Extreme Values of Infinite-Measure Processes

Abstract

We study the statistics of the maximum and minimum of a set of $N$ random variables whose dynamical and statistical properties fall within the scope of infinite ergodic theory. These non-stationary yet recurrent systems are described, in the long-time limit, by a non-normalizable infinite invariant density. Extreme events in such systems emerge in a joint limit where the observation time $t$ is long and the number of variables $N$ is large. We show that the resulting extreme value statistics are controlled by the return exponent $α$ and the infinite invariant measure, and therefore depart from the classical Fréchet, Gumbel, and Weibull universality classes. We illustrate the theory for weakly chaotic intermittent maps, overdamped diffusion in an asymptotically flat potential, and a stochastic model of sub-recoil laser cooling, and show how measurements of extremes can be used to infer the infinite-density structure.

Figures (13)

• Figure 1: Convergence of the rescaled single-particle PDF to an infinite invariant density for a particle in an asymptotically flat Lennard-Jones potential in Eq. (\ref{['eq:LJ_tail']}) with the parameters in Eq. (\ref{['eq:parameterspot']}) aghion2020infinite. As $t$ increases, the distribution converges to the infinite-density scaling form $\mathcal{I}(x)$ given in Eq. (\ref{['eq:boltexample']}), which is the Boltzmann state in a non normalized form.
• Figure 2: Numerical test of the prediction in Eq. (\ref{['eq:boltzmannQN']}) for the minimum CDF $Q_N^{min}(m,t)$ for overdamped diffusion in an asymptotically flat potential of Lennard-Jones type in Eq. (\ref{['eq:LJ_tail']}). Shown is $-\ln[1-Q_N^{min}(m,t)]/\rho$ versus $m$ for several effective densities $\rho=N/\sqrt{\pi Dt}$ obtained from Langevin simulations. In the limit $N,t\to\infty$ at fixed $\rho$, all curves collapse onto the integrated infinite density $\mathcal{J}(\cdot)$ (black dashed line), where $\mathcal{I}(\cdot)$ is the non-normalizable Boltzmann weight given in Eq. (\ref{['eq:boltexample']}). Here we used $D=1$ and $k_BT=1$, with the parameters in Eq. (\ref{['eq:parameterspot']}).
• Figure 3: Probability density $q_N^{min}(m,t)$ of the minimum position for $N$ random variables, shown for several $N$ (with $t$ chosen such that $\rho=2$ is fixed). The symbols, represent how data obtained from Langevin simulations, approach the limiting extreme-value form (black dashed line). The model uses an Lennard-Jones potential in Eq. (\ref{['eq:LJ_tail']}). We see that observing very small $m$ is very unlikely due to diverging character of the potential field at $x\rightarrow 0$. We use the same parameters as in Fig. \ref{['fig:boltzmanncollapse']}.
• Figure 4: Comparison of the limiting distribution $q_N^{min}(m,t)\to \rho\,\mathcal{I}(m)\exp[-\rho \mathcal{J}(m)]$ (solid lines) for different temperatures for $N=100,t=100$ for the overdamped Langevin model. The figure shows that at lower temperatures, the PDF of the minimum position has a peak close to the minimum point of the potential. The distribution becomes progressively flatter as $T$ increases, approaching the free-diffusion form in the high-temperature limit. The dashed lines indicate the high-temperature limit approximation in Eq. (\ref{['eq:hightwithc']}). Agreement with this approximation improves as $T$ increases, while deviations persist at small $m$ due to the divergent repulsive core of $V(x)$.
• Figure 5: Rescaled density $t^{1-\alpha}p(x,t)$ of the Thaler map with $z=3$, evaluated at $t=10^2,10^3,10^4$ compared to the infinite invariant density $\mathcal{I}(x)$ (black dashed curve) given in Eq. (\ref{['eq:thaler_I']}). Solid curves are obtained from a numerical iteration of the Frobenius–Perron operator (see Appendix \ref{['appendix:FP']} for details). Dots show direct Monte Carlo simulations. See korabel2013numerical for further details.