optimization in statistical physics

OPTIMIZATION IN STATISTICAL PHYSICSKwan-Yuet HoInstitute for Physical Science and Technology & Department of PhysicsUniversity of Maryland9/27/2012

PhD (Physics), University of MarylandThesis: Properties of Metallic HelimagnetsField: Condensed Matter Physics, Statistical Physics

BSc (Physics & Math), Chinese University of Hong KongThesis: Quantum Entanglement of Continuous SystemsField: Quantum Physics, Mathematical Physics

Other projects: Two-dimensional Bose gas

(Condensed Matter Physics, Statistical Physics, Atomic Physics, Quantum Physics)

Ultra-high Energy Cosmic Rays(Particle Astrophysics)

PHYSICS

Classical Mechanics

Relativistic Mechanics

Quantum Mechanics(Bose-Einstein condensate)

PHYSICS

String Theory (Calabi-Yau space)

Superfluid

Liquid Crystals

PHYSICS Classical vs

Quantum

Deterministic vs probabilistic

Continuous vs Discrete

I think it is safe to say that no one understands Quantum Mechanics. –Richard Feynman

PHYSICS Microscopic vs Macroscopic N << or ~ 1023

Few-body vs Many-body

PHYSICS

macroscopicmicroscopic

classical

quantum

NewtonianBlack hole

Plasma

Liquid crystalsHelimagnets

SuperconductorSuperfluid

Bose-Einstein condensate

SemiconductorAtomsParticle physicsQuantum bits (qubits)

Kinetic theory

A CRASH COURSE OF STATISTICAL PHYSICS Statistical Physics/Mechanics: the study of a

system containing many (N~1023) particles, using probability theory and statistics

Fixed N, V and T, probability of a state m is given by

€

pm =Ce−βEm

€

β =1kBT

€

p1 + p2 +K + pm +K = Ce−βEm

m∑ =1

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C = e−βEm

m∑ ⎛

⎝ ⎜

⎞

⎠ ⎟−1

=1Z

€

Z = e−βEm

m∑ Partition Function

Information such as T, <E> and other measurable quantities

Normalization constant

A CRASH COURSE OF STATISTICAL PHYSICS

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Z = e−βF = e−βEm

m∑€

Z = e−βEm

m∑

€

Z = dd x⋅ e−βE x( )∫

Helmholtz free energy F

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Z = e−βF = dd x⋅ e−βE x( )∫

We model the free energy F, a summary of all the information about the system!

Appropriate F is the minimized E(x) one with respect to m or x, to get the expected measured value of m and x.

Method of steepest descent / mean field theory / equation of motion


Phase diagram for water


Stability matters!Fluctuations (standard deviation or variance)


Fluctuations and stability are studied by perturbation:

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x = x* +δxPutting it back to the free energy, and studying its variance.


€

Z = e−βEm

m∑

€

Z = dd x⋅ e−βE x( )∫

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Z = DM x( )⋅ exp −β dd x∫ ⋅H M x( )[ ]( )∫

True kind of problem I am dealing with.

Perturbation does not only lead to variance but also correlations, <M(x) M(x’)>.

Hamiltonian functional

HELIMAGNETS Leonard: What would you be if you were

attached to another object by an inclined plane, wrapped helically around an axis?

Sheldon: Screwed.

HELIMAGNETS Helimagnets, or helical magnets, are

magnets with magnetic dipoles aligned helically.

Good for computer memories because of its non-volatility.

HELIMAGNETS Landu-Ginzburg-Wilson (LGW) functional

Minimizing H: mean-field theory (for SFM and SDM)

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SFM M[ ] = d3xr2M2 +

a2∇M( )2 +

u4M2( )

2−H⋅M

⎡ ⎣ ⎢

⎤ ⎦ ⎥∫

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SDM M[ ] =c2

d3x M⋅ ∇ ×M( )∫

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Scf M[ ] = d3xb2∂iM i[ ]

2+b1

2∂iM i+1[ ]

2+v4M i

4 ⎡ ⎣ ⎢

⎤ ⎦ ⎥

i=1

3

∑∫€

H M[ ] = SFM M[ ] + SDM M[ ] + Scf M[ ]

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rM − a∇2M + c∇ ×M + uM2M −H = 0

HELIMAGNETS Optimize the solution for M to minimize the

energy Phase diagram But the full solution is very difficult to

find!!!!!! We identify the solutions from measurement

(ansatz).

HELIMAGNETS Ansatz 1 - Something like bar magnets:

M=Mz Equation: rM+uM3-H=0 Numerically find all the “optimized” solutions,

and reject those that are invalid and that does not give the minimum energy.

HELIMAGNETSEquation to solve

Finding minimum energy

Finding optimized M

HELIMAGNETS Ansatz 2 - Helical phase:

M=m0 (cos(qz)x+sin(qz)y) Ansatz 3 - Conical phase:

M=m0 (cos(qz)x+sin(qz)y)+mlz

Solve for m0 and ml. Closed forms available.

HELIMAGNETS Ansatz 4 - Perpendicular helix: tested,

and found not to be valid. No closed form available. Numerically solve three equations, discarding

invalid data, discarding data with larger energies.

Equations to solve

HELIMAGNETS

Finding minimum energy

Finding optimized M

HELIMAGNETS Ansatz 5 - Hexagonal columnar

structure Several choices of guess solutions Minimizing energy to obtain the parameters.

HELIMAGNETS In my program, all the postulated solutions

are considered. The code decides the solution by checking

which has the lowest energy.

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HELIMAGNETS

(Thessieu et al 1997)

MnSi

(Ishimoto et al 1995)

Fe0.8Co0.2Si

(Ho, Kirkpatrick, Belitz, 2011)

VEHICULAR TRAFFIC FLOW Traffic is a big problem in Washington DC.

VEHICULAR TRAFFIC FLOW

time

VEHICULAR TRAFFIC FLOW Nagel-

Schreckenberg (NaSch) rule

Step 1: Accelerationvn -> min(vn+1, vmax)

Step 2: Brakingvn -> min(vn, gn)

Step 3: Randomizationvn -> max(vn-1,0) with a probability p

Next node(s)

VEHICULAR TRAFFIC FLOW 2-lane highway Lane-switching rule:

At cell i, find the distance of the next barrier (a car, a red traffic light, the end of a road) ahead for both lanes 0 (d0) and 1 (d1).

On lane 0, switch to lane 1 if d1>d0, and vice versa.

VEHICULAR TRAFFIC FLOW

CONCLUSION Statistical physics is the study of many-body

physics using probability theory and statistics.

The phase of the matter is the minimized energy of the system. Finding the phase is an optimization problem.

The stability of the system depends on its variance and correlation.

The flow of traffic can be verified by microscopic simulation by implementing linked list.

ACKNOWLEDGMENTS Theodore Kirkpatrick (University of Maryland) Dietrich Belitz (University of Oregon) Bei-lok Hu (University of Maryland) Esteban Calzetta (Universidad de Buenos

Aires) Yan Sang (University of Oregon) Chi Kwong Law (Chinese University of Hong

Kong) Lin Tian (University of California, Merced) Robert McKweon (Jefferson Lab)