You are not a Boltzmann brain

According to the “Boltzmann brain” hypothesis, we popped into existence as a thermal fluctuation in an otherwise chaotic universe, with our brains replete with spurious memories of a fictitious, orderly past. The hypothesis extends less ambitious argumentation by Ludwig Boltzmann in the late 19th century, but it lacks the physical foundation of Boltzmann’s original arguments. We are assured of neither the recurrence nor the reversibility of the time developments of the applicable physics. The Boltzmann brain scenario is much more likely to produce a physically spurious “batty brain” whose memories fail to conform to the scientifically well-behaved regularities of our brains

Quantum carpets in a leaky box: Poincaré's recurrences in the continuous spectrum

The freedom to define branch cuts of the complex function is used to derive an integral representation of the quantum carpet, thus producing a generalization of the Poincaré recurrence theorem in the case of the continuous spectrum. This approach provides a different way to renormalize resonant states to be both space and time convergent. The coherence of quantum carpets was related to the properties of the Wigner function in the canonical time-frequency phase space. It has been shown that the distortion of the Wigner function shape is directly responsible for the lack of the ability of the dynamics to produce revivals equally as sharp as the initial wave packet

Fractal dimension of a random invariant set

AbstractIn recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations