The Fokker-Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker-Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time-discrete, iterative variational scheme whose solutions converge to the solution of the Fokker-Planck equation. The major novelty of this iterative scheme is that the time step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker-Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.

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We present a statistical equilibrium model of self-organization in a class of focusing, nonintegrable nonlinear Schrodinger (NLS) equations. The theory predicts that the asymptotic-time behavior of the NLS system is characterized by the formation and persistence of a large-scale coherent solitary wave, which minimizes the Hamiltonian given the conserved particle number (L2-norm squared), coupled with small-scale random fluctuations, or radiation. The fluctuations account for the difference between the conserved value of the Hamiltonian and the Hamiltonian of the coherent state. The predictions of the statistical theory are tested against the results of direct numerical simulations of NLS, and excellent qualitative and quantitative agreement is demonstrated. In addition, a careful inspection of the numerical simulations reveals interesting features of the transitory dynamics leading up to the long-time statistical equilibrium state starting from a given initial condition. As time increases, the system investigates smaller and smaller scales, and it appears that at a given intermediate time after the coalescense of the soliton structures has ended, the system is nearly in statistical equilibrium over the modes that it has investigated up to that time.

A continuum model of coherent structures in two-dimensional magnetohydrodynamic turbulence is developed. These structures are macroscopic states which persist amongst the turbulent microscopic fluctuations, typically as magnetic islands with flow. They are modeled as statistical equilibrium states for the ideal (nondissipative) dynamics, which conserves energy and families of cross-helicity and flux integrals. The model predicts that an ideal magnetofluid will evolve into a turbulent relaxed state having steady mean magnetic and velocity fields, and Gaussian local fluctuations in these fields. Excellent qualitative and quantitative agreement is found with the results of direct numerical simulations. A rigorous justification of the theory is also provided, in the sense that the continuum model is derived from a lattice model in a fixed-volume, small-spacing limit. This construction uses the discrete Fourier transform to link the discretization of x-space with the truncation of k-space. The lattice model is defined by the most probable distribution on the discretized phase space that respects the approximated dynamical constraints. A concentration property shows that this distribution is equivalent to the microcanonical distribution in the continuum limit.

Medical student wellness is of great concern in the health care field. A growing number of studies point to increases in suicide, depression, anxiety, mood disorders, and burnout related to physician lifestyles. Mental health issues commencing in medical school have been suggested to have a significant impact on future physician lifestyle and burnout. Tracking the mental health of medical students at the University of Toledo College of Medicine and Life Sciences (UTCOMLS) with standardized indices will help elucidate triggers of poor mental health. Anonymous surveys were developed and distributed to preclinical medical students at five strategic time points throughout the 2018 2019 academic year. Surveys collected basic demographic information as well as inventories measuring perceived stress, burnout, resilience, and mindfulness. 172 M1s (83 males and 89 females) were included in the study and average response rate for the first 4 (out of 5) surveys averaged 74.8%. M1 males and females had on average increased personal burnout over time with females consistently scoring higher. Both males and females had an increase in stress from August to each subsequent month (p<0.05). Females reported a higher level of perceived stress than males in the beginning and middle of the academic year (p<0.05). Both males and females report a gradual decrease in resiliency throughout the academic year. These surveys demonstrated over half of males and females in medical school reported higher perceived stress scores than their gender-matched peers in the general United States population. Our study strengthens documented trends in resiliency, perceived stress, and burnout amongst medical students. More study in designing targeted approaches to ameliorate these findings in the medical student population is warranted.

Abstract-The air transportation system is a network of many interacting, capacity-constrained elements. When the demand for airport and airspace resources exceed the available capacities of these resources, delays occur. The state of the air transportation system at any time can be represented as a weighted directed graph in which the nodes correspond to airports, and the weight on each arc is the delay experienced by departures on that origin-destination pair. Over the course of any day, the state of the system progresses through a timeseries, where the state at any time-step is the weighted directed graph described above.This paper presents algorithms for the clustering of air traffic delay network data from the US National Airspace System, in order to identify characteristic delay states (i.e., weighted directed graphs) as well as characteristic types-ofdays (i.e., sequences of such weighted directed graphs) that are experienced by the air transportation system. The similarity of delay states during clustering are evaluated on the basis of not only the in-and out-degrees of the nodes (the total inbound and outbound delays), but also network-theoretic properties such as the eigenvector centralities, and the hub and authority scores of different nodes. Finally, the paper looks at community detection, that is, the grouping of nodes (airports) based on their similarities within a system delay state. The type of day is found to have an impact on the observed community structures.

A statistical equilibrium theory is developed to characterize the large-scale coherent svuctures in a turbulent two-dimensional magnetofluid. Macrostates are defined as local joint probability distributions, or Young measures, on the values of the fluctuating magnetic field and velocity field at each point in the spatial domain. The most probable macrostate is found by maximizing a Kullback enmpy functional subject to constnints dicated by the conserved integrals of the ideal dynamics. This maximum envopy macrosate is, for each point in the spatial domain, a Gaussian probability distribution, whose local mean is an exact stationary solution of the equations of ideal magnetohydrodynamics. n e predictions of the model are in excellent qualitative and quantitative agreement with recent high resolution numerical simulations of turbulence in slightly dissipative two-dimensional magnetofluids.