Entropic and gradient flow formulations for nonlinear diffusion

Nonlinear diffusion $\partial_t \rho = \Delta(\Phi(\rho))$ is considered for a class of nonlinearities $\Phi$. It is shown that for suitable choices of $\Phi$, an associated Lyapunov functional can be interpreted as thermodynamics entropy. This information is used to derive an associated metric, here called thermodynamic metric. The analysis is confined to nonlinear diffusion obtainable as hydrodynamic limit of a zero range process. The thermodynamic setting is linked to a large deviation principle for the underlying zero range process and the corresponding equation of fluctuating hydrodynamics. For the latter connections, the thermodynamic metric plays a central role

Firm-Level Exposure to Epidemic Diseases: Covid-19, SARS, and H1N1

Using tools described in our earlier work (Hassan et al., 2019, 2020), we develop text-based measures of the costs, benefits, and risks listed firms in the US and over 80 other countries associate with the spread of Covid-19 and other epidemic diseases. We identify which firms expect to gain or lose from an epidemic disease and which are most affected by the associated uncertainty as a disease spreads in a region or around the world. As Covid-19 spreads globally in the first quarter of 2020, we find that firms’ primary concerns relate to the collapse of demand, increased uncertainty, and disruption in supply chains. Other important concerns relate to capacity reductions, closures, and employee welfare. By contrast, financing concerns are mentioned relatively rarely. We also identify some firms that foresee opportunities in new or disrupted markets due to the spread of the disease. Finally, we find some evidence that firms that have experience with SARS or H1N1 have more positive expectations about their ability to deal with the coronavirus outbreak

Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States

The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We introduce the notion of relaxation height in a general energy landscape and we prove sufficient conditions which are valid even in presence of multiple metastable states. We show how these results can be used to approach the problem of multiple metastable states via the use of the modern theories of metastability. We finally apply these general results to the Blume--Capel model for a particular choice of the parameters ensuring the existence of two multiple, and not degenerate in energy, metastable states

Heat flow in chains driven by thermal noise

We consider the large deviation function for a classical harmonic chain composed of N particles driven at the end points by heat reservoirs, first derived in the quantum regime by Saito and Dhar and in the classical regime by Saito and Dhar and Kundu et al. Within a Langevin description we perform this calculation on the basis of a standard path integral calculation in Fourier space. The cumulant generating function yielding the large deviation function is given in terms of a transmission Green's function and is consistent with the fluctuation theorem. We find a simple expression for the tails of the heat distribution which turn out to decay exponentially. We, moreover, consider an extension of a single particle model suggested by Derrida and Brunet and discuss the two-particle case. We also discuss the limit for large N and present a closed expression for the cumulant generating function. Finally, we present a derivation of the fluctuation theorem on the basis of a Fokker-Planck description. This result is not restricted to the harmonic case but is valid for a general interaction potential between the particles.Comment: Latex: 26 pages and 9 figures, appeared in J. Stat. Mech. P04005 (2012

Entropy production and fluctuation relations for a KPZ interface

We study entropy production and fluctuation relations in the restricted solid-on-solid growth model, which is a microscopic realization of the KPZ equation. Solving the one dimensional model exactly on a particular line of the phase diagram we demonstrate that entropy production quantifies the distance from equilibrium. Moreover, as an example of a physically relevant current different from the entropy, we study the symmetry of the large deviation function associated with the interface height. In a special case of a system of length L=4 we find that the probability distribution of the variation of height has a symmetric large deviation function, displaying a symmetry different from the Gallavotti-Cohen symmetry.Comment: 21 pages, 5 figure

N-Terminal Pro–B-Type Natriuretic Peptide in the Emergency Department: The ICON-RELOADED Study

Background Contemporary reconsideration of diagnostic N-terminal pro–B-type natriuretic peptide (NT-proBNP) cutoffs for diagnosis of heart failure (HF) is needed. Objectives This study sought to evaluate the diagnostic performance of NT-proBNP for acute HF in patients with dyspnea in the emergency department (ED) setting. Methods Dyspneic patients presenting to 19 EDs in North America were enrolled and had blood drawn for subsequent NT-proBNP measurement. Primary endpoints were positive predictive values of age-stratified cutoffs (450, 900, and 1,800 pg/ml) for diagnosis of acute HF and negative predictive value of the rule-out cutoff to exclude acute HF. Secondary endpoints included sensitivity, specificity, and positive (+) and negative (−) likelihood ratios (LRs) for acute HF. Results Of 1,461 subjects, 277 (19%) were adjudicated as having acute HF. The area under the receiver-operating characteristic curve for diagnosis of acute HF was 0.91 (95% confidence interval [CI]: 0.90 to 0.93; p < 0.001). Sensitivity for age stratified cutoffs of 450, 900, and 1,800 pg/ml was 85.7%, 79.3%, and 75.9%, respectively; specificity was 93.9%, 84.0%, and 75.0%, respectively. Positive predictive values were 53.6%, 58.4%, and 62.0%, respectively. Overall LR+ across age-dependent cutoffs was 5.99 (95% CI: 5.05 to 6.93); individual LR+ for age-dependent cutoffs was 14.08, 4.95, and 3.03, respectively. The sensitivity and negative predictive value for the rule-out cutoff of 300 pg/ml were 93.9% and 98.0%, respectively; LR− was 0.09 (95% CI: 0.05 to 0.13). Conclusions In acutely dyspneic patients seen in the ED setting, age-stratified NT-proBNP cutpoints may aid in the diagnosis of acute HF. An NT-proBNP <300 pg/ml strongly excludes the presence of acute HF

The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class

We explain the exact solution of the 1+1 dimensional Kardar-Parisi-Zhang equation with sharp wedge initial conditions. Thereby it is confirmed that the continuum model belongs to the KPZ universality class, not only as regards to scaling exponents but also as regards to the full probability distribution of the height in the long time limit.Comment: Proceedings StatPhys 2

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference