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Optimization in Statistical Physics
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Transcript of 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
€
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
€
Z = e−βF = e−βEm
m∑€
Z = e−βEm
m∑
€
Z = dd x⋅ e−βE x( )∫
Helmholtz free energy F
€
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
A CRASH COURSE OF STATISTICAL PHYSICS
Phase diagram for water
A CRASH COURSE OF STATISTICAL PHYSICS
Stability matters!Fluctuations (standard deviation or variance)
A CRASH COURSE OF STATISTICAL PHYSICS
Fluctuations and stability are studied by perturbation:
€
x = x* +δxPutting it back to the free energy, and studying its variance.
A CRASH COURSE OF STATISTICAL PHYSICS
A CRASH COURSE OF STATISTICAL PHYSICS
€
Z = e−βEm
m∑
€
Z = dd x⋅ e−βE x( )∫
€
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)
€
SFM M[ ] = d3xr2M2 +
a2∇M( )2 +
u4M2( )
2−H⋅M
⎡ ⎣ ⎢
⎤ ⎦ ⎥∫
€
SDM M[ ] =c2
d3x M⋅ ∇ ×M( )∫
€
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[ ]
€
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.
26
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)