Party is over

First thing first, I want to say that I really should have graduated earlier. The past year unfolded a big scene that I could not be in, and today officially announced was the news equivalent to saying “babies, party is over.”

Goldman Sachs and Morgan Stanley, the last two standing i-banks on the street, will no longer be i-banks on Monday. See the news here. Everything is to be regulated and Uncle Same says no more laissez-faire banking. Maybe it was the repeal of Glass-Stegall that should be blamed, after all.

妈的我还真是适合考试

刚才有人发给我一个据说是今年很难的一道高考数学题：
证明对任意n，存在一个长度为n的非平凡等差数列，其中任意三项不构成等比数列。
我10s就做出来了；仰天长叹一声，我这种人才真是太适合高考了。

Things to do regarding last project

1, for the zero momentum surface - I'd better do the sampling on two leaves separately.
2, for the high density along orbits - I don't think wavepacket is sufficient to explain everything (the 2nd-order approximation is very inaccurate, and base everything on an expansion-contraction balance sounds shaky to me). I think some diffusion argument can be made - so we know how to calculate the relative intensity.
3, for the fraction of regular trajectories - that result should be double-checked. I need to look into the high energy limits, and plot a few fLI vs. time to see if it is logrithmatic or linear. If no concrete results can be found, I will drop it.

I am proud of myself today

This afternoon my college classmate called me on MSN and asked if I had any ideas that can help with the current rescue work in Sichuan. He told me that he had connections so my ideas can be actually passed to people on the frontier. I supplied mine: to use truck tire tubes or air bags and either compressed air cans or even manual pumps to elate them underneath the fallen concretes for gradual lifting. I knew some basic facts about tires and I did ballpark calculations to find this idea actually quite feasible.

Later this evening when I got back on MSN my classmate told me that his connection thought highly of my idea and it was already passed to the people in charge.

I literally burst into tears, for that my ideas might be able to actually save lives. I never felt so proud of myself.

Update on May 19, I am really really happy today:
[01:51:29] 子之魂魄兮，为鬼雄 says:
你的气囊救到了人命
[01:51:41] 子之魂魄兮，为鬼雄 says:
李承光师长已经表扬了
[01:52:01] 子之魂魄兮，为鬼雄 says:
成都前指的反馈
[01:52:14] 子之魂魄兮，为鬼雄 says:
自从传达到了以后部队已经开始用高压气囊
[02:06:20] 子之魂魄兮，为鬼雄 says:
别张扬出去，咱们自己乐就可以了，毕竟出点子的人还是很多的

Possible thesis title?

The dynamics to localization in high-dimensional multi-well systems: a comparative study

Why being patient or why not?

Our subject is a wolf. He is thrown into a world comprised of sites. There are two kinds of sites: easy sites, where a rabbit is coming every T1 hours, and hard sites, where a rabbit is coming every T2 hours, T2>T1. The wolf knows that the density of easy sites is p, 0<p<1. How long should the wolf wait at one site before he decides to move around? Some extreme cases: if T2=infinite, the wolf should wait no longer than T1, and if T2=T1, he should probably just keep waiting. This decision also clearly depends on the value of p. For example, if p is 0, the wolf is better off just waiting. Note that the same strategy doesn't apply to p=1. This can explain why different animals have different behavioral patterns in terms of patience; it is an adaptation to their individual environments.

Some preliminary results show that, after waiting time of T1, the wolf is better off moving if p>T1/(T2-T1), and he should keep waiting otherwise.

Level repulsion, Landau levels, and quantum dot

SAR sent me a paper (Science, 320, p.356, 2008) which told a very intriguing story (if not conceptually new). Basically they can actually see the quantum chaos of Dirac billiard by the Coulomb blockade measurements. A relevant question would be: how to test the quantum chaos in our system?

As described here, we can make a strong analogue of our system, at the vicinity of wells, to swing spring plus a kinetic coupling. It would be interesting to study the swing spring system under magnetic field, and see how the levels quantize. As a started, there have been work done on the Diamagnetic hydrogen... Is it possible to employ similar approaches to actually identify the states in experiments?

A possible way to simplify (in steps) the potential surface

V = V(r,θ,φ), exact, useless
V = V(r) + V(θ,φ) + δV(r,θ,φ), separation + coupling, not very representative
V = V(θ,φ) + ½k[r-r₀(θ,φ)]², closest to reality, note that k, if treated as a variable of (θ,φ), has a variance less than 6%.

A possible approach to the third potential: V = ½k[r-r₀(θ,φ)]² + ε*V(θ,φ). The first part resembles a study of the Monodromy problem (of course it is much more complicated, because now we have 3 DOF, however locally we can approximate r₀(θ,φ)≈r₀(φ), for example, and from there we can further separate the phase space into tiles and continue the study.

Notes on integrators

R-K 4th order: about 4.5 times of computation time of the same step V-V.

However, it has about the same conservation of energy (a gauge of accuracy) as V-V with 1/50 time step. Overall speaking, it's about 10 times more efficient. RK4 for E = 1000 cm^-1, dt = 0.005 t₀ for t=100 ps integration gives ~ 10^-11 eV of total energy shift, for E = 500 cm^-1, it is even better (~3×10^-12 eV). Doubling dt (to 0.01 t₀) leads to ~35 times higher E_shift.

We still ought to check the symplecticity of the integrator, i.e. whether it preserves the two-form:
ω² = ∑ dp_i ^ dq_j. PS: as it turns out, the SIA4 integrator proposed by J. Candy in J. Comput. Phys, 92, 230 (1991) is a better (and symplectic) one than the plain RK4 method.

Quantum control, diffusion

It might be worthwhile to look into H=H₀+μ·E. The driving term cause the diffusion perpendicular to energy surface, and H₀ has diffusion by itself on the energy surface. If D_∥»D_⊥, it can be argued that the quasienergy (Floquet energy) eigenfunctions will be localized exponentially (see D.L. Shepelyansky, Physica D, 28, 103-114 (1987)). Also read Wigner distribution, Arnol'd diffusion.

Flashback: in the XCN isomerization studies (also between-tile hopping in the current study), everything might be broiled down to the comparison between diffusion rates along different directions (in the phase space). If D_∥»D_⊥, then the particle has a good chance to "isomerize" than to "dissociate".

For d>2 degrees of freedom systems, the invariant tori in phase space, being d-dimensional, can not slice the 2d-1 dimensional space of the surface of constant energy. This means for (d>2)-DOF systems the phase space is essentially "ergodic" (Arnol'd diffusion).

swing spring, 1:1:2 resonance, etc

It can be argued that although the radial mode is not exactly double the doubly degenerate peripheral mode frequency, a substantial coupling term (manifested by the strong overtone peak) could make an analog to the 1:1:2 resonance of swing spring valid. It would be interesting to actually observe the pulsation and precession effects described in Lynch's paper.

Regarding the polygon (Sinai) billiard analog - basically we treat the whole motion as motion in 'triangles' that tiled the icosahedron, with a reflection rule at the boundaries. However, since the triangle is actually on the S² surface, special treatment is needed; for example, a rotation around the center of a pentagon should be mapped as a "bouncing" motion between adjacent two boundaries - which is quite ill-defined in the planar case. In fact, how much efforts are warranted to just carry out this simulation is unknown, as we can always do the whole simulation and mapped it back to the sub-region. The question is more interesting, only in analytical terms (since it reflects both the symmetry and/or interconnectivity).

Lastly, a measurement of 1/20 sub-region "revisit" frequency could be highly valuable. Say we have N trajectories originating from inside the boundaries, we want to calculate the number of trajectories that remain in the boundary as a function of time t. To check if this follows a polynomial or exponential decay law can be very enlightening on the stickiness of the phase space.

PS: assuming equal 'diffusion' toward the three adjacent tiles, in the long run, the decay should follow an exponential law of 0.95*exp(-0.255*r*t) + 0.05, in which r is the diffusion rate. (-0.254644 is the three leading exponents other than 0 of the eigenvalues of the diffusion matrix.) Update: preliminary simulations show a similar decay curve (the decay constant drops quickly) to an end value slightly above 0.05 and with a very fluctuating tail, even with 1000 trajectories. Despite being consistent with most of our speculations, the enormous fluctuation around the 0.05 tail is unaccounted for.

A trivial result

This explains why normal mode analysis always gives doubly-degenerate results at the bottom of the wells. Freshman level proof, but the results are pretty unexpected.

A possible way to investigate the effects of symmetry breaking

With the five-fold symmetry about z-axis intact, the Poincaré surface of section of (φ=0, p_φ>0) will look identical to (φ=2nπ/5, p_φ>0), n=1, 2, 3, 4, ... but won't if the symmetry is broken. What happens if there is a strong localization? One may guess the density on the “right” surface of section is going to be much denser, but again the measure of the section points is always zero; one needs to think about a way to go about that.

Another issue: on generating initial conditions evenly on the surface of identical energy: if we first pick up r then v, the number of v's given r should be proportional to E⁰ in 2d and E^1/2 in 3d.

Bogged

The singularity of the map S²→R² makes any straightforward simulation impossible. A possible turnaround: rotate the axis every time θ goes close to 0 or π.

Another idea: what if the surface is a, say, tetrahedron? Keep in mind the relationship between the kicked rotator and tight binding. Both manifests the lack of a continuous symmetry, but both are solvable.

SAR suggests investigating the dynamics with a perturbed potential surface. Stöckmann points out the kinship between the Anderson (spatial) localization and dynamical localization. Heller's paper provides a good ground for studying the semiclassical aspects.

It is likely that the localization effect is absent in the classical limit (strong diffusion). If S¹ can be taken as an analog of the kicked rotator (loosely), what about S²?

This can relate the current project to both previous projects. Also valuable would be to re-evaluate the results with a different separable potential profile (can be simply ½k(r-r₂)² + V(r=r₀,θ,φ)) against the Morse potential.

Edit:

A possible approach:
Consider the Hamiltonian of the form:
H_ε = H₀ + ε*V(θ,φ)
H₀ is integrable, then Kolmogorov–Arnold–Moser theorem can be applied.

Binary choice experiment and so on

So I made 4 bets on NVDA in just 7 days: 3 shorts & 1 long. Fortunately I got all of them worked out with positive returns (one as small as $20), but in retrospect, I blame myself for an extreme irrationality when making the choices. Let me explain.

I recently read about this from a book (Kuenne, Readings in Applied Microeconomic Theory). It is not really anything “economics”, but rather some psychology experiment helping to understand why people do not always maximize their expected returns or even minimax the losses.

The subject is given two choices A, B in each round and then asked to pick one. If he picks the right one, he gets a reward, and if he picks the wrong one, he gets nothing. The experimenter sets A or B to be correct on a completely random basis, for example, for 2/3 of the times A is the correct answer and for the rest 1/3 B is. Interestingly enough, the subject can soon realize this distribution, but he would not always bet on A, which is the optimal strategy that rewards 2/3 of the time. Instead, he bets on A 2/3 of the times and on B 1/3 of the times, randomly. This strategy rewards 5/9 of the time, therefore less optimal than the aforementioned one.

So why? Actually nobody knows for sure why subjects would choose to be both random and pattern-following. It is plausible that men have a tendency to be competitive in situations, against “nature” in this experiment. Men want to “outguess” his opponents, and in this case, he assumes the nature is playing against him. It is like poker: if you know your opponent is bluffing 2/3 of the times, you don't want to call always, because that would change his strategy - so you probably want to exploit this to the full and consequently randomize your responses.

Now you see why the hidden rationale behind my bets is so irrational. I predicted the market goes down 75% of time and up 25% of time. So I made 3 bets of it going down and 1 bet of it going up. Exactly the same mistake made by the subjects in the experiment. I was trying to outguess the market (who doesn't even care what I have guessed).

I can not help linking this to the philosophical ideas of Lao-Zi. “In harmony with the natural laws, not against them.” I also think now the news in which SAC traders were forced to take female hormones to enhance their performance, makes perfect sense. Spirit of competition (or in my favorite Nietzsche's words, The Will to Power) is probably innate to the male aggression. And it is a good thing in general - just that we have to put a check to it so it doesn't act foolish.

N-well tunneling

Basically t~2Zγ=ZE_tun. General to arbitrary dimensions or symmetries (rotational/translational doesn't matter). See my write-up.

Recommending a book

It is not “thoughtful” and it is no fiction. I go practical here. It is for anybody currently in school but in desperate need of financial awareness. Take my word: I only regret that I did not read it earlier.

It is All You Need To Know About The City by Christopher Stoakes (City means London).

It is completely readable and friendly to neophytes of all levels (if the ground state is degenerate at all...). Most importantly it is loaded with information essential to a coherent understanding of the industry.

Questions to be answered

1, On tunneling vs. randomness, how is our argument compared to Thouless’ in his famous scaling law paper (relating g to sensitivity of levels to boundary conditions)?

2, Does a study on just the ground state suffice a claim of localization, which in general deals with transport theories (more on band structure)?

3, Is our system essentially an analogue of the 2d crystalline (equivalent of Bloch wave in translational symmetry in rotational symmetry such as Y_lm?)? What to make out of the established conductivity results?

4, Possible to map the randomness to the breathing modes (an analogue of electro-phonon interaction)?

5, Check if wave packets actually follow the localized-delocalized scheme (fourth moment, exponential decay etc)? Above all, how to define it? We have limited number of “sites”. How to differentiate between the “extended” and the “localized”?

6, Explore the relationship with Hall effect and magnetoresistance?

“Crap,” says M, “I hate the most exciting part of writing the conclusion.”
“Because it is going to take forever to close it.”