Revisiting Schrödinger's fourth-order, real-valued wave equation and the implication from the resulting energy levels.

In his seminal part IV, Annalen der Physik vol. 81, 1926 paper, Schrödinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is fourth-order in space and second-order in time. In the view of the mathematical difficulties associated with the eigenvalue analysis of a fourth-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrödinger splits the fourth-order real operator into the product of two, second-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic second-order, complex-valued wave equation. In this paper, we show that Schrödinger's original fourth-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his second-order, complex-valued wave equation that predicts with remarkable accuracy the energy levels observed in the atomic line spectra of the chemical elements. Accordingly, the fourth-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that quantum mechanics can only be described with the less stiff, second-order, complex-valued wave equation.

Mechanical analogue for cities.

Motivated from the increasing need to develop a science-based, predictive understanding of the dynamics and response of cities when subjected to natural hazards, in this paper, we apply concepts from statistical mechanics and microrheology to develop mechanical analogues for cities with predictive capabilities. We envision a city to be a matrix where cell-phone users are driven by the city's economy and other incentives while using the collection of its infrastructure networks in a similar way that thermally driven Brownian particles are moving within a complex viscoelastic material. Mean-square displacements of thousands of cell-phone users are computed from GPS location data to establish the creep compliance and the resulting impulse response function of a city. The derivation of these time-response functions allows the synthesis of simple mechanical analogues that model satisfactorily the city's behaviour under normal conditions. Our study concentrates on predicting the response of cities to acute shocks (natural hazards) that are approximated with a rectangular pulse; and we show that the derived solid-like mechanical networks predict that cities revert immediately to their pre-event response suggesting an inherent resilience. Our findings are in remarkable good agreement with the recorded response of the Dallas metroplex following the February 2021 North American winter storm.

Impulse response function for Brownian motion.

Motivated from the central role of the mean-square displacement and its second time-derivative - that is the velocity autocorrelation function in the description of Brownian motion and its implications to microrheology, we revisit the physical meaning of the first time-derivative of the mean-square displacement of Brownian particles. By employing a rheological analogue for Brownian motion, we show that the time-derivative of the mean-square displacement of Brownian microspheres with mass m and radius R immersed in any linear, isotropic viscoelastic material is identical to , where h(t) is the impulse response function (strain history γ(t), due to an impulse stress τ(t) = δ(t - 0)) of a rheological network that is a parallel connection of the linear viscoelastic material with an inerter with distributed inertance . The impulse response function of the viscoelastic material-inerter parallel connection derived in this paper at the stress-strain level of the rheological analogue is essentially the response function of the Brownian particles expressed at the force-displacement level by Nishi et al. after making use of the fluctuation-dissipation theorem. By employing the viscoelastic material-inerter rheological analogue we derive the mean-square displacement and its time-derivatives of Brownian particles immersed in a viscoelastic material described with a Maxwell element connected in parallel with a dashpot and we show that for Brownian motion of microparticles immersed in such fluid-like materials, the impulse response function h(t) maintains a finite constant value in the long term.

Viscous-viscoelastic correspondence principle for Brownian motion.

Motivated by the classical expressions of the mean-square displacement and the velocity autocorrelation function of Brownian particles either suspended in a Newtonian viscous fluid or trapped in a harmonic potential, we show that for all timescales the mean-square displacement of Brownian microspheres with mass m and radius R suspended in any linear, isotropic viscoelastic material is identical to the creep compliance of a linear mechanical network that is a parallel connection of the linear viscoelastic material with an inerter with distributed inertance m_{R}=m/6πR. The synthesis of this mechanical network leads to the statement of a viscous-viscoelastic correspondence principle for Brownian motion which simplifies appreciably the calculations of the mean-square displacement and the velocity autocorrelation function of Brownian particles suspended in viscoelastic materials where inertia effects are non-negligible at longer timescales. The viscous-viscoelastic correspondence principle established in this paper by introducing the concept of the inerter is equivalent to the viscous-viscoelastic analogy adopted by Mason and Weitz [T. G. Mason and D. A. Weitz, Phys. Rev. Lett. 74, 1250 (1995)10.1103/PhysRevLett.74.1250].

Amygdala volume associated with alcohol abuse relapse and craving.

OBJECTIVE: Amygdala volume has been associated with drug craving in cocaine addicts, and amygdala volume reduction is observed in some alcohol-dependent subjects. This study sought an association in alcohol-dependent subjects between volumes of reward-related brain regions, alcohol craving, and the risk of relapse. METHOD: Besides alcohol craving, the authors assessed amygdala, hippocampus, and ventral striatum volumes in 51 alcohol-dependent subjects and 52 age- and education-matched healthy comparison subjects after detoxification. After imaging and clinical assessment, patients were followed for 6 months and alcohol intake was recorded. RESULTS: Alcohol-dependent subjects showed reduced amygdala, hippocampus, and ventral striatum volumes and reported stronger craving in relation to healthy comparison subjects. However, only amygdala volume and craving differentiated between subsequent relapsers and abstainers. A significant decrease of amygdala volume in alcohol-dependent subjects was associated with increased alcohol craving before imaging and an increased alcohol intake during the 6-month follow-up period. CONCLUSIONS: These findings suggest a relationship between amygdala volume reduction, alcohol craving, and prospective relapse into alcohol consumption.