# 9.4 Classical Chaotic Scattering

TLDRThis lecture delves into the applications of differential equations in classical chaotic scattering, drawing parallels with pinball machines to illustrate hypersensitivity to initial conditions. It introduces a model potential for scattering and outlines the theoretical framework based on Newton's laws. The lecture encourages students to solve the resulting differential equations using the Runge-Kutta method, explore phase space, and calculate scattering angles and cross-sections. The practical implications and potential for chaos in scattering experiments are highlighted, with an optional exploration of time delay, offering students a hands-on approach to understanding complex physical phenomena.

###### Takeaways

- π The lectures cover applications of differential equations, focusing on classical chaotic scattering and the quantum eigenvalue problem from Chapter 9 of a referenced book.
- π Classical chaotic scattering involves systems that are hypersensitive to initial conditions, leading to significant changes in outcomes from minor alterations.
- π― The concept is illustrated using the example of a pinball machine, demonstrating the idea of multiple scatterings and the resulting unpredictability.
- π§© The lecture introduces a model using a potential with four 'bumps' to simulate the chaotic scattering, which is likened to bumpers in a pinball game.
- π The potential is represented mathematically as a function of x and y, with a Gaussian shape in each direction, creating peaks for scattering.
- βοΈ The theoretical framework for the model is based on classical dynamics, specifically Newton's law (F=ma), to determine the force and acceleration of a particle.
- π¬ In a scattering experiment, the particle's known initial velocity and the impact parameter are key variables, alongside the observation of the scattering angle at 'infinity'.
- π The differential cross-section (Ο(ΞΈ)) is highlighted as the measurable quantity in scattering experiments, essential for comparing experimental results with theoretical predictions.
- π The script emphasizes the importance of understanding the physics behind the model, including the calculation of the scattering angle as a function of the impact parameter.
- π The use of numerical methods, specifically the fourth-order Runge-Kutta method, is recommended for solving the system of differential equations that describe the system's dynamics.
- π§ The script concludes by encouraging students to explore the topic further, noting its relevance to ongoing research and the potential for deeper study, such as master's and PhD theses.

###### Q & A

### What are the two lectures mentioned in the script about?

-The two lectures are about applications of differential equations, specifically classical chaotic scattering and the quantum eigenvalue problem, both corresponding to chapter 9 from a book.

### What does the term 'classical chaotic scattering' refer to?

-Classical chaotic scattering refers to a system in classical dynamics that is hypersensitive to different parameters or initial conditions, leading to significant changes in outcomes even from minor alterations in input.

### How is the concept of a pinball machine related to the idea of chaotic scattering?

-The pinball machine serves as an analogy for chaotic scattering, illustrating how multiple scatterings within the machine can lead to unpredictable and highly sensitive outcomes based on small changes in the initial conditions, such as the force of the impact or the angle of the shot.

### What is the significance of the potential model used in the script for chaotic scattering?

-The potential model, represented by a two-dimensional function with four bumps, is used to simulate the 'bumpers' in a pinball machine scenario. It is a simplified representation to study the effects of multiple scatterings and the resulting chaotic behavior.

### What is the fundamental difference between studying bound states and scattering in physics?

-Bound states, like the motion of planets, are stable and predictable, whereas scattering involves the study of particles that are not bound and can be influenced by external forces, leading to a variety of outcomes, making it a different kind of experiment.

### What is meant by the term 'differential cross section' in the context of scattering experiments?

-The differential cross section (Ο_theta or Ο_dΞ©) is a measure used in scattering experiments to quantify the number of particles scattered into a specific solid angle at a given angle, normalized to be independent of experimental details such as detector size and incident particle flux.

### How does one determine the scattering angle in a classical scattering experiment?

-The scattering angle can be determined by observing the trajectory of the particle after it has passed the potential region and using trigonometric functions, such as atan2(vy/vx), to account for all four quadrants and avoid division by zero issues.

### What is the significance of the derivative d_theta/db in the context of the script?

-The derivative d_theta/db represents how the scattering angle varies with the impact parameter. Discontinuities or zeros in this derivative can lead to unusual behavior in the cross section, which is indicative of chaotic scattering.

### How does the script suggest using the Runge-Kutta method (RK-4) for solving the problem of chaotic scattering?

-The script suggests using RK-4 for its accuracy and robustness in solving the two simultaneous second-order differential equations that describe the motion in the x and y directions, by converting them into four first-order equations.

### What is the role of the impact parameter 'b' in the scattering experiment described in the script?

-The impact parameter 'b' is a measure of the distance from the center of the potential to the point where the particle would have traveled if it were not deflected. Varying 'b' allows for the exploration of different scattering outcomes and is crucial for observing the effects of chaotic scattering.

###### Outlines

##### π Introduction to Differential Equations Applications

The script begins with an introduction to two lectures on applications of differential equations, both related to chapter 9 of a referenced book. The lectures are not introducing new algorithms but will delve into solving ordinary differential equations within the context of classical chaotic scattering and the quantum eigenvalue problem. The speaker emphasizes the importance of these applications in practical scenarios and encourages study, suggesting that understanding these topics can lead to innovative uses of known tools.

##### π Exploring Classical Chaotic Scattering

This paragraph delves into the concept of classical chaotic scattering, contrasting it with quantum chaotic scattering and emphasizing its accessibility at a classical level despite being a complex research subject quantum mechanically. Chaos in this context is described as a system's extreme sensitivity to initial conditions, leading to dramatically different outcomes from minor changes. The speaker uses the analogy of a pinball machine to illustrate the unpredictable and complex behavior of a system undergoing multiple scatterings, setting the stage for a deeper exploration of the physics behind such phenomena.

##### π Modeling Chaotic Scattering with a Static Potential

The script continues with a discussion on modeling chaotic scattering using a static potential, rather than active bumpers, to simulate the behavior observed in a pinball machine. A specific potential model is introduced, visualized as a landscape with four 'bumps', serving as a simplified representation of the complex interactions within a chaotic scattering scenario. The potential is described mathematically as a Gaussian function in x and y, with additional terms to create peaks, allowing for both scattering and reflection of particles. The paragraph concludes with an introduction to the theoretical framework, which is rooted in classical dynamics and Newton's laws of motion.

##### π Understanding Scattering Experiments and Observations

The fourth paragraph explains the process of a scattering experiment, highlighting the differences between studying bound states and scattering states in physics. The focus is on high-energy scattering where particles do not get trapped but are projected, interact with a potential, and then leave the area. The concept of impact parameter 'b' is introduced as a variable in the experiment, along with the discussion of how scattering is observed at large distances, effectively at 'infinity,' where the potential's influence is negligible. The paragraph concludes with an explanation of how experimental data is collected by measuring the scattering angle and the number of particles scattered at various angles.

##### π Analyzing Data with Differential Cross-Sections

This paragraph discusses the analysis of scattering experiment data, introducing the concept of the differential cross-section (Ο(theta) or Ο(ΞΟ)), which is a measure independent of experimental setup details. The speaker explains the importance of this measure for comparing experimental results across different setups. The paragraph provides a detailed explanation of how the differential cross-section is derived from the raw data of scattered particles and how it is used to normalize the results based on the detector's solid angle, the number of incident particles, and the illuminated target area.

##### π§² Solving Newton's Law for Chaotic Scattering

The script moves into the theoretical aspect of solving the problem of chaotic scattering by applying Newton's law in two dimensions, resulting in a system of two simultaneous second-order differential equations. The equations are derived from the forces obtained by taking the gradient of the potential. The paragraph explains the process of solving these equations using numerical methods, specifically the fourth-order Runge-Kutta method, which is adept at handling such systems. The goal is to find the trajectory of the particle as a function of time, which can then be analyzed to understand the scattering behavior.

##### π¬ Visualizing Chaotic Scattering with an Applet

The sixth paragraph introduces an interactive applet that visually demonstrates the chaotic scattering phenomenon by solving the differential equations in real-time. The applet allows users to manipulate variables such as the potential type, mass, velocity, and impact parameters to observe different scattering outcomes. The speaker discusses the importance of precision in the numerical solution and encourages exploring various impact parameters to witness the sensitive dependence on initial conditions that characterizes chaotic scattering.

##### π Phase Space Analysis and Scattering Angle Determination

This paragraph focuses on the analysis of phase space trajectories, which are expected to show open figures characteristic of scattering, as opposed to the closed orbits of bound systems. The speaker advises on the use of the atan2 function to accurately determine the scattering angle, taking care to avoid division by zero and to correctly identify the quadrant of the scattering angle. The paragraph also touches on the optional exploration of time delay as a function of impact parameter, which can provide further insights into the chaotic nature of the scattering process.

##### π Conclusion and Encouragement for Practical Exploration

The final paragraph concludes the script by encouraging students to engage with the practical aspects of the chaotic scattering problem, using the provided applet and numerical methods to explore the physics involved. The speaker highlights the research potential of the topic, mentioning that several students have been inspired to pursue further studies in this area, even at the graduate level. The paragraph ends with a motivational note to 'make some chaos in the lab' and a reminder to explore both lectures on the subject.

###### Mindmap

###### Keywords

##### π‘Differential Equations

##### π‘Classical Chaotic Scattering

##### π‘Quantum Eigenvalue Problem

##### π‘Multiple Scatterings

##### π‘Potential

##### π‘Impact Parameter

##### π‘Scattering Angle

##### π‘Differential Cross Section

##### π‘Runge-Kutta Method

##### π‘Phase Space

##### π‘Time Delay

###### Highlights

Introduction to two lectures on applications of differential equations from Chapter 9, focusing on classical chaotic scattering and the quantum eigenvalue problem.

Recommendation to study both lectures for their importance in practical applications, despite not introducing new algorithms.

Explanation of classical chaotic scattering, emphasizing its hypersensitivity to initial conditions leading to unpredictable outcomes.

Analogy of classical chaotic scattering to pinball machines, illustrating the concept of multiple scatterings and their chaotic nature.

Discussion on whether classical chaotic scattering can be modeled with a static potential or requires active elements.

Introduction of a model potential resembling the Grand Tetons for scattering, highlighting its role in simulating chaotic scattering.

Differentiation between scattering and bound states in physics, with emphasis on the importance of scattering in understanding microscopic phenomena.

Description of the experimental setup for scattering, detailing the initial conditions and the measurement of scattering angles.

Introduction of the differential cross-section as a key concept for comparing experimental and theoretical results in scattering.

Theoretical approach to calculating scattering angles as a function of impact parameters using classical dynamics.

Use of the Runge-Kutta fourth-order method (RK-4) for solving the system of differential equations in the scattering model.

Visualization of scattering trajectories using an applet, demonstrating the physics of chaotic scattering.

Observation of discontinuities in the scattering angle versus impact parameters, indicative of chaotic behavior.

Instructions for students to use RK-4 to solve the scattering problem with precision, varying impact parameters to observe chaotic effects.

Analysis of phase space for scattering, contrasting open orbits with the closed orbits of bound states.

Method for determining the scattering angle using the arctangent function, avoiding division by zero issues.

Optional exploration of time delay in scattering as a function of impact parameter, offering a measurable experimental comparison.

Encouragement for students to engage with the material, highlighting its relevance to research-level applications.

###### Transcripts

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