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I have some real measurements from a process and I happened to look at the mutual distribution of (x(t), xdot(t)). I found that they seem to follow 2d normal distribution around (mu, 0). See image, the y-axis is xdot.
Is there some well known process which exhibit this property? What kind of dynamic equations on x(t) cause this? |
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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 the two distributions will be only approximate. To the extend that one of the distributions is Gaussian, you would expect the other to be Gaussian as well, as this is a property of the Gaussian under a Fourier transform. |
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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 $p=\dot x$ are gaussians. Including masses, etc... you can get different width for the gaussians. |
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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 shows this is called the "central limit theorem".) So if your process has something that keeps more or less randomly perturbing x and xdot then all those perturbations will be added together, and this is the sort of result you should expect to see. |
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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 Gaussian distribution (but xdot exists only as a distribution). More generally, you might have an Ornstein–Uhlenbeck process. See http://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_process The small deviations from normality might mean that your process is actually slightly nonlinear. |
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An equation for $x(t)$ of the form $$\ddot{x} = -\kappa (x-\mu) - \dot{x} + \sqrt{2T} \eta(t)$$ where $\eta(t)$ is a zero-mean, unit-variance, Gaussian white noise, i.e., $$\langle \eta(t) \rangle = 0; \qquad \langle \eta(t) \eta(t') \rangle = \delta(t-t')$$ will generate normal distributions for both $x$ and $\dot{x}$ with $$\langle x(t) \rangle = \mu; \qquad \langle x^2(t) \rangle - \mu^2 = \frac{T}{\kappa}$$ and $$\langle \dot{x}(t) \rangle = 0; \qquad \langle \dot{x}^2(t) \rangle = T$$. From fitting the data you can extract $\mu$, $\kappa$ and $T$. This is nothing but the Langevin equation for a particle in a harmonic trap. |
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