Hot answers tagged complex-systems
5
Yes. The same system can - at least in many cases - be described by either a stochastic process with memory or by a Markov process. The point is that in order to write it as a Markov process, one must add enough variables encoding the memory. For example, an autoregressive moving average (ARMA) process is defined as a process with memory, but each such ...
4
It doesn't look that much like a normal distribution to me - particularly on the x axis, the right-hand tail looks heavier than the left, whereas the left one is much longer.
But, generally speaking, normal distributions tend to arise when lots of small, independently distributed random numbers (of any distribution) are added together. (The theorem that ...
4
An harmonic oscillator.
When evolving with time, its joint distribution in (p,x) is given by the Boltzman distribution: $e^{-H(p,x)}$, but the energy along a trajectory is constant. Nevertheless if write explicitly the hamiltonian you will find that
$e^{-H} = e^{-p^2/2 - x^2/2}$
and although the energy is constant the individual distributions of $x$ and ...
3
What you are seeing are artifacts caused by the hard-wall type kernel you are using. These patterns consist of horizontal and vertical streaks with a definite periodicity, which gets more pronounced as you make the averaging box bigger. The streaks occur because the Fourier transform of a hard-wall box has zeros at certain wavenumbers. To get rid of them, ...
3
No, I would say it is wrong to immediately conclude that there is no scale just because the variance diverges. Only functions of $x$ of the form $x^n$, a power law, have a chance to be considered scale-free; none of these functions may be considered a probability distribution because the integral diverges. Any other function – and therefore any normalizable ...
3
The critical point of a general statistical system is a point in the space parameterized by intensive quantities, especially temperature and pressure, at which there exist no boundaries between two different phases of the material even though the boundaries exist at an infinitesimally nearby point. It's the end of a co-existence curve for two phases.
...
3
It a widely known and experimentally useful fact in nuclear and particle physics that the position and momentum distributions of bound systems are related to one another by a Fourier transform.
Is the system you are inspecting bound?
The tails in the data that Nathaniel notes suggest that it is not fully bound, which means the Fourier relationship between ...
2
If x(t) is a random process it is quite unlikely that the derivative xdot(t) exists. So your description looks somewhat problematic.
It seems that you have a Wiener process (= random walk, Brownian motion).
See http://en.wikipedia.org/wiki/Wiener_process
Here the changes in x are Gaussian and uncorrelated with x itself.
Then x itself also follows a ...
2
In general, it can be hard to tell if a given set of equations of motion (eom) are part of a (possibly larger) set of eom that can be put on Hamiltonian (or on Lagrangian) form.
Specifically, OP asks about the Kuramoto model with eom
$$\tag{1} \dot{\theta}_j -\omega_j
~=~\frac{K}{N}\sum_{k=1}^N\sin(\theta_k-\theta_j)
~\equiv~ K ~{\rm Im} \left( ...
2
As Michael Brown asked for an elaboration, here's a short introduction into the geometry of Hamiltonian mechanics:
While the traditional formulation works with special sets of coordinates (the 'canonical' ones), the differential-geometric approach de-emphasizes these and instead introduces the symplectic form as the characteristic structure of Hamiltonian ...
1
I like the book Energy Landscapes by David Wales. It deals with various classes of complex systems (clusters, glasses, proteins) in the context of chemistry.
I want to add - emergence is fraught with flaky ideas; a lot of appeals to ignorance are rooted from the idea of irreducible complexity. So because we can't, say, predict the weather from $F=ma$, this ...
1
Maybe these three lectures about emergence could be interesting to get a first overview of the topic. Therein Prof. De Deo explains for example that emergence has a lot to to with what new phenomena can occurre when coarse graining (or renormalizing) microscopic degrees of freedom of a large system to obtain an effective (possibly including emergent ...
1
No, or at least not in the sense the phrase "butterfly effect" is normally used. Well, possibly, but only if the critical system is chaotic.
The phrase is normally applied to systems that show chaotic behaviour. In such systems the trajectory of the system is very sensitive to the starting point i.e. if you take two points very close together in phase space ...
1
My immediate reaction is that the condensation of water vapour to form water droplets is unlikely to be a critical phenomenon, but a quick Google suggests that various authors have claimed critical behaviour in clouds. However from my cursory glance at the Google results it isn't clear if they are talking about the condensation process or some other ...
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 ...
1
First of all, I don't think the usual terminology is invariant, but rather homogeneous (see Homogeneous function). In particular, we are usually only talking about functions $f$ that, for a fixed $d$, satisfy
$$
f(ax)=a^df(x),
$$
and such functions are said to be homogeneous of degree $d$.
Now, on to the actual question. It seems as if you want to ...
1
I'm still not entirely sure I understand what you're asking, so I'm not sure I'm answering the right question, but consider this: the property of scale invariance is something that applies to functions in general, not just distributions. In essence, scale invariant functions are those which do not require a specific unit for their argument. I've written a ...
1
There is absolutely no relation between the length/time scale relation and the complexity of the phenomenon. The graph you are looking for has a log-axis for L and for T, and a black region for $L>T/c$ which is the speed-of-light bound on the allowed time scales for change in a system of size L. You can make complexity happen in our universe in the ...
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