Optimization is an important topic in physics, operations research and computer science. In the current era of big data, text analytics and data mining involve a lot of machine learning algorithms, which is essentially optimization algorithms. This slideshare shows how optimization is being used in statistical physics. This sheds a light of how ideas in statistical physics can be applied in computer science.
2. PhD (Physics), University of Maryland
Thesis: Properties of Metallic Helimagnets
Field: Condensed Matter Physics, Statistical Physics
BSc (Physics & Math), Chinese University of Hong
Kong
Thesis: Quantum Entanglement of Continuous Systems
Field: 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)
5. 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
8. 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
Partition Function
Information such as T, <E>
and other measurable
quantities
Normalization
constant
9. A CRASH COURSE OF STATISTICAL PHYSICS
Helmholtz free energy F
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
10. A CRASH COURSE OF STATISTICAL PHYSICS
Phase diagram for water
11. A CRASH COURSE OF STATISTICAL PHYSICS
Stability matters!Fluctuations (standard deviation or variance)
12. A CRASH COURSE OF STATISTICAL PHYSICS
Fluctuations and stability are studied by perturbation:
Putting it back to the free energy, and studying its variance.
14. A CRASH COURSE OF STATISTICAL PHYSICS
True kind of problem I am dealing with.
Perturbation does not only lead to variance but also
correlations, <M(x) M(x’)>.
Hamiltonian functional
15. HELIMAGNETS
Leonard: What would you be if you were attached
to another object by an inclined plane, wrapped
helically around an axis?
Sheldon: Screwed.
16. HELIMAGNETS
Helimagnets, or helical magnets, are magnets with
magnetic dipoles aligned helically.
Good for computer memories because of its non-
volatility.
18. 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).
19. 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.
21. 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.
22. 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
29. VEHICULAR TRAFFIC FLOW
Nagel-Schreckenberg
(NaSch) rule
Step 1: Acceleration
vn -> min(vn+1, vmax)
Step 2: Braking
vn -> min(vn, gn)
Step 3: Randomization
vn -> max(vn-1,0) with a
probability p
Next node(s)
30. 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.
32. 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.
33. 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)