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I'm not a physicist, my training is in math and CS. If anything in this question is ill defined or doesn't make sense, say so in the comments and Ill try to fix it.

Suppose I have a discrete dynamical system $x_{t+1}=f(x_t)$ (I don't think discreteness should matter too much. If I'm wrong and the answer for continuous systems is very different do let me know). I'm uncertain about the value of the initial value $x_0$, so I model it as a random variable $X_0$ with an appropriate distribution.

Question 1: My understanding is that it is standard to assume that $X_0$ is uniform on some ball. e.g. if my scalar valued measurement is $x_0=10\pm1$ then I'm implicitly assuming $X_0\sim Uni(9,11)$. Are there any notable exceptions?

Next, I'd like to know what my uncertainty should be over $X_t$ (which is again a random variable because I'm uncertain about it's precise value. This is obviously a property of $f$. It may be wider than the my original uncertainty (e.g. if $f$ is chaotic it grows, at least initially, exponentially) or narrower (e.g. a system which I know converges to a fixed point for any initial value).

Question 2: If the system is Hamiltonian, then phase-space volume is conserved under time evolution, so does this mean that my uncertainty (the STD of $X_t$s distribution) necessarily remains constant?

Finally, $f$ itself may be stochastic. So we have $x_{t+1}=f(x_t, \xi_t)$ where $\xi_t$ is some stochastic driving of the system. In a simple case where the dynamics are a Gaussian random walk ($x_{t+1}=x_t+\xi_t$ and $\xi_{t}=\mathcal{N}\left(0,\sigma^{2}\right)$ are independent Gaussians), the uncertainty grows like $\sigma\sqrt{t}$.

Question 3: Are there any other common functions that describe growth of uncertainty in stochastic systems besides the square root? Specifically, are there cases in which uncertainty grows linearly or polynomially with $t$?

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2 Answers 2

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Yes, being a discrete or continuous system makes no big difference here - except when solving it numerically, where there'll be the additional error from the numerical integration method.

Semoi already answers well Q1, namely that in physics errors are most often considered to be Gaussian, not step functions.

Q2: No, the uncertainty doesn't necessarily remain constant. In a chaotic 2D Hamiltonian system, your ball of initial conditions (or uncertainty) will typically be "spaghettified", i.e., exponentially stretched in one dimension and compressed in the other (not fixed directions, but, in general, along the stable and unstable manifolds of the chaotic attractor). The uncertainty therefore grows exponentially.

Here it's important to remember that in dynamical systems we're usually concerned with orbits that remain in a finite region of the phase space (a notable exception being scattering processes), a feature which, together with its defining exponential divergence, leads to the paradigmatic horseshoe mechanism. And which also means that the uncertainty can't grow unboundedly - but in physics an uncertainty as large as the space of possibilities might often be considered effectively "infinite".

Q3: Yes, there are. Due to ballistic modes, ratchet effects, KAM barriers or other sources of stickiness, delays or other types of memory effects, among other possibilities, the uncertainty growth can strongly deviate from the diffusive or normal ($\sqrt{t}$) behavior. This anomalous behaviors are generically called subdiffusive when slower and superdiffusive when faster than normal diffusion.

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  • $\begingroup$ Thank you, especially for Q3, superdiffusive behavior was precisely what I was looking for. Re. Q2 - If I understand correctly, in a chaotic Hamiltonian system phase space volume is conserved, but in a way that makes keeping track of it difficult? $\endgroup$ Feb 21, 2021 at 17:06
  • $\begingroup$ @H.Rappeport Yes, exponentially difficult. If the uncertainty in the $x_1$ direction is $\Delta x_1 \to \infty$, the total uncertainty $\Delta \vec{x}$ also diverges, even if $\Delta x_2 \to 0$. We could consider the dynamics along the unstable dimension as asymptotically described by a map such as the Bernoulli map - in that it stretches geometrically without diverging. $\endgroup$
    – stafusa
    Feb 21, 2021 at 21:26
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Q1: If $x_0=10$ is your measured value, $\epsilon = \pm1$ is the resolution of the measurement device, and you do not take into account any other error contribution, then you are right in assuming $\epsilon \sim \rm{Unif}(-1,1)$. However, if you have many error contributions of approx. equal size, and $\epsilon = \pm1$ is the resulting uncertainty (sum of all errors), then $\epsilon$ will be normally distributed. Putting this into the context of Bayesian data analysis: You should choose the prior error distribution according to your current state of knowledge. There is not single distribution which is best for all problems -- no free lunch, available.

Q2: Use the transformation law of random variables to calculate how the uncertainty in X transforms into the uncertainty of Y $$ Y = f(X) \Rightarrow dY = \left|\frac{\partial f}{\partial X}\right| dX $$ Using this formula and assuming that the derivative is not constant, I don't see why the uncertainty of Y should be constant even if the uncertainty of X is constant. Of course, here $Y$ is a single random variable and not a phase space volume.

Q3: Your model assumes that the future value $x_{t+1}$ depends only on the present value $x_t$ and a random impulse. There exists a branch of statistic called time series analysis, which includes many different terms (correlation & moving averages) and discusses the outcome.

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