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

Affine constraints in non-reversible diffusions with degenerate noise

This paper deals with the realisation of affine constraints on nonreversible stochastic differential equations (SDE) by strong confining forces. We prove that the confined dynamics converges pathwise and on bounded time intervals to the solution of a projected SDE in the limit of infinitely strong confinement, where the projection is explicitly given and depends on the choice of the confinement force. We present results for linear Ornstein-Uhlenbeck (OU) processes, but they straightforwardly generalise to nonlinear SDEs. Moreover, for linear OU processes that admit a unique invariant measure, we discuss conditions under which the limit also preserves the long-term properties of the SDE. More precisely, we discuss choices for the design of the confinement force which in the limit yield a projected dynamics with invariant measure that agrees with the conditional invariant measure of the unconstrained processes for the given constraint. The theoretical findings are illustrated with suitable numerical examples

Integrating Agent-Based and Compartmental Models for Infectious Disease Modeling: A Novel Hybrid Approach

This study investigates the spatial integration of agent-based models (ABMs) and compartmental models for infectious disease modeling, presenting a novel hybrid approach and examining its implications. ABMs offer detailed insights by simulating interactions and decisions among individuals but are computationally expensive for large populations. Compartmental models capture population-level dynamics more efficiently but lack granular detail. We developed a hybrid model that aims to balance the granularity of ABMs with the computational efficiency of compartmental models, offering a more nuanced understanding of disease spread in diverse scenarios, including large populations. This model spatially couples discrete and continuous populations by integrating an ordinary differential equation model with a spatially explicit ABM. Our key objectives were to systematically assess the consistency of disease dynamics and the computational efficiency across various configurations. For this, we evaluated two experimental scenarios and varied the influence of each sub-model via spatial distribution. In the first, the ABM component modeled a homogeneous population; in the second, it simulated a heterogeneous population with landscape-driven movement. Results show that the hybrid model can significantly reduce computational costs but is sensitive to between-model differences, highlighting the importance of model equivalence in hybrid approaches. The code is available at: git.zib.de/ibostanc/hybrid_abm_ode

Toward Grid-Based Models for Molecular Association

This paper presents a grid-based approach to model molecular association processes as an alternative to sampling-based Markov models. Our method discretizes the six-dimensional space of relative translation and orientation into grid cells. By discretizing the Fokker–Planck operator governing the system dynamics via the square-root approximation, we derive analytical expressions for the transition rate constants between grid cells. These expressions depend on geometric properties of the grid, such as the cell surface area and volume, which we provide. In addition, one needs only the molecular energy at the grid cell center, circumventing the need for extensive MD simulations and reducing the number of energy evaluations to the number of grid cells. The resulting rate matrix is closely related to the Markov state model transition matrix, offering insights into metastable states and association kinetics. We validate the accuracy of the model in identifying metastable states and binding mechanisms, though improvements are necessary to address limitations like ignoring bulk transitions and anisotropic rotational diffusion. The flexibility of this grid-based method makes it applicable to a variety of molecular systems and energy functions, including those derived from quantum mechanical calculations. The software package MolGri, which implements this approach, offers a systematic and computationally efficient tool for studying molecular association processes

Filtered data based estimators for stochastic processes driven by colored noise

We consider the problem of estimating unknown parameters in stochastic differential equations driven by colored noise, which we model as a sequence of Gaussian stationary processes with decreasing correlation time. We aim to infer parameters in the limit equation, driven by white noise, given observations of the colored noise dynamics. We consider both the maximum likelihood and the stochastic gradient descent in continuous time estimators, and we propose to modify them by including filtered data. We provide a convergence analysis for our estimators showing their asymptotic unbiasedness in a general setting and asymptotic normality under a simplified scenario

Approximation of time-periodic flow past a translating body by flows in bounded domains

We consider a time-periodic incompressible three-dimensional Navier-Stokes flow past a translating rigid body. In the first part of the paper, we establish the existence and uniqueness of strong solutions in the exterior domain Ω ⊂ R3 that satisfy pointwise estimates for both the velocity and pressure. The fundamental solution of the time-periodic Oseen equations plays a central role in obtaining these estimates. The second part focuses on approximating this exterior flow within truncated domains Ω∩BR, incorporating appropriate artificial boundary conditions on ∂BR. For these bounded domain problems, we prove the existence and uniqueness of weak solutions. Finally, we estimate the error in the velocity component as a function of the truncation radius R, showing that, as R → ∞, the velocities of the truncated problems converge, in an appropriate norm, to the velocity of the exterior flow

Finite-size effects in molecular simulations: A physico-mathematical view

Molecular simulation of condensed matter systems has always been characterized by the aim for an optimal balance between a precise physical description of the simulated substance, and the efficient use of computational resources. A major challenge for the accurate representation of a physical system in a simulation, therefore, consists in determining the appropriate size of the simulated sample. The latter must be sufficiently large in order to represent the bulk of the substance, and thus to reproduce its characteristic thermodynamic features. This problem is known under the name of “finite-size effects”, and several criteria have been adopted in order to determine these effects, thereby inferring about the validity of a simulation study. In this article, we discuss the application of a rigorous mathematical theorem, the so-called “two-sided Bogoliubov inequality”, to estimate the finite-size effects. The theorem provides upper and lower bounds for the free energy cost of partitioning a system into equivalent, non-interacting subsystems, and it can be used to obtain a rigorous definition of the minimal size of a system with its full thermodynamic features. The corresponding criterion based on this theorem is complementary to those existing in the literature, and it can be applied to both classical and quantum systems. The need for accurate and physically consistent results of current simulations is enormously increased by the use of simulation data in machine learning procedures. Physically inconsistent data, produced by simulations of insufficient size, results in a substantial error in the modeling procedure that propagates further into the study of several other systems or larger scales beyond the molecular one. Furthermore, the statistical nature of machine learning implies questions about the number of parameters and the size of the training set. Such problems are the equivalent of the size effects discussed in the first part of the review. Here this feature is treated employing the same statistical mechanics framework developed for the first problem

Multi-Grid Reaction-Diffusion Master Equation: Applications to Morphogen Gradient Modelling

The multi-grid reaction-diffusion master equation (mgRDME) provides a generalization of stochastic compartment-based reaction-diffusion modelling described by the standard reaction-diffusion master equation (RDME). By enabling different resolutions on lattices for biochemical species with different diffusion constants, the mgRDME approach improves both accuracy and efficiency of compartment-based reaction-diffusion simulations. The mgRDME framework is examined through its application to morphogen gradient formation in stochastic reaction-diffusion scenarios, using both an analytically tractable first-order reaction network and a model with a second-order reaction. The results obtained by the mgRDME modelling are compared with the standard RDME model and with the (more detailed) particle-based Brownian dynamics simulations. The dependence of error and numerical cost on the compartment sizes is defined and investigated through a multi-objective optimization problem

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