From Colliding Atoms to Colliding Galaxies

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

• T. P. Devereaux
• Students C. M. Palmer, M. Gallamore G.
  McCormack

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Many-Body Physics at Many Length Scales
1026 - 1015 m
102 10-4 m
Universe evolve? Galaxy formation? Cosmic
strings?
Phases of matter? Neural networks?
10-4 10-8 m
1015 - 108 m
Cell dynamics? Protein folding? Magnetic vortices
in superconductors?
Black holes? Star formation? Are orbits stable?
108 - 102 m
10-8 10-16 m
Global warming? Predict weather? Population
biology?
Electron transport? Ultra-cold atoms? Forces
inside the nucleus?
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The Many-Body Problem

• What cannot be explained in terms of
  non-interacting particles
• Collective behavior of many particles (galaxies,
  proteins, metals, etc.).
• Phase transitions (e.g. solid-liquid,
  ferromagnet-paramagnet).
• Structures and conformations (crystals, polymers,
  biopolymers, etc.).
• Instabilities of particles or fields (1D
  Luttinger liquid, black holes, cosmic strings).

Solving for a particles path

• Start out with 1 particle
• Fma or
• -ih??/?t H ?
• - determines particles path.

• Add another particle
• add V(r1-r2)
• - path of particle 1 depends on path of
  particle 2.

• 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
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Simulation N-Body Tree Codes

• Fma for all coupled particles (106).

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

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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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Lets Look at a Specific Problem
1026 - 1015 m
102 10-4 m
Universe evolve? Galaxy formation? Cosmic
strings?
Phases of matter? Neural networks?
10-4 10-8 m
1015 - 108 m
Cell dynamics? Protein folding? Magnetic vortices
in superconductors?
Are orbits stable? Star formation? Black holes?

• How do structures order?
• How are they affected by defects?
• How do they respond to external forces?

What are magnetic vortices in superconductors?
Dynamics of Extended Floppy Objects
108 - 102 m
10-8 10-16 m

• Lipids, proteins
• DNA
• Magnetic vortices

Predict weather? Global warming? Population
biology?
Electron transport? Ultra-cold atoms? Forces
inside the nucleus?
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Real applications of superconductors
Mag-lev
Transmission lines
Biomedical applications

• 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 gt KE kills
  superconductivity.

• SOLUTION Rather than kill superconductivity
  altogether, let magnetic field penetrate in
  isolated places -gt 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
More and more vortices appear as the magnetic
field increases
and the vortices begin to order into a lattice.
J
F
Lorentz force causes vortices to move -gt 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.
Columnar defects
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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 lt-gt vortex avalanches

• STRATEGY
• Use defects to pin, block channel flow.
• Take advantage of repulsion.

• So we must pin all vortices.
• Identified main ingredient blocking channel
  flow.

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A Wall of Defects?
A wall of defects can stop channel flow
but causes too much damage to sample.
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Splaying (tilting) Defects
Vortices stuck on tilted defects.
But vortices have difficulty accommodating to
defects for large angles of splay.

• Stuck vortices block interstitials.
• Channels of flow eliminated.

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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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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 -gt Raman (light) scattering.
• Onset of avalanches.
• Behavior as a qubit (quantum computing).