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

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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

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SLIDE 2

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?

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SLIDE 3

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?

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SLIDE 4

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?

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SLIDE 5

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?

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SLIDE 6

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?

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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?

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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?

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SLIDE 9

Model and Theory

Static 2-D Potential V(x, y) = ± x2y2e−(x2+y2)

v V(x,y) x y b v'

Scatter (high energy)

±: repulsive/attractive 4 peaks ⇒ internal reflections? Theory: Classical Dynamics F = ma

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SLIDE 10

Model and Theory

Static 2-D Potential V(x, y) = ± x2y2e−(x2+y2)

v V(x,y) x y b v'

Scatter (high energy)

±: repulsive/attractive 4 peaks ⇒ internal reflections? Theory: Classical Dynamics F = ma

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SLIDE 11

Model and Theory

Static 2-D Potential V(x, y) = ± x2y2e−(x2+y2)

v V(x,y) x y b v'

Scatter (high energy)

±: repulsive/attractive 4 peaks ⇒ internal reflections? Theory: Classical Dynamics F = ma

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SLIDE 12

Model and Theory

Static 2-D Potential V(x, y) = ± x2y2e−(x2+y2)

v V(x,y) x y b v'

Scatter (high energy)

±: repulsive/attractive 4 peaks ⇒ internal reflections? Theory: Classical Dynamics F = ma

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SLIDE 13

Model and Theory

Static 2-D Potential V(x, y) = ± x2y2e−(x2+y2)

v V(x,y) x y b v'

Scatter (high energy)

±: repulsive/attractive 4 peaks ⇒ internal reflections? Theory: Classical Dynamics F = ma

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SLIDE 14

Model and Theory

Static 2-D Potential V(x, y) = ± x2y2e−(x2+y2)

v V(x,y) x y b v'

Scatter (high energy)

±: repulsive/attractive 4 peaks ⇒ internal reflections? Theory: Classical Dynamics F = ma

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SLIDE 15

Model and Theory

Static 2-D Potential V(x, y) = ± x2y2e−(x2+y2)

v V(x,y) x y b v'

Scatter (high energy)

±: repulsive/attractive 4 peaks ⇒ internal reflections? Theory: Classical Dynamics F = ma

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SLIDE 16

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Particle at +∞

v V(x,y) x y b v'

Know velocity v

Vary impact parameter b Observe scattering θ No target recoil ⇒ no ∆v Measure N(θ) scattered

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SLIDE 17

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Particle at +∞

v V(x,y) x y b v'

Know velocity v

Vary impact parameter b Observe scattering θ No target recoil ⇒ no ∆v Measure N(θ) scattered

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SLIDE 18

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Particle at +∞

v V(x,y) x y b v'

Know velocity v

Vary impact parameter b Observe scattering θ No target recoil ⇒ no ∆v Measure N(θ) scattered

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SLIDE 19

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Particle at +∞

v V(x,y) x y b v'

Know velocity v

Vary impact parameter b Observe scattering θ No target recoil ⇒ no ∆v Measure N(θ) scattered

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SLIDE 20

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Particle at +∞

v V(x,y) x y b v'

Know velocity v

Vary impact parameter b Observe scattering θ No target recoil ⇒ no ∆v Measure N(θ) scattered

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SLIDE 21

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Particle at +∞

v V(x,y) x y b v'

Know velocity v

Vary impact parameter b Observe scattering θ No target recoil ⇒ no ∆v Measure N(θ) scattered

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SLIDE 22

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Particle at +∞

v V(x,y) x y b v'

Know velocity v

Vary impact parameter b Observe scattering θ No target recoil ⇒ no ∆v Measure N(θ) scattered

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SLIDE 23

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Projectile at +∞

v V(x,y) x y b v'

Measure N(θ) → ⇒ σ(θ)

Differential cross section σ(θ) Independent experiment details

σ(θ) = lim Nscatt(θ)/∆Ω Nin/∆Ain (1)

Compare theory

σ(θ) = b

dθ

db

sin θ(b)

(2)

Unusual dθ(b)/db

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SLIDE 24

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Projectile at +∞

v V(x,y) x y b v'

Measure N(θ) → ⇒ σ(θ)

Differential cross section σ(θ) Independent experiment details

σ(θ) = lim Nscatt(θ)/∆Ω Nin/∆Ain (1)

Compare theory

σ(θ) = b

dθ

db

sin θ(b)

(2)

Unusual dθ(b)/db

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SLIDE 25

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Projectile at +∞

v V(x,y) x y b v'

Measure N(θ) → ⇒ σ(θ)

Differential cross section σ(θ) Independent experiment details

σ(θ) = lim Nscatt(θ)/∆Ω Nin/∆Ain (1)

Compare theory

σ(θ) = b

dθ

db

sin θ(b)

(2)

Unusual dθ(b)/db

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SLIDE 26

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Projectile at +∞

v V(x,y) x y b v'

Measure N(θ) → ⇒ σ(θ)

Differential cross section σ(θ) Independent experiment details

σ(θ) = lim Nscatt(θ)/∆Ω Nin/∆Ain (1)

Compare theory

σ(θ) = b

dθ

db

sin θ(b)

(2)

Unusual dθ(b)/db

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SLIDE 27

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Projectile at +∞

v V(x,y) x y b v'

Measure N(θ) → ⇒ σ(θ)

Differential cross section σ(θ) Independent experiment details

σ(θ) = lim Nscatt(θ)/∆Ω Nin/∆Ain (1)

Compare theory

σ(θ) = b

dθ

db

sin θ(b)

(2)

Unusual dθ(b)/db

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SLIDE 28

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Projectile at +∞

v V(x,y) x y b v'

Measure N(θ) → ⇒ σ(θ)

Differential cross section σ(θ) Independent experiment details

σ(θ) = lim Nscatt(θ)/∆Ω Nin/∆Ain (1)

Compare theory

σ(θ) = b

dθ

db

sin θ(b)

(2)

Unusual dθ(b)/db

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SLIDE 29

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Projectile at +∞

v V(x,y) x y b v'

Measure N(θ) → ⇒ σ(θ)

Differential cross section σ(θ) Independent experiment details

σ(θ) = lim Nscatt(θ)/∆Ω Nin/∆Ain (1)

Compare theory

σ(θ) = b

dθ

db

sin θ(b)

(2)

Unusual dθ(b)/db

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SLIDE 30

Scattering Experiment: What’s Measured?

Project from y = −∞, Observe Scattered Projectile at +∞

v V(x,y) x y b v'

Measure N(θ) → ⇒ σ(θ)

Differential cross section σ(θ) Independent experiment details

σ(θ) = lim Nscatt(θ)/∆Ω Nin/∆Ain (1)

Compare theory

σ(θ) = b

dθ

db

sin θ(b)

(2)

Unusual dθ(b)/db

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SLIDE 31

Theory: Equations to Solve

Newton’s Law in x-y Plane

F = ma

v V(x,y) x y b v'

− ∂V

∂x ˆ i − ∂V ∂y ˆ j = md2x dt2 (1) ∓ 2xye−(x2+y2) y(1 − x2)ˆ i + x(1 − y2)ˆ j

= md2x

dt2 ˆ i + md2y dt2 ˆ j (2)

Simultaneous 2nd-order ODEs md2x dt2 = ∓ 2y2x(1 − x2)e−(x2+y2) (3) md2y dt2 = ∓ 2x2y(1 − y2)e−(x2+y2) (4)

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SLIDE 32

Theory: Equations to Solve

Newton’s Law in x-y Plane

F = ma

v V(x,y) x y b v'

− ∂V

∂x ˆ i − ∂V ∂y ˆ j = md2x dt2 (1) ∓ 2xye−(x2+y2) y(1 − x2)ˆ i + x(1 − y2)ˆ j

= md2x

dt2 ˆ i + md2y dt2 ˆ j (2)

Simultaneous 2nd-order ODEs md2x dt2 = ∓ 2y2x(1 − x2)e−(x2+y2) (3) md2y dt2 = ∓ 2x2y(1 − y2)e−(x2+y2) (4)

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SLIDE 33

Theory: Equations to Solve

Newton’s Law in x-y Plane

F = ma

v V(x,y) x y b v'

− ∂V

∂x ˆ i − ∂V ∂y ˆ j = md2x dt2 (1) ∓ 2xye−(x2+y2) y(1 − x2)ˆ i + x(1 − y2)ˆ j

= md2x

dt2 ˆ i + md2y dt2 ˆ j (2)

Simultaneous 2nd-order ODEs md2x dt2 = ∓ 2y2x(1 − x2)e−(x2+y2) (3) md2y dt2 = ∓ 2x2y(1 − y2)e−(x2+y2) (4)

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SLIDE 34

Theory: Equations to Solve

Newton’s Law in x-y Plane

F = ma

v V(x,y) x y b v'

− ∂V

∂x ˆ i − ∂V ∂y ˆ j = md2x dt2 (1) ∓ 2xye−(x2+y2) y(1 − x2)ˆ i + x(1 − y2)ˆ j

= md2x

dt2 ˆ i + md2y dt2 ˆ j (2)

Simultaneous 2nd-order ODEs md2x dt2 = ∓ 2y2x(1 − x2)e−(x2+y2) (3) md2y dt2 = ∓ 2x2y(1 − y2)e−(x2+y2) (4)

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SLIDE 35

Theory: Equations to Solve

Newton’s Law in x-y Plane

F = ma

v V(x,y) x y b v'

− ∂V

∂x ˆ i − ∂V ∂y ˆ j = md2x dt2 (1) ∓ 2xye−(x2+y2) y(1 − x2)ˆ i + x(1 − y2)ˆ j

= md2x

dt2 ˆ i + md2y dt2 ˆ j (2)

Simultaneous 2nd-order ODEs md2x dt2 = ∓ 2y2x(1 − x2)e−(x2+y2) (3) md2y dt2 = ∓ 2x2y(1 − y2)e−(x2+y2) (4)

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Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 37

Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 38

Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 39

Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 41

Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 42

Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 43

Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 44

Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 45

Implementation

Simultaneous x(t), y(t) rk4: Simultaneous 1st O ⇒ Trajectory [x(t), y(t)] ⇒ 4 1st O ODEs: dy(t) dt = f(t, y), (1) y(0) def = x(t), y(1) def = y(t), (2) y(2) def = dx dt , y(3) def = dy dt (3) Force Function f (0) = y(2), f (1) = y(3), (4) f (2) = ∓2y2x(1 − x2)e−(x2+y2)/m (5) f (3) = ∓2x2y(1 − y2)e−(x2+y2)/m (6)

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SLIDE 46

Assessment

Apply rk4 With 4-D Force Function

1

Initial Conditions: vx = 0, x = b (impact param)

2

m = 0.5, vy(0) = 0.5, −1 ≤ b ≤ 1

3

Plot trajectories [x(t), y(t)]

4

Look for ∼backward, multiple scattering

5

Phase space trajectories [x(t), ˙ x(t)], [y(t), ˙ y(t)]

6

How scattering, bound problem differ?

7

Determine θ = atan2(Vx,Vy) (y = +∞)

8

When discontinuous dθ/db?

9

Run attractive & repulsive potentials

10 Run range E [Vmax = exp(−2)] 11 Time Delay* T(b) 46 / 47

SLIDE 47

Assessment

Apply rk4 With 4-D Force Function

1

Initial Conditions: vx = 0, x = b (impact param)

2

m = 0.5, vy(0) = 0.5, −1 ≤ b ≤ 1

3

Plot trajectories [x(t), y(t)]

4

Look for ∼backward, multiple scattering

5

Phase space trajectories [x(t), ˙ x(t)], [y(t), ˙ y(t)]

6

How scattering, bound problem differ?

7

Determine θ = atan2(Vx,Vy) (y = +∞)

8

When discontinuous dθ/db?

9

Run attractive & repulsive potentials

10 Run range E [Vmax = exp(−2)] 11 Time Delay* T(b) 47 / 47