Long-time Behavior Research Articles

Learning Markovian dynamics with spectral maps.

The long-time behavior of many complex molecular systems can often be described by Markovian dynamics in a slow subspace spanned by a few reaction coordinates referred to as collective variables (CVs). However, determining CVs poses a fundamental challenge in chemical physics. Depending on intuition or trial and error to construct CVs can lead to non-Markovian dynamics with long memory effects, hindering analysis. To address this problem, we continue to develop a recently introduced deep-learning technique called spectral map [J. Rydzewski, J. Phys. Chem. Lett. 14, 5216-5220 (2023)]. Spectral map learns slow CVs by maximizing a spectral gap of a Markov transition matrix describing anisotropic diffusion. Here, to represent heterogeneous and multiscale free-energy landscapes with spectral map, we implement an adaptive algorithm to estimate transition probabilities. Through a Markov state model analysis, we validate that spectral map learns slow CVs related to the dominant relaxation timescales and discerns between long-lived metastable states.

The Fast Recurrent Subspace on an N-Level Quantum Energy Transport Model

The fast recurrent subspace (the biggest support of all invariant states) of a weak coupling limit type quantum Markov semigroup modeling a quantum transport open system of [Formula: see text]-energy levels is determined. This is achieved by characterizing the structure of all the invariant states and their spectra in terms of a natural generalization of the discrete Fourier transform operator. Finally, the attraction domains and long-time behaviour of the evolution are studied on hereditary subalgebras where faithful invariant states exist.

Data-driven discovery of invariant measures

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification.

On the long-time behavior of compressible fluid flows excited by random forcing

We are concerned with the long-time behavior of the stochastic Navier–Stokes system for compressible fluids in dimensions two and three. In this setting, the part of the phase space occupied by the solution depends sensitively on the choice of the initial state. Our main results are threefold. (i) The kinetic energy of a solution is universally and asymptotically bounded, independent of the initial datum. (ii) Time shifts of a solution with initially controlled energy are asymptotically compact and generate an entire solution defined for all t\in R . (iii) Every solution with initially controlled energy generates a stationary solution and even an ergodic stationary solution on the closure of the convex hull of its \omega -limit set on the space of measures on the path space.

Energy-optimal control of adaptive structures

Abstract Adaptive structures are equipped with sensors and actuators to actively counteract external loads such as wind. This can significantly reduce resource consumption and emissions during the life cycle compared to conventional structures. A common approach for active damping is to derive a port-Hamiltonian model and to employ linear-quadratic control. However, the quadratic control penalization lacks physical interpretation and merely serves as a regularization term. Rather, we propose a controller, which achieves the goal of vibration damping while acting energy-optimal. Leveraging the port-Hamiltonian structure, we show that the optimal control is uniquely determined, even on singular arcs. Further, we prove a stable long-time behavior of optimal trajectories by means of a turnpike property. Last, the proposed controller’s efficiency is evaluated in a numerical study.

Global well-posedness and decay estimates for the one-dimensional models of blood flow with a general parabolic velocity profile

In this paper, we study the one-dimensional models of blood flow arising from the hemodynamics of aorta, which are derived from the averaging of the Navier–Stokes equations. We establish the global well-posedness and long-time behavior of the viscid 1D models of blood flow in the Sobolev space framework, where a general parabolic velocity profile is considered. Precisely speaking, we prove the global existence of smooth solution when the initial data is sufficiently small. Moreover, by combining the time-weighted energy estimates with the Green function method, we obtain the optimal time decay rate in Lp (2≤p≤∞) norm. In addition, one can see the area-averaged axial velocity decays faster than the cross-sectional area of vessel from the Green function of the linearized system. This observation is essential to study the decay rates of our nonlinear system.

Dynamics for the diffusive logistic equation with a sedentary compartment and free boundary

This paper investigates the diffusive logistic equation with a sedentary compartment and free boundary whose dynamics has been considered by Wang and Cao (2015) when the intrinsic rate r of reproduction in the stationary class is less than the rate c of switching from stationary to mobile. However, the case r≥c is left as an open problem in Wang and Cao (2015). We shall show that spreading happens if r≥c and give the longtime behavior.

Uniform attractors of non-autonomous suspension bridge equations with memory

In this article, we investigate the long-time dynamical behavior of non-autonomous suspension bridge equations with memory and free boundary conditions. We first establish the well-posedness of the system by means of the maximal monotone operator theory. Secondly, the existence of uniformly bounded absorbing set is obtained. Finally, asymptotic compactness of the process is verified, and then the existence of uniform attractors is proved for non-autonomous suspension bridge equations with memory term. For more information see https://ejde.math.txstate.edu/Volumes/2024/16/abstr.html

Maturation-dependent changes in the size, structure and seeding capacity of Aβ42 amyloid fibrils

Many proteins self-assemble to form amyloid fibrils, which are highly organized structures stabilized by a characteristic cross-β network of hydrogen bonds. This process underlies a variety of human diseases and can be exploited to develop versatile functional biomaterials. Thus, protein self-assembly has been widely studied to shed light on the properties of fibrils and their intermediates. A still open question in the field concerns the microscopic processes that underlie the long-time behaviour and properties of amyloid fibrillar assemblies. Here, we use atomic force microscopy with angstrom-sensitivity to observe that amyloid fibrils undergo a maturation process, associated with an increase in both fibril length and thickness, leading to a decrease of their density, and to a change in their cross-β sheet content. These changes affect the ability of the fibrils to catalyse the formation of new aggregates. The identification of these changes helps us understand the fibril maturation processes, facilitate the targeting of amyloid fibrils in drug discovery, and offer insight into the development of biocompatible and sustainable protein-based materials.

Dynamic correlations: exact and approximate methods for mutual information.

Proteins are dynamic entities that undergo conformational changes critical for their functions. Understanding the communication pathways and information transfer within proteins is crucial for elucidating allosteric interactions in their mechanisms. This study utilizes mutual information (MI) analysis to probe dynamic allostery. Using two cases, Ubiquitin and PLpro, we have evaluated the accuracy and limitations of different approximations including the exact anisotropic and isotropic models, multivariate Gaussian model, isotropic Gaussian model, and the Gaussian Network Model (GNM) in revealing allosteric interactions. Our findings emphasize the required trajectory length for capturing accurate mutual information profiles. Long molecular dynamics trajectories, 1 ms for Ubiquitin and 100 µs for PLpro are used as benchmarks, assuming they represent the ground truth. Trajectory lengths of approximately 5 µs for Ubiquitin and 1 µs for PLpro marked the onset of convergence, while the multivariate Gaussian model accurately captured mutual information with trajectories of 5 ns for Ubiquitin and 350 ns for PLpro. However, the isotropic Gaussian model is less successful in representing the anisotropic nature of protein dynamics, particularly in the case of PLpro, highlighting its limitations. The GNM, however, provides reasonable approximations of long-range information exchange as a minimalist network model based on a single crystal structure. Overall, the optimum trajectory lengths for effective Gaussian approximations of long-time dynamic behavior depend on the inherent dynamics within the protein's topology. The GNM, by showcasing dynamics across relatively diverse time scales, can be used either as a standalone method or to gauge the adequacy of MD simulation lengths. Mutual information codes are available at https://github.com/kemaldemirtas/prc-MI.git.

The Fokker–Planck–Boltzmann equation in the finite channel

In this paper, we establish the existence of small-amplitude unique solutions near the Maxwellian for the Fokker–Planck–Boltzmann equation in a finite channel with specular reflection boundary conditions. The solution space we consider is denoted as Lk̄1LT∞Lx1,v2, introduced in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. In addition, we investigate the long-time behavior of solutions for both hard and soft potentials.

A chemotaxis system with singular sensitivity for burglaries in the higher-dimensional settings: generalized solvability and long-time behavior

A chemotaxis system with singular sensitivity for burglaries in the higher-dimensional settings: generalized solvability and long-time behavior

Subdiffusion in an array of solid obstacles

More than a decade ago, Goychuk reported on a universal behavior of subdiffusive motion (as described by the generalized Langevin equation) in a one-dimensional bounded periodic potential (Goychuk 2009 Phys. Rev. E 80 046125) where the numerical findings show that the long-time behavior of the mean squared displacement is not influenced by the potential, so that the behavior in the potential, under homogenization, is the same as in its absence. This property may break down if the potential is unbounded. In the present work, we report on the results of simulations of subdiffusion in a two-dimensional (2D) periodic array of solid obstacles (i.e. in an unbounded potential) with different packing fractions. It is revealed that the universal subdiffusive behavior at long times is not influenced by the presence of solid scatterers, whose presence influences the behavior at intermediate times only. This result is discussed as having possible relations to the emerging problem of interpretation of results on trajectories of tracers spreading in the brain’s extracellular space.

Global Existence and Long-Time Behavior in the 1 + 1-Dimensional Principal Chiral Model with Applications to Solitons

Global Existence and Long-Time Behavior in the 1 + 1-Dimensional Principal Chiral Model with Applications to Solitons

Analyzing a class of stochastic SIRS models under imperfect vaccination

This work is devoted to a class of stochastic infectious disease models, namely, stochastic SIRS models under imperfect vaccination. Because of the inclusion of the vaccination, the underlying models are more difficult to analyze than that of the previously considered in the literature. Our main effort is devoted to overcoming the difficulties and deriving a complete characterization of longtime behavior of systems. Numerical examples are provided to present computational evidence. These examples also provide insight that the discrete event process (the random switching) can reverse persistence to extinction, and vice versa.

Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data

We characterize the long-time asymptotic behavior of the solution of the initial value problem for the coupled complex short-pulse equation associated with the 4×4 matrix spectral problem. The spectral analysis of the 4×4 matrix spectral problem is very difficult because of the existence of energy-dependent potentials and the WKI type. The method we adopted is a combination of the inverse scattering transform and Deift-Zhou nonlinear steepest descent method. Starting from the Lax pair associated with the coupled complex short-pulse equation, we derive a basic Riemann-Hilbert problem by introducing some appropriate spectral function transformations, and reconstruct the potential parameterized from the solution of the basic Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at k→0. We finally obtain the leading order asymptotic behavior of the solution of the coupled complex short-pulse equation through a series of Deift-Zhou contour deformations.

Long-time oxidation behavior of the nearly stoichiometric polycrystalline SiC fibers under air atmosphere at different temperatures

SiC fibers are important reinforcements due to their excellent oxidation and high-temperature resistance. In this work, near-stoichiometric polycrystalline SiC fibers were oxidized in air at 1200 °C to 1600 °C for up to 100 h. The thickness of the oxide layer reached 1.9 µm after oxidation at 1400 °C for 100 h. Wide cracks appeared on the surface due to residual stress. Activation energy was comparable to that of commercial SiC fibers. The results in this work could provide important reference for such SiC fibers to be applied in the fields of aviation, aerospace, and nuclear power.

Long-time behavior for evolution processes associated with non-autonomous nonlinear Schrödinger equation

We consider non-autonomous nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in a bounded smooth domain and time-dependent forcing that models the motion of waves in a quantum-mechanical system. We address the problem of the local and global well posedness and using rescaling of time we prove the existence of a compact pullback attractor for the associated evolution process. Moreover, we prove that the pullback attractor has finite fractal dimension. To the best of our knowledge, our approach has not been used in the literature to treat the non-autonomous Schrödinger equation.

Time decay of solutions for compressible isentropic non-Newtonian fluids

In this paper, we consider the Cauchy problem of a compressible Navier–Stokes system of Eills-type non-Newtonian fluids. We investigate the time decay properties of classical solutions for the compressible non-Newtonian fluid equations. More specifically, we construct a new linearized system in terms of a combination of the solutions, and then we investigate the long-time behavior of the Cauchy problem for the three-dimensional isentropic compressible Eills-type non-Newtonian fluids with an initial perturbation.

Self-interacting diffusions: Long-time behaviour and exit-problem in the uniformly convex case

We study a class of time-inhomogeneous diffusion: the self-interacting one. We show a convergence result with a rate of convergence that does not depend on the diffusion coefficient. Finally, we establish a so-called Kramers’ type law for the first exit-time of the process from domain of attractions when the landscapes are uniformly convex.