# Tag Info

8

These kinds of proportionality questions are often best answered with dimensional analysis. You want to know a form a quantity with the units of time in terms of what you have. You have a quantity $k$ with units $\frac{\text{Energy}}{\text{Distance}^3} = \frac{\text{Mass}}{\text{Distance} \times \text{Time}^2}$. You also have the mass $m$ (units of Mass) ...

6

David Bar Moshe's answer is fine, but I wanted to go into more detail. The main reason that random matrices show up in dynamical systems is because they describe the level statistics of classically chaotic motions. In classically integrable systems, there is a semiclassical formula for the level-spacing, determined by the Bohr-Sommerfeld rule. If you know ...

6

The basic idea is that statistical properties of complex physical systems fall into a small number of universal classes. A very known example of this phenomenon is the universal law implied by the central limit theorem where the sum of a large number of random variables belonging to a large class of distrubutions converges to the normal distribution. Please ...

4

In general, no. It is possible to recognise a (system of) differential equation(s) as being nonlinear purely by inspection, but there are plenty of non-chaotic nonlinear systems. Chaos is a stronger (and, unfortunately, not well-defined) condition. Also, many (most?) systems, including the famed Lorenz attractor, only exhibit chaos under certain conditions. ...

3

zkf gives you enough to answer this question but I would like to make a few extra points: The absolute value operation in the potential makes this a nonlinear problem, which are generally pretty difficult to deal with. I got impatient waiting for Mathematica to come up with a closed form solution for $x(t)$, so there probably isn't one. This is the typical ...

3

The dimensional analysis in zkf's answer completely solves the exercise. Nevertheless, it is possible to give a closed formula for the period $$T~=~ 4 ~\sqrt{\frac{m}{2k}} \int_0^a\! \frac{dx}{\sqrt{a^3-x^3}} ~\stackrel{x=au}{=}~ 4 ~\sqrt{\frac{m}{2ka}} \int_0^1\! \frac{du}{\sqrt{1-u^3}}.$$ Can you see why? Unsurprisingly, this just confirms zkf's ...

3

No, it is not possible, and the argument is simple--- there is no dimensional parameter with unit of length, so if there were a stable equilibrium at one radius, there would be many such equilibria obtained by rescaling the original solution to a one-parameter family of solutions. In fact, it is easier to see that the stable solution is for the electron to ...

3

Although,the set of solutions of the nonlinear Schrodinger equation (NLS) is not a Hilbert space and the field $\psi$ cannot be interpreted as a wave function, this does not mean that the NLS cannot be quantized. It can if we interpret $\psi$ as a classical field. In this case the space of solutions or equivalently, the space of initial conditions ...

3

This is not a question about physics. As has been stressed numerous times here, solutions of the NLS cannot be interpreted as quantum mechanical wave-functions. Their evolution is not unitary. As a consequence, the solution space has much less physical relevance. The cubic NLS you wrote down appears in various approximations to nonlinear dispersive waves ...

3

1) Some of the assumptions of the Gross-Pitaevskii equation (GPE) are: all atoms are in the same condensate wave function, the condensate is at $T=0$, collisions between atoms are sufficiently low energy that the interactions can be well described by the $s$-wave scattering length, so that the interaction can be written ...

3

A stepping mortor or magnetic field lines in a Tokamak when two magnetic islands overlap are simple examples of chaotic behaviour (really different from turbulent) that can be easily controlled. http://www.epj.org/_pdf/HP_EPJB_slowly_rocking.pdf http://www-student.elec.qmul.ac.uk/people/josh/documents/ReissAlinSandlerRobert-ICIT2002.pdf Lyshevski S., ...

2

Yes, there is investigation. Some random names on the field (more on the physics side, NO specific order): Carl Dettmann, Tamás Tél, Ott, Ying-Cheng Lai, Adilson Motter, Celso Grebogi, Holger Kantz, Alessandro Moura, Eduardo G. Altmann, etc, etc, etc. A quick search on some of these names should help you to find some recent papers on what is being done ...

2

I) The question formulation (v4) leaves out some important implicit assumptions$^1$ of Theorem 5.2 in Ref.1. These are, among other things, the following four items. The word topologically equivalent should be replaced by locally topologically equivalent, i.e. in some local neighborhood. The (vector-valued) functions $g(w)$ and $h(w)$ are $o(w)$ for $w\to ... 2 In this PhD thesis (unfortunately in German :-/, a follow up paper can be found here), it is shown how for an action of a field theory containing even and Grassman fields, the renormalization group equation can be solved numerically (after expanding the action in derivatives and the fields). To investigate the corresponding renormalization group flow in the ... 2 Suppose you have a small ball in a periodic potential. The period of the potential is much larger than the ball size. (You can see it as a bead in a wash-board). This system is in a water (or any viscous liquid). Then minima (maxima) correspond to nodes (saddles).If you tilt the wash-board, then at angles larger than some threshold, there are no minima and ... 1 If you are talking about small displacements, where$\sin\theta\approx\theta$is a reasonable approximation for the angle from horizontal, then I suggest looking at the linear model. Start with the one-dimensional case, which is a string suspended between two points, uniform load$p$force/length, and tension$T$. Then a small piece$\Delta x$long has ... 1 I strongly suggest Steven H. Strogatz' book Nonlinear Dynamics and Chaos, which has several simple examples of all basic bifurcations in a few different fields. An interesting one in biology is that of fireflies synchronizing their light-flashing. As a 1D physical example (for which the term saddle-node bifurcation seems a bit odd because saddles and nodes ... 1 Since time evolution is linear for a quantum system, this rules out classical chaos in terms of hypersensivity to initial conditions. The question of how quantum theory could explain classical chaos is adressed from the perspective of decoherence. This interesting document may be of some help: http://www.iqc.ca/publications/tutorials/chaos.pdf 1 If the dissipative system has a thermodynamic equilibrium state, then in general, the set of microscopic initial conditions is larger than the set of microscopic states in the thermodynamic equilibrium state. Imagine a melting ice with a final state of water at 10°C. The initial state (some microscopic configuration corresponding to ice, or in general a set ... 1 If Mathematica refuses to solve this, it is likely that no exact solution exists. You might look for approximate solutions then, maybe expanding in power series (even though that wouldn't allow you to keep track of the delta term). Or you could generalize whatyou do in simple one-dimension potential scattering. WKB method could help, too. What have you ... 1 A nonlinear field or a nonlinear theory is, well, a field or a theory that is not linear. There are two obstructions to something being linear: a equation is said to be linear if, whenever$\phi$and$\psi$are solutions to the equation and$a,b$are constant scalars, so is$a\phi + b\psi$. So for the definition to make sense, you need (a) a way to ¨add two ... 1 The argument they are using is the uniqueness of the stationary distribution of a Markov chain. The master-equation description produces a Markov chain approximation of the dynamics, and then if you find a stationary solution, it must be the unique global attractor. If you have a Markov process which takes independent steps, you can think of it as a random ... 1 OK, a little bit like zephyr's answer, I started with white noise, took the FFT, and then scaled each frequency component up or down according to the square-root of the power at that frequency, then did inverse-FFT to get the time-domain signal. An equivalent approach would have been to generate the FFT directly by giving each component the appropriate ... 1 You might be interested in in Pulse-coupled oscillators. See for example Mirrollo&Strogatz,1990: They investigate a set of identical oscillators each described by a single phase variable$\phi_i \in [0,1]$with$\dot{\phi_i}=1$. When$\phi_i = 1\$ the oscillator resets to zero and sends out a spike which causes an instantaneous phase jump in all other ...

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