Hydrodynamically Inspired Pilot-Wave Theory: An Ensemble Interpretation

This chapter explores a deterministic hydrodynamically-inspired ensemble interpretation for free relativistic particles, following the original pilot wave theory conceptualized by de Broglie in 1924 and recent advances in hydrodynamic quantum analogs. We couple a one-dimensional periodically forced Klein-Gordon wave equation and a relativistic particle equation of motion, and simulate an ensemble of multiple uncorrelated particle trajectories. The simulations reveal a chaotic particle dynamic behavior, highly sensitive to the initial random condition. Although particles in the simulated ensemble seem to fill out the entire spatiotemporal domain, we find coherent spatiotemporal structures in which particles are less likely to cross. These structures are characterized by de Broglie's wavelength and the relativistic modulation frequency kc. Markedly, the probability density function of the particle ensemble correlates to the square of the absolute wave field, solved here analytically, suggesting a classical deterministic interpretation of de Broglie's matter waves and Born's rule

A generalized theory of Brownian particle diffusion in shear flows

This study presents a generalized theory for the dynamics and diffusion of Brownian particles in shear flows. By solving the Langevin equations using stochastic instead of classical calculus, we propose a new mathematical formulation that resolves the particle MSD at all time scales for any two-dimensional shear flow described by a polynomial velocity profile. The theory is validated for the cases of Couette and Plane-Poiseuille flows while neglecting the effect of boundaries on particle diffusion. We show that for times much longer than the relaxation time scale, $\tau_p$, the time dependency of the MSD is $t^{n+2}$, where $n$ is the polynomial order of the transverse velocity profile. The coupling between Brownian motion in the transverse direction and the flow velocity gradients significantly alters the particle diffusion in the streamwise direction. Finally, we generalize the theory to resolve any polynomial shear flow at all time scales, including the order of $\tau_p$, which is not resolved in current theories. Thus, higher temporal and spatial resolution for diffusion processes in shear flows may be realized, suggesting a more accurate analytical approach for the diffusion of Brownian particles

Trading Information Complexity for Error

We consider the standard two-party communication model. The central problem studied in this article is how much can one save in information complexity by allowing a certain error. * For arbitrary functions, we obtain lower bounds and upper bounds indicating a gain that is of order Omega(h(epsilon)) and O(h(sqrt{epsilon})). Here h denotes the binary entropy function. * We analyze the case of the two-bit AND function in detail to show that for this function the gain is Theta(h(epsilon)). This answers a question of Braverman et al. [Braverman, STOC 2013]. * We obtain sharp bounds for the set disjointness function of order n. For the case of the distributional error, we introduce a new protocol that achieves a gain of Theta(sqrt{h(epsilon)}) provided that n is sufficiently large. We apply these results to answer another of question of Braverman et al. regarding the randomized communication complexity of the set disjointness function. * Answering a question of Braverman [Braverman, STOC 2012], we apply our analysis of the set disjointness function to establish a gap between the two different notions of the prior-free information cost. In light of [Braverman, STOC 2012], this implies that amortized randomized communication complexity is not necessarily equal to the amortized distributional communication complexity with respect to the hardest distribution. As a consequence, we show that the epsilon-error randomized communication complexity of the set disjointness function of order n is n[C_{DISJ} - Theta(h(epsilon))] + o(n), where C_{DISJ} ~ 0.4827$ is the constant found by Braverman et al. [Braverman, STOC 2012]

Dispersion of Airborne Respiratory Saliva Droplets by Indoor Wake Flows

A new analysis of the interaction between vortical wake structures and exhaled saliva droplets is conducted in the presented study. We demonstrate how wake flows may alter the droplet and aerosol dispersion, exceeding previously reported settling distances. A dipolar vortex is used to model the wake flow, self propelling through a cloud of micron sized evaporating saliva droplets, which are tracked using a Lagrangian method. The droplet spatial location, velocity, diameter, and temperature are traced, while coupled to their local flow field thereof. Contrary to previous studies of droplet dispersion by vortical flows, a vortex viscous decay model is included. This proves to be essential for accurately predicting the dispersion and settling distances of droplets and aerosols. The non volatile saliva components are also considered, allowing for properly capturing the droplet aerosol transition while predicting the equilibrium diameter of post evaporation residual aerosols. Moreover, the model is verified with previously published results, yielding an accurate prediction of the droplet evaporation falling curves. Using this model, our theoretical analysis reveals non intuitive interactions between wake flows, droplet relaxation time, gravity, and mass transfer rates. The underlying physical mechanism responsible for the increased dispersion is studied, where aerosols originating from saliva droplets entrap within the vortical structure and subsequently translate to distances two orders of magnitude larger than the size of the vortex. Thus, a new mechanism enhancing the transmission of airborne pathogens is suggested, offering a new outlook on the spreading of airborne-carried pathogens.Comment: arXiv admin note: text overlap with arXiv:2108.0706