Open quantum systems beyond equilibrium: Lindblad equation and path integral molecular dynamics

The Lindblad equation determines the time evolution of the density operator of open quantum systems. While valid for any system size, its use is, in practice, restricted to prototype and surrogate models with the aim of tackling specific aspects of the overall quantum complexity of a multiatomic system. Path integral molecular dynamics (PIMD) instead provides static and dynamical quantum statistical averages of physical observables for systems in equilibrium composed of up to thousands of atoms over timescales up to nanoseconds, under the condition that short-time quantum coherence is not relevant for the properties of interest. PIMD relies on the well-established technique of molecular dynamics with its associated classical trajectories. However, it cannot describe a direct time evolution of a system and its convergence to a stationary state in situations out of equilibrium. In this work we analyze the link between the Lindblad equation and PIMD; specifically, we will discuss how PIMD can actually be used to calculate the time evolution of ensemble-averaged physical observables and their convergence to a stationary state for situations out of equilibrium, bypassing the need for explicitly solving the Lindblad equation. Yet, at the same time, the Lindblad equation and PIMD are linked to one another through a formal relation of equivalence, which provides an argument for the consistency of PIMD results, namely, the positivity of the density operator at any time. A numerical study of a prototype system, which is of interest in chemical physics, will be used to showcase the method

Analysis of a Three-Dimensional Rapidly Rotating Convection Model Without Thermal Diffusion

We study a three-dimensional rapidly rotating convection model featuring tall columnar structures, in the absence of thermal diffusion. We establish the global existence and uniqueness of weak solutions, as well as the Hadamard well-posedness of global strong solutions to this model. The lack of thermal diffusion introduces significant challenges in the analysis. To overcome these challenges, we first investigate the regularized model with thermal diffusion and establish delicate estimates that are independent of the thermal diffusion coefficient, and consequently justify the vanishing diffusivity limit. This work serves as a continuation of our previous paper Cao(JAMA 21:2923–2954 (2021)

A singular perturbation analysis for the Brusselator

In this work we study the Brusselator – a prototypical model for chemical oscillations– under the assump- tion that the bifurcation parameter is of order O(1/ϵ) for positive ϵ ≪ 1. The dynamics of this mathematical model exhibits a time scale separation visible via fast and slow regimes along its unique attracting limit cy- cle. This limit cycle accumulates at infinity as ϵ → 0, so that appropriate coordinates (w, z) are used to analyse the dynamics near the line at infinity, corresponding to the set {z = 0}. This object becomes a non- hyperbolic invariant manifold for which we use a desingularising rescaling, in order to study the closeby dynamics. Further use of geometric singular perturbation techniques allows us to give a decomposition of the Brusselator limit cycle in terms of four different fully quantified time scales for small ϵ

Combined effects of evaporation, sedimentation and solute crystallization on the dynamics of aerosol size distributions on multiple length and time scales

We investigate three aspects of aerosol-mediated air-borne viral infection mechanisms on different length and time scales. First, we address the evolution of the size distribution of a non-interacting ensemble of droplets that are subject to evaporation and sedimentation using a sharp droplet-air interface model. From the exact solution of the evolution equation we derive the viral load in the air and show that it depends sensitively on the relative humidity. Secondly, from Molecular Dynamics simulations we extract the molecular reflection coefficient of single water molecules from the air-water interface. This parameter determines the water condensation and evaporation rate at a liquid droplet surface and therefore the evaporation rate of aqueous droplets. We find the reflection of water to be negligible at room temperature but to rise significantly at elevated temperatures and for grazing incidence angles. Thirdly, we derive a thermodynamically consistent three-dimensional diffuse-interface model for solute-containing droplets that is formulated as a three-phase Cahn-Hilliard/Allen-Cahn system. By numerically solving the coupled system of equations, we explore representative scenarios that show that this model reproduces and generalizes features of the sharp-interface model. These interconnected studies on the dynamics of aerosol droplet evaporation are relevant in order to quantitatively assess the airborne infection risk under varying environmental conditions

Dream-stellar: parallel and space efficient exact local alignment

Background: Searching large genomic data sets for local alignments poses a computational challenge. A particular obstacle is the handling of repetitive sequences that appear in various contexts and incur a high runtime cost. For practical homology search, it is important to develop a specific but sensitive filter. Good filters reduce the search space before alignment without missing significant matches. Results: We introduce DREAM-Stellar, a parallelized, updated version of the pairwise local aligner Stellar. The new aligner, DREAM-Stellar, is composed of four steps: preprocessing the queries and references, building a data structure for distributing the queries, computing in parallel the results and finally combining them. For distributing the queries we use the IBF data structure and a new prefilter for local alignments. We present our comparison of five local aligners on simulated and real genomic data and conclude that heuristic tools like BLAST miss a large percentage of significant local alignments or "drown" them in millions of less significant matches. This new version of Stellar is up to 900 times faster on 32 parallel threads than its single-threaded predecessor and can find all alignments between a pair of genomes in minutes. With that, the runtime of DREAM-Stellar is on par with tools like BLAST etc. Conclusions: DREAM-Stellar is very practical and fast on very long sequences which makes it a suitable new tool for finding local alignments between genomic sequences under the edit distance model. The software is freely available for Linux and Mac OS X at https://github.com/seqan/dream-stella

Toward Dynamic Phase-Field Fracture at Finite Strains

We investigate the evolution of dynamic phase-field fracture in the finite-strain setting, extending our previous work in the small-strain viscoelastodynamic regime. The elastodynamic equations are coupled with a dissipative damage evolution for the phase-field variable . The material response is described with a polyconvex stored energy density , where denotes the gradient of the deformation, its cofactor, and its determinant. This ensures compatibility with the principles of nonlinear elasticity. A fully discrete time-staggered approximation scheme is proposed, along with associated stability of discrete solutions. We present compactness results and analyze the convergence of the discrete approximations. While convergence of the phase-field variable and the compatibility of the kinematic variables can be demonstrated, the identification of the limit stress in the momentum balance remains open. To address this, two strategies are outlined: an extension of the classical (weak) framework using generalized Young or defect measures, and an alternative formulation via energetic-variational solutions that avoids the explicit measure construction. Partial results on existence and the structure of the limit system are discussed

Time-asymptotic self-similarity of the damped compressible Euler equations in parabolic scaling variables

We study the long-time behavior of solutions to the compressible Euler equations with frictional damping in the whole space, where we prescribe direction-dependent values for the density at spatial infinity. To this end, we transform the system into parabolic scaling variables and derive a relative entropy inequality, which allows to conclude the convergence of the density towards a self-similar solution to the porous medium equation while the associated limit momentum is governed by Darcy's law. Moreover, we obtain convergence rates that explicitly depend on the flatness of the limit profile. While we focus on weak solutions in the one-dimensional case, we extend our results to energy-variational solutions in the multi-dimensional setting

Asymptotic stability of the equilibrium for the free boundary problem of a compressible atmospheric primitive model with physical vacuum

This paper concerns the large time asymptotic behavior of solutions to the free boundary problem of the compressible primitive equations in atmospheric dynamics with physical vacuum. Up to second order of the perturbations of an equilibrium, we have introduced a model of the compressible primitive equations with a specific viscosity and shown that the physical vacuum free boundary problem for this model system has a global-in-time solution converging to an equilibrium exponentially, provided that the initial data is a small perturbation of the equilibrium. More precisely, we introduce a new coordinate system by choosing the enthalpy (the square of sound speed) as the vertical coordinate, and thanks to the hydrostatic balance, the degenerate density at the free boundary admits a representation with separation of variables in the new coordinates. Such a property allows us to establish horizontal derivative estimates without involving the singular vertical derivative of the density profile, which plays a key role in our analysis

Rough homogenization for Langevin dynamics on fluctuating Helfrich surfaces

In this article, we study different scaling rough path limit regimes in space and time for the Langevin dynamics on a quasi-planar fluctuating Helfrich surface. The convergence results of the processes were already proven in [Citation1]. We extend this work by proving the convergence of the Itô and Stratonovich rough path lift. For the rough path limit, there appears, typically, an area correction term to the Itô iterated integrals and, in certain regimes, to the Stratonovich iterated integrals. This yields additional information on the homogenization limit and enables us to conclude on homogenization results for diffusions driven by the Brownian motion on the membrane using the continuity of the Itô-Lyons map in rough paths topology