Classical Chaotic Scattering Direct ODE Application Rubin H Landau - PowerPoint PPT Presentation

Classical Chaotic Scattering Direct ODE Application Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II 1 / 47

What Does Classical Chaotic Scattering Look Like? Recall Troubled Youth Pinball machines: multiple scattering Classical scattering ⇒ continuous? Enough reflection ⇒ memory loss Model with static potential? Need active bumpers? 2 / 47

What Does Classical Chaotic Scattering Look Like? Recall Troubled Youth Pinball machines: multiple scattering Classical scattering ⇒ continuous? Enough reflection ⇒ memory loss Model with static potential? Need active bumpers? 3 / 47

What Does Classical Chaotic Scattering Look Like? Recall Troubled Youth Pinball machines: multiple scattering Classical scattering ⇒ continuous? Enough reflection ⇒ memory loss Model with static potential? Need active bumpers? 4 / 47

What Does Classical Chaotic Scattering Look Like? Recall Troubled Youth Pinball machines: multiple scattering Classical scattering ⇒ continuous? Enough reflection ⇒ memory loss Model with static potential? Need active bumpers? 5 / 47

What Does Classical Chaotic Scattering Look Like? Recall Troubled Youth Pinball machines: multiple scattering Classical scattering ⇒ continuous? Enough reflection ⇒ memory loss Model with static potential? Need active bumpers? 6 / 47

What Does Classical Chaotic Scattering Look Like? Recall Troubled Youth Pinball machines: multiple scattering Classical scattering ⇒ continuous? Enough reflection ⇒ memory loss Model with static potential? Need active bumpers? 7 / 47

What Does Classical Chaotic Scattering Look Like? Recall Troubled Youth Pinball machines: multiple scattering Classical scattering ⇒ continuous? Enough reflection ⇒ memory loss Model with static potential? Need active bumpers? 8 / 47

Model and Theory V ( x , y ) = ± x 2 y 2 e − ( x 2 + y 2 ) Static 2-D Potential V(x,y) Scatter (high energy) ± : repulsive/attractive y 4 peaks ⇒ internal � v' reflections? Theory : Classical Dynamics v b F = m a x 9 / 47

Model and Theory V ( x , y ) = ± x 2 y 2 e − ( x 2 + y 2 ) Static 2-D Potential V(x,y) Scatter (high energy) ± : repulsive/attractive y 4 peaks ⇒ internal � v' reflections? Theory : Classical Dynamics v b F = m a x 10 / 47

Model and Theory V ( x , y ) = ± x 2 y 2 e − ( x 2 + y 2 ) Static 2-D Potential V(x,y) Scatter (high energy) ± : repulsive/attractive y 4 peaks ⇒ internal � v' reflections? Theory : Classical Dynamics v b F = m a x 11 / 47

Model and Theory V ( x , y ) = ± x 2 y 2 e − ( x 2 + y 2 ) Static 2-D Potential V(x,y) Scatter (high energy) ± : repulsive/attractive y 4 peaks ⇒ internal � v' reflections? Theory : Classical Dynamics v b F = m a x 12 / 47

Model and Theory V ( x , y ) = ± x 2 y 2 e − ( x 2 + y 2 ) Static 2-D Potential V(x,y) Scatter (high energy) ± : repulsive/attractive y 4 peaks ⇒ internal � v' reflections? Theory : Classical Dynamics v b F = m a x 13 / 47

Model and Theory V ( x , y ) = ± x 2 y 2 e − ( x 2 + y 2 ) Static 2-D Potential V(x,y) Scatter (high energy) ± : repulsive/attractive y 4 peaks ⇒ internal � v' reflections? Theory : Classical Dynamics v b F = m a x 14 / 47

Model and Theory V ( x , y ) = ± x 2 y 2 e − ( x 2 + y 2 ) Static 2-D Potential V(x,y) Scatter (high energy) ± : repulsive/attractive y 4 peaks ⇒ internal � v' reflections? Theory : Classical Dynamics v b F = m a x 15 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Particle at + ∞ V(x,y) Know velocity v Vary impact parameter b y � Observe scattering θ v' No target recoil ⇒ no ∆ v Measure N ( θ ) scattered v b x 16 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Particle at + ∞ V(x,y) Know velocity v Vary impact parameter b y � Observe scattering θ v' No target recoil ⇒ no ∆ v Measure N ( θ ) scattered v b x 17 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Particle at + ∞ V(x,y) Know velocity v Vary impact parameter b y � Observe scattering θ v' No target recoil ⇒ no ∆ v Measure N ( θ ) scattered v b x 18 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Particle at + ∞ V(x,y) Know velocity v Vary impact parameter b y � Observe scattering θ v' No target recoil ⇒ no ∆ v Measure N ( θ ) scattered v b x 19 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Particle at + ∞ V(x,y) Know velocity v Vary impact parameter b y � Observe scattering θ v' No target recoil ⇒ no ∆ v Measure N ( θ ) scattered v b x 20 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Particle at + ∞ V(x,y) Know velocity v Vary impact parameter b y � Observe scattering θ v' No target recoil ⇒ no ∆ v Measure N ( θ ) scattered v b x 21 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Particle at + ∞ V(x,y) Know velocity v Vary impact parameter b y � Observe scattering θ v' No target recoil ⇒ no ∆ v Measure N ( θ ) scattered v b x 22 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Projectile at + ∞ Measure N ( θ ) → ⇒ σ ( θ ) Differential cross section σ ( θ ) V(x,y) Independent experiment details y σ ( θ ) = lim N scatt ( θ ) / ∆Ω (1) � N in / ∆ A in v' Compare theory b v b σ ( θ ) = (2) x � sin θ ( b ) � � d θ � db Unusual d θ ( b ) / db 23 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Projectile at + ∞ Measure N ( θ ) → ⇒ σ ( θ ) Differential cross section σ ( θ ) V(x,y) Independent experiment details y σ ( θ ) = lim N scatt ( θ ) / ∆Ω (1) � N in / ∆ A in v' Compare theory b v b σ ( θ ) = (2) x � sin θ ( b ) � � d θ � db Unusual d θ ( b ) / db 24 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Projectile at + ∞ Measure N ( θ ) → ⇒ σ ( θ ) Differential cross section σ ( θ ) V(x,y) Independent experiment details y σ ( θ ) = lim N scatt ( θ ) / ∆Ω (1) � N in / ∆ A in v' Compare theory b v b σ ( θ ) = (2) x � sin θ ( b ) � � d θ � db Unusual d θ ( b ) / db 25 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Projectile at + ∞ Measure N ( θ ) → ⇒ σ ( θ ) Differential cross section σ ( θ ) V(x,y) Independent experiment details y σ ( θ ) = lim N scatt ( θ ) / ∆Ω (1) � N in / ∆ A in v' Compare theory b v b σ ( θ ) = (2) x � sin θ ( b ) � � d θ � db Unusual d θ ( b ) / db 26 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Projectile at + ∞ Measure N ( θ ) → ⇒ σ ( θ ) Differential cross section σ ( θ ) V(x,y) Independent experiment details y σ ( θ ) = lim N scatt ( θ ) / ∆Ω (1) � N in / ∆ A in v' Compare theory b v b σ ( θ ) = (2) x � sin θ ( b ) � � d θ � db Unusual d θ ( b ) / db 27 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Projectile at + ∞ Measure N ( θ ) → ⇒ σ ( θ ) Differential cross section σ ( θ ) V(x,y) Independent experiment details y σ ( θ ) = lim N scatt ( θ ) / ∆Ω (1) � N in / ∆ A in v' Compare theory b v b σ ( θ ) = (2) x � sin θ ( b ) � � d θ � db Unusual d θ ( b ) / db 28 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Projectile at + ∞ Measure N ( θ ) → ⇒ σ ( θ ) Differential cross section σ ( θ ) V(x,y) Independent experiment details y σ ( θ ) = lim N scatt ( θ ) / ∆Ω (1) � N in / ∆ A in v' Compare theory b v b σ ( θ ) = (2) x � sin θ ( b ) � � d θ � db Unusual d θ ( b ) / db 29 / 47

Scattering Experiment: What’s Measured? Project from y = −∞ , Observe Scattered Projectile at + ∞ Measure N ( θ ) → ⇒ σ ( θ ) Differential cross section σ ( θ ) V(x,y) Independent experiment details y σ ( θ ) = lim N scatt ( θ ) / ∆Ω (1) � N in / ∆ A in v' Compare theory b v b σ ( θ ) = (2) x � sin θ ( b ) � � d θ � db Unusual d θ ( b ) / db 30 / 47

Theory: Equations to Solve Newton’s Law in x-y Plane F = m a V(x,y) y j = md 2 x � v' − ∂ V i − ∂ V ˆ ˆ (1) v b x ∂ x ∂ y dt 2 = md 2 x i + md 2 y ∓ 2 xye − ( x 2 + y 2 ) � � y ( 1 − x 2 )ˆ i + x ( 1 − y 2 )ˆ dt 2 ˆ dt 2 ˆ j j (2) Simultaneous 2 nd -order ODEs md 2 x dt 2 = ∓ 2 y 2 x ( 1 − x 2 ) e − ( x 2 + y 2 ) (3) md 2 y dt 2 = ∓ 2 x 2 y ( 1 − y 2 ) e − ( x 2 + y 2 ) (4) 31 / 47