the limit of instantaneous collisions (infinitely hard spheres), the trajectories ( q ( t ) , p ( t )) are piecewise smooth , with free motion intercepted by a finite number of collisions at which the positions are kept fixed but the velocities “jump” according to the rules of elastic collisions: If particles n = 1 and n ′ = 2 collide, then ( p 1 , p 2 ) → ( p ′ 1 , p ′ 2 ) with p ′ p ′ 1 = p 1 − [( p 1 − p 2 ) · n ] n , 2 = p 2 + [( p 1 − p 2 ) · n ] n where n denotes the unit vector pointing from the centre of the sphere 1 to the centre of the sphere 2 Jani Lukkarinen

Intro Billiard Waves Summary One-particle density 6 1 Suppose that the positions and velocities are random (for simplicity) with a probability density ρ N ( q , p ) 2 Particles are identical ⇒ ρ N is label permutation invariant Then the average number of particles inside a volume V ⊂ R d is given by � N � � � � = N E [ ✶ { q 1 ∈V} ] = d q R d d p ρ 1; N ( q , p ) E ✶ { q n ∈V} V n =1 where ρ 1; N denotes the first reduced one-particle density function � ρ 1; N ( q 1 , p 1 ) = N d q 2 d p 2 · · · d q N d p N ρ N ( q , p ) Central question of transport theory How does ρ 1; N evolve with time? Jani Lukkarinen

Intro Billiard Waves Summary How does the particle density evolve between collisions? 7 As show earlier, between collisions q n ( t ) = q n ( t 0 ) + ( t − t 0 ) p n ( t 0 ) , p n ( t ) = p n ( t 0 ) ⇒ ρ N ( q , p ; t ) = ρ N ( q − ( t − t 0 ) p , p ; t 0 ) Denote f t ( q , p ) := ρ 1; N ( q , p ; t ) ⇒ f t ( q , p ) = f t 0 ( q − ( t − t 0 ) p , p ) Evolution of one-particle density under free transport ∂ t f t ( q , p ) + p · ∇ q f t ( q , p ) = 0 Jani Lukkarinen

Intro Billiard Waves Summary Free Schr¨ odinger evolution 8 Free Schr¨ odinger equation ( � = 1, m = 1) i ∂ t ψ t ( x ) = − 1 2 ∇ 2 x ψ t ( x ) Probability density for finding the particle = | ψ t ( x ) | 2 Can be solved explicitly by Fourier transform: � d x e − i p · x ψ t ( x ) and then Denote � ψ t ( p ) = 2 p 2 � ψ t ( p ) = e − i t 1 � ψ 0 ( p ) Jani Lukkarinen

Intro Billiard Waves Summary Wigner function and evolution of particle density 9 Wigner function of ψ � d p � v − 1 � ∗ � � v + 1 � (2 π ) d e i x · p � W [ ψ ]( x , v ) = ψ ψ 2 p 2 p R d Controls density both in spatial and in the Fourier variables: � � � � d v 2 (2 π ) d W [ ψ ]( x , v ) = | ψ ( x ) | 2 , � � � � d x W [ ψ ]( x , v ) = ψ ( v ) � Hence for free evolution, W t ( x , v ) := W [ ψ t ]( x , v ) = W 0 ( x − t v , v ) Free Sch¨ odinger transport ∂ t W t ( x , v ) + v · ∇ x W t ( x , v ) = 0 Jani Lukkarinen

Intro Billiard Waves Summary General wave equations 10 Also more general wave-type evolution equations can be solved via Fourier transform: ψ t ( p ) = e − i t ω ( p ) � � ψ 0 ( p ) where the function ω : R d → R is called the dispersion relation For example, standard wave equation has ω ± ( p ) = ±| p | Large-scale transport can be solved via the Wigner function, using ∂ t W t ( x , k ) + ∇ k ω ( k ) · ∇ x W t ( x , k ) = 0 Boundary conditions still need to be properly implemented Jani Lukkarinen

Intro Billiard Waves Summary Summary 11 f t ( q , p ) := ρ 1; N ( q , p ; t ) f t ( q , p ) := W [ ψ t ]( q , p ) f t controls # of particles in f t controls spatial and a phase space volume Fourier densities Ignore boundary and Ignore boundary conditions particle collisions and interactions ∂ t f t ( q , p ) + p · ∇ q f t ( q , p ) = 0 Jani Lukkarinen

Intro Billiard Waves Summary Dominant corrections ⇒ kinetic theory (rarefied gas) 12 Boltzmann–Grad scaling limit : 1 Particle radius r 0 → 0 2 Assume N (2 r 0 ) 2 → c 0 > 0 ( rarefied gas ) ⇒ N → ∞ 3 Assume that the particle positions and velocities are initially ( t = 0) “independently distributed” , each according to f 0 ( x , v ) Define the collision operator � C b [ h ]( v 0 ) = 2 c 0 ( R d ) 3 d v 1 d v 2 d v 3 δ ( v 0 + v 1 − v 2 − v 3 ) × δ ( | v 0 | 2 + | v 1 | 2 − | v 2 | 2 − | v 3 | 2 ) ( h ( v 2 ) h ( v 3 ) − h ( v 0 ) h ( v 1 )) Theorem (Lanford) Then there is t 0 > 0 such that the r 0 → 0 limit of the “one-particle distribution function” f t satisfies for 0 < t < t 0 ∂ t f t ( x , v ) + v · ∇ x f t ( x , v ) = C b [ f t ( x , · )]( v ) (Scholarpedia / Boltzmann–Grad limit) Jani Lukkarinen

Intro Billiard Waves Summary Dominant corrections ⇒ kinetic theory (waves) 13 Add a non-linear interaction term “+ λ | ψ t ( x ) | 2 ψ t ( x )” to the Schr¨ odinger equation ( nonlinear Schr¨ odinger equation ) 1 Consider a limit of weak interactions, λ → 0 2 Take t λ 2 → τ , x λ 2 → q 3 Consider initial data which have sufficiently fast decay of correlations and for which the initial Wigner function has a limit Define the collision operator � C [ h ]( p 1 ) = 2 d p 2 d p 3 d p 4 δ ( p 1 + p 2 − p 3 − p 4 ) π × δ ( p 2 1 + p 2 2 − p 2 3 − p 2 4 ) ( h 2 h 3 h 4 + h 1 h 3 h 4 − h 1 h 2 h 4 − h 1 h 2 h 3 ) with h j = h ( p j ), j = 1 , 2 , 3 , 4 ∂ τ f τ ( q , p ) + p · ∇ q f τ ( q , p ) = C [ f τ ( q , · )]( p ) (as yet unproven) Jani Lukkarinen