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31

Gravity can, of course, become turbulent if it is coupled to a turbulent fluid. The interesting question is thus, as John Rennie points out, whether a vacuum solution can be "turbulent". As far as I'm aware this is not known. If turbulence does occur in vacuum gravity, it is remarkably hard to stir up. Even in very extreme situations like colliding black ...


22

None of the interesting equations in physics can be derived from simpler principles, because if they could they wouldn't give any new information. That is, those simpler principles would already fully describe the system. Any new equation, whether it's the Navier-Stokes equations, Einstein's equations, the Schrodinger equation, or whatever, must be ...


19

They are derivable from classical mechanics using either the continuum or molecular points of view. Starting with a continuum view, one applies conservation of mass, momentum, and energy to a control volume and the result is the Navier Stokes equations. The Navier Stokes equations, in the usual form, apply to Newtonian fluids, that is fluids whose stress ...


15

Thanks to holography, we now know that solutions to the Einstein equation in certain $d+1$ dimensional spaces are equivalent (dual) to solutions of the Navier-Stokes equation in $d$ dimensions. This is the fluid-gravity correspondence. As a result, turbulence can be studied using the Einstein equations, see, for example, http://arxiv.org/abs/1307.7267.


13

Not all nonlinear systems are chaotic. However a chaotic system is necessarily nonlinear. There doesn't exists a definition for chaos but using the one given by Strogatz, ref 1: Chaos is aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Like explained in the text: aperiodic long-termed ...


12

I once asked Putterman after a similar colloquium what he meant by this statement, and his answer was "long time tails". Long time tails are fractional powers that appear in the long time behavior of correlation functions, see, for example, here and here. These fractional powers are seen in molecular dynamics (they are more difficult to see experimentally), ...


9

The interplay of Hamiltonian and Lagrangian theory is based on the following general identities, where $L$ is the Lagrangian function of the system, $$\dot{q}^k = \frac{\partial H}{\partial p_k}\:,\qquad(1)$$ $$\frac{\partial L}{\partial q^k} = -\frac{\partial H}{\partial q^k}\:.\qquad(2)$$ Above, the RH sides are functions of $t,q,p$ whereas the LH sides ...


8

Numerical simulations are not always meaningful, as chaos theory belongs to the large subject of dynamical systems theory. Although the definitions differ, chaos generally occurs in three contexts: Sensitive dependence on initial conditions (SDIC). The set is topologically transitive. Periodic points are dense in the set. Think of two particles having ...


7

If you want to generalize a potential to a class that's broader than the simple $\frac12 k_2 x^2$, it is tempting as a first step to include a small perturbation of the form $\frac13k_3x^3$. Unfortunately, this drastically changes the structure of the potential, because it becomes unbounded from below. Thus, you might get a slightly perturbed behaviour ...


7

There are indeed vacuum solutions to the Einstein field equations which are unstable under perturbations. A famous example is the result of Gregory and Laflamme for black strings, which essentially have the geometry of $\mathrm{Sch}_d \times \mathbb{R}$. For example, a five dimensional black string could have a metric, $$ds^2 = \left( ...


6

Nonlinear optical elements are called nonlinear precisely because of the behaviour you note: because the optical response of the material does not depend linearly on the driving fields. The response may then have a quadratic or higher dependence on the driver, which is usually written in the form $$ \mathbf P =\varepsilon_0 \chi^{(1)} \mathbf E + ...


6

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. ...


5

@Ron Maimon has given the canonical answer to this: the wavefunction is probabilities, and to preserve probabilities one must have a linear equation (indeed, also a norm-preserving evolution operator). I offer another viewpoint, in the style of how Einstein thought about relativity, i.e. two postulates. The postulate is that it is not possible to solve ...


5

Three different points of views on essentially the same thing: Chaotic systems are not only sensitive to numerical errors, but also to any other small perturbations, such as dynamical noise, which may simulate real conditions. Though tiny perturbations affect the detailled, microscopic future of a system, its qualitative dynamics is unaffected. And the ...


5

The fluid-gravity correspondence that Thomas referred to in his answer is a very concrete set-up where we can import intuition from fluid dynamics to suggest how we might get turbulence in vacuum GR (with negative cosmological constant). I thought it deserved more explanation. First, fluid dynamics is a universal description, applicable in any system (e.g. ...


5

(1) In general, what is meant by non-linear system in classical mechanics? A linear system is described by a set of differential equations that are a linear combination of the dependent variable and its derivatives. Some examples of linear systems in classical mechanics: A damped harmonic oscillator, $$m \frac{d^2 x(t)}{dt^2} + c \frac{d x(t)}{dt} + k ...


5

The essential idea of a Poincaré map is to boil down the way you represent a dynamical system. For this, the system has to have certain properties, namely to return to some region in it’s state space from time to time. This is fulfilled if the dynamics is periodic, but it also works with chaotic dynamics. To give a simple example, instead of analysing the ...


4

A linear system is one whose dynamics obeys linear differential equations, in contrast with those that are non-linear whose dynamics obeys non-linear differential equations. So if the dyanmics of the variable $x(t)$ obeys a a differential equation $$f\left(x(t),\frac{d}{dt}x(t),\dots,\frac{d^n}{dt^n}x(t),t\right)=0,$$ if $x_1(t)$ and $x_2(t)$ are differente ...


4

We are considering a discrete time evolution $$x_{n}~=~f(x_{n-1})~=~f^{n\circ}(x_0), \qquad n~\in~\mathbb{N},$$ in a $2N$-dimensional symplectic manifold $(M,\omega)$, where $f$ is a symplectomorphism. Let us for simplicity work in local coordinates. Define the Jacobian matrix as $$\tag{1} A(x,n)^{i}{}_{j}~:=~\frac{\partial (f^{n\circ} (x))^i}{\partial ...


4

At some point, whatever quantities you're talking about in "experimental sciences" refer directly or indirectly to something measurable with a precision limited to a certain number of siginificant digits. Anything smaller than the quantities in the frame of reference of the discussion by more orders of magnitude than this number of significant digits may be ...


4

It is an interesting problem. Usually you would find the infinitesimal oscillations by setting $p_i = b + \delta p_i$, $q_i = a + \delta q_i$ and expanding the Hamiltonian to second order in the $\delta p_i$, $\delta q_i$. Here though, this doesn't work as you just get the same Hamiltonian, $$ H = \sum_{i,j} M_{ij} \sqrt{(\delta p_i - \delta p_j)^2 + (\delta ...


4

Since the rate of change of $ x $ is the same as the rate of change of $y $ you really only a single equation of with one variable. We write, \begin{equation} x = y + c \end{equation} where the constant $ c $ is determined by your initial conditions, \begin{equation} c = x (0) - y (0) \end{equation} (in your case it is the difference between the ...


4

Let’s consider this incarnation of the FitzHugh–Nagumo model: $$\begin{matrix} \dot{V} & = & V-V^3/3 - W + I \\ \dot{W} & = & 0.08(V+0.7 - 0.8W) \end{matrix}$$ This describes an excitable system. In the following I will explain what this means, how this relates to the above equation and use a small patch of forest as an example. A ...


4

1D Burger's equation is not meant to model a physical phenomenon. Rather, it is a simplification of homogeneous incompressible Navier-Stokes equations that preserves (some of) its mathematical structure: the non-linear convection term and the second order derivative of viscous forces. It was initially intended as a useful simplification to try to ...


4

The obvious example of a chaotic solution to the Einstein equations is the Mixmaster metric. However this is not a vacuum solution, and when matter is present it shouldn't be any surprise that it can evolve in a chaotic fashion. The more interesting question is whether a vacuum solution can evolve in a chaotic fashion. I can offer only a vague recollection ...


3

Will there be a symbolic sequence for each dimension or will a symbol be assigned to a point $(x,y)$? This depends what you eventually want to do with your symbol sequence, but for typical applications, such as determining the entropy or modelling, you want to assign one symbol to the point. The general reason behind this is that (for a proper ...


3

One of the simplest measurements you can do is to characterize the spatial mode coming out of the laser. This can be done with a simple photodiode and a scanning slit (optical chopper or razor blade mounted on a movable stage). I wrote a brief note on making such a measurement here. This isn't the most interesting of measurements if you aren't going to ...


3

The other answers address how to solve this analytically, but I like numerical solutions to things so here goes: $$\frac{d}{dt} \begin{bmatrix}x \\ y \end{bmatrix} = -\begin{bmatrix}kxy\\kxy\end{bmatrix}$$ which can be solved using any number of numerical methods. For simplicity, we can take the second-order Runge-Kutta method where $i$ is the time index. ...


3

Hints: Conclude that $y-x=c$ is a constant. Use separation of variables $-k\int \!\mathrm{d}t= \int \!\frac{\mathrm{d}x}{x(x+c)}$.



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