From Colliding Atoms to Colliding Galaxies – The Complex Dynamics of Interacting Systems

T. P. Devereaux Students: C. M. Palmer, M. Gallamore & G. McCormack PHYSICS 10, 2002 1

Many-Body Physics at Many Length Scales

10 26 - 10 15 m 10 2 – 10 -4 m Universe evolve?

Phases of matter?

Galaxy formation?

Neural networks?

Cosmic strings?

10 -4 – 10 -8 m 10 15 - 10 8 m Cell dynamics?

Black holes?

Protein folding?

Star formation?

Are orbits stable?

Magnetic vortices in superconductors?

10 8 - 10 2 m 10 -8 – 10 -16 m Global warming?

Electron transport?

Predict weather?

Ultra-cold atoms?

Population biology?

Forces inside the nucleus?

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  

The Many-Body Problem

What cannot be explained in terms of non-interacting particles: Solving for a particle’s path Collective behavior of many particles (galaxies, proteins, metals, etc.).

Phase transitions (e.g. solid-liquid, ferromagnet-paramagnet).

Structures and conformations (crystals, polymers, biopolymers, etc.).

  Start out with 1 particle: F=ma or -iħ∂Ψ/∂t = H Ψ - determines particle’s path.

Add another particle: add V(r 1 -r 2 ) - path of particle 1 depends on path of particle 2.

 Instabilities of “particles” or “fields” (1D Luttinger liquid, black holes, cosmic strings).

 Add one more particle… NOT EXACTLY SOLVABLE!

(except in special cases) PROBLEM – How can we approach real systems?

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What is Computational Physics?

Computation v. Experiment v. Theory in Physics  The goal of computational physics is not to replace theory or experiment, but to enhance our understanding of physical processes.

     “Create experiments”.

Visualize physics in action.

Multi-disciplinary.

Cost effective research.

Very accessible.

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Different Computational Approaches

    Enumeration (e.g. Monte Carlo).

Simulation (molecular dynamics).

Algebraic manipulations (Maple, Mathematica).

Solution of approximate equations (dynamical mean field theory).

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  Enumeration – Monte Carlo methods Enumerate all the states of a system and determine their energy.

Evolve towards a ground state.

Used widely in chemistry, materials physics, and biophysics: Example: Simulated Annealing, Lattice Melting Low Temperature High Temperature PHYSICS 10, 2002 6

PHYSICS 10, 2002 Simulation: N-Body Tree Codes F=ma for all coupled particles (~10 6 ).

Widely used in astronomy and condensed matter: Example: Galaxy merger C. Mihos, CWRU ` 7

Approach to Modeling Real Systems

    Work on either exact problems or toy models.

Do “experiments” with basic fundamental ideas.

Determine dynamics – macroscopic behavior reproduced?

Determine essential physics ingredient.

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Let’s Look at a Specific Problem…

10 26 - 10 15 m 10 2 – 10 -4 m Universe evolve?

Galaxy formation?

Cosmic strings?

Phases of matter?

Neural networks?

10 15 - 10 8 m Are orbits stable?

Star formation?

Dynamics of Extended Floppy Objects

10 8 • • Lipids, proteins DNA Global warming?

Magnetic vortices Cell dynamics?

• • 10 -4 – 10 -8 How do structures order?

m How are they affected by defects?

• How do they respond to external forces?

Forces inside the nucleus?

-8 – 10 -16 m PHYSICS 10, 2002 9

Mag-lev

Real applications of superconductors

Transmission lines Biomedical applications PHYSICS 10, 2002

Further applications?

•

peta-flop supercomputer?

•

nanoscale devices?

•

quantum computation?

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Vortices in Superconductors

• Electrons pair to lower their energy when cooled to superconducting state.

• Electrons carry current without resistance and expel magnetic fields.

• Electrons swirl in magnetic field –> KE kills superconductivity.

• SOLUTION: Rather than kill superconductivity altogether, let magnetic field penetrate in isolated places ->

VORTICES (tubes of swirling electrons).

EXTENDED FLOPPY OBJECT (you can choose another if you like)!

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Visualization of Increasing Applied Magnetic Field

B B Now if an external current J is applied…

J

More and more vortices appear as the magnetic field increases…

F

Lorentz force causes vortices to move -> EMF produced and we get resistance!

NO LONGER A SUPERCONDUCTOR!

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Solution: Create defects to pin vortices • Krusin-Elbaum et al (1996).

Vortices lower their energy by sitting on defects.

• Critical current enhanced over “virgin” material.

• Splayed defects better than straight ones.

• Optimal splaying angle ~ 4 degrees.

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Molecular Dynamics Simulations of Vortices

     Vortices = elastic strings under tension.

Vortices repel each other.

Temperature treated as Langevin noise.

Solve equations of motion for each vortex.

Calculate current versus applied Lorentz force, determine critical current.

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Animation: Pinning of Vortices

Different types of pinning: • straight • stretched • collective …would be missed if vortices were treated individually.

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Pinning Principles (fixed field)

At low T, a few pins can stop the whole lattice.

At larger T, pieces of lattice shear away.

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Pinning principles (fixed temperature) For small fields, pinned vortices may trap others.

But “channels” of vortex flow appear at larger fields.

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Depinning <-> vortex avalanches

• So we must pin all vortices.

• Identified main ingredient – blocking channel flow.

STRATEGY • Use defects to pin, block channel flow.

• Take advantage of repulsion.

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A wall of defects can stop channel flow…

A Wall of Defects?

…but causes too much damage to sample.

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Splaying (tilting) Defects

Vortices “stuck” on tilted defects.

• Stuck vortices block interstitials.

• Channels of flow eliminated.

But vortices have difficulty accommodating to defects for large angles of splay.

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Reproducing Experiments

• Two-stage depinning for columnar defects (squares) – channel flow and onset of bulk flow. Splayed defects (circles) eliminate channels of flow.

• Used our simulations to identify main physical ingredient (blocking channel flow) to reproduce experimental behavior.

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PHYSICS 10, 2002

Ending the story…

Computational many body physics is diverse and applicable to many important problems across many fields.

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  

Summary

Many-body problem touches all length scales, many areas of physics.

Computational physics is a powerful and cost effective tool to complement theory/experiment.

Many roads to follow:  Use N-body tree codes to simulate galaxies and larger scale systems.

   Unzipping transitions in DNA; Pathways of protein folding -> Raman (light) scattering.

Onset of avalanches.

Behavior as a qubit (quantum computing).

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