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
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Many-Body Physics at Many Length
Scales
10 - 10 m
10 – 10 m
26
15
2
Universe evolve?
Phases of matter?
Galaxy formation?
Neural networks?
-4
Cosmic strings?
10-4 – 10-8 m
1015 - 108 m
Black holes?
Star formation?
Are orbits stable?
108 - 102 m
Cell dynamics?
Protein folding?
Magnetic vortices in
superconductors?
Global warming?
Electron transport?
Predict weather?
Ultra-cold atoms?
Population biology?
Forces inside the nucleus?
PHYSICS 10, 2002
10-8 – 10-16 m
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The Many-Body Problem
What cannot be explained in
terms of non-interacting
particles:
Solving for a particle’s path
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Collective behavior of many particles
(galaxies, proteins, metals, etc.).
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Phase transitions (e.g. solid-liquid,
ferromagnet-paramagnet).
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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(r1-r2)
- path of particle 1 depends
on path of particle 2.
Instabilities of “particles” or “fields”

Add one more particle…
(1D Luttinger liquid, black holes,
NOT EXACTLY SOLVABLE! (except in special cases)
cosmic strings).
PROBLEM – How can we approach real systems?
PHYSICS 10, 2002
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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.
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PHYSICS 10, 2002
“Create experiments”.
Visualize physics in action.
Multi-disciplinary.
Cost effective research.
Very accessible.
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Different Computational
Approaches
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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

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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
PHYSICS 10, 2002
High Temperature
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Simulation: N-Body
Tree Codes
F=ma for all
coupled particles
(~106).
Widely used in astronomy and
condensed matter:
Example: Galaxy merger
C. Mihos, CWRU
`
PHYSICS 10, 2002
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Approach to Modeling Real Systems
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Work on either exact problems or toy
models.
Do “experiments” with basic fundamental
ideas.
Determine dynamics – macroscopic
behavior reproduced?
Determine essential physics ingredient.
PHYSICS 10, 2002
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Let’s 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?
Are orbits stable?
Protein folding?
• How do structures order?
Star formation?
in
Whatholes?
are magnetic Magnetic
vortices vortices
in superconductors?
Black
• How are they affected by
superconductors?
Dynamics of Extended Floppy Objects
8 - 102 m
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• Lipids,
proteins
Predict weather?
• DNA
Global warming?
Population biology?
• Magnetic vortices
PHYSICS 10, 2002
defects?
10-8 – 10-16 m
How do they respond to external
Electron•transport?
forces?
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?
PHYSICS 10, 2002
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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)!
PHYSICS 10, 2002
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Visualization of Increasing Applied
Magnetic Field B
Now if an external
current
B
J is applied…
More and more
…and
the vortices
vortices
appear
begin
to magnetic
“order”
as the
into
a lattice.
field
increases…
J
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.
PHYSICS 10, 2002
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Molecular Dynamics Simulations of
Vortices
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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
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…would be
missed if
vortices were
treated
individually.
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Pinning Principles (fixed field)
At low T, a few pinsColumnar
can stop the
defects
whole lattice.
PHYSICS 10, 2002
At larger T, pieces of lattice
shear away.
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Pinning principles (fixed temperature)
For small fields, pinned vortices
may trap others.
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But “channels” of vortex flow
appear at larger fields.
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Depinning <-> vortex avalanches
• So we must
pin all vortices.
STRATEGY
• Identified
• Use defects to
pin, block channel
flow.
– blocking
channel flow.
• Take advantage
of repulsion.
main ingredient
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A Wall of Defects?
A wall of
defects can
stop channel
flow…
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…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.
PHYSICS 10, 2002
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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Ending the story…
Computational
many body
physics is diverse
and applicable to
many important
problems across
many fields.
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Summary
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Many-body problem touches all length scales,
many areas of physics.
Computational physics is a powerful and costeffective tool to complement theory/experiment.
Many roads to follow:
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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).
PHYSICS 10, 2002
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