Cascades of Scales: Applications and Mathematical Methodologies

Abstract

The present special collection presents reviews and genuine research articles addressing real-life applications involving many interacting scales and related methodological developments. Specifically, applications from the biomolecular, material, atmospheric, and geophysical sciences form the core of the dedicated collection, while this does not exclude selected ventures into other areas. Of particular interest here are problems involving couplings across their scales from small to large and vice versa,1–3 and mathematical concepts and methodologies that have proven to be transferable from specific applications to a generalized framework guided by mathematical abstraction.4–9 In many cases, these abstractions result in related computational codes and software libraries for specific applications of interest to the different disciplines involved.10–13 Being, in this way, inherently interdisciplinary in nature, contributions to this volume give high priority to explaining relations between the application-oriented domain-specific languages used to frame their scientific problem in an application of mathematical physics on the one hand, and the complementary unifying abstract mathematical language that enables the generalization of methodologies to other fields on the other hand. Thus, the mission of the special issue is two-pronged: It reports on exciting recent methodological developments for challenging problems in mathematical physics involving cascades of interacting scales, and it showcases the importance of precise language(s) and the power abstraction in fostering efficient interdisciplinary research. May it become a well-visited point of reference and point of departure for further conceptual developments. With solid foundations in concrete applications and phrased in the general language of mathematics, they may gain substantial momentum beyond their original discipline of application. The contributions are published as regular articles of the Journal of Mathematical Physics, while they are linked at the same time in the present themed (online) collection. Physical systems featuring multiple spatio-temporal scales have posed notoriously difficult challenges to theory and mathematical analysis in past decades, as witnessed, e.g., by a myriad of articles published in the present Journal of Mathematical Physics (JMP), by the establishment in 2003 of the Journal of Multiscale Modeling and Simulation of the Society of Industrial and Applied Mathematics (SIAM), or the references listed below. For a long time, the complexities of real-life physical problems were beyond the reach of mathematical analysis, both formal and rigorous. Thus, theoreticians developed and honed their tools and techniques on physically motivated but substantially simplified models, while researchers from the applied sciences (Physics, Chemistry, Biology, and the Geo-Sciences) pragmatically bridged gaps in theoretical knowledge by well-informed and intuitive closure schemes that would address the ubiquitous multiscale-effects. Over the past 1 1/2 decades, however, mathematical/theoretical multiscale techniques have matured to levels which increasingly allow their transfer to real-life applications, see, e.g., Refs. 4, 6, 7, 14, and 15. In addition, the wealth of observational data available today for physical processes from laboratory setups to the Earth system jointly with the exciting capabilities of recent machine learning techniques have lended tremendous thrust to data-based modeling activities.16–18 The recent rapid rise of data-based modeling in the physical and chemical sciences offers a promising path to accessing a large class of problems across scales. This path comes with the danger, however, of artificial results due to insufficient and/or corrupted data—especially in the face of the curse of dimensionality. For this reason, current research calls strongly for physical models that must be as rigorous as possible to ensure scientific consistency and avoidance of any misuse of data (see, e.g., Refs. 17 and 18). One aim of the present collection is to highlight this aspect, among the others, and build a path to a sustainable research based also on data but with the control of rigor and consistency of physico-mathematical models. These and similar developments have motivated JMP’s special collection on “Cascades of Scales: Applications and Mathematical Methodologies” of which we provide an overview in this paper. A very important aspect in this context, besides impressive recent methodological advances and particularly successful real-life applications, concerns the challenges of interdisciplinary cooperation. To a large extent, these challenges are due to the fact that the scientific disciplines participating in a project rely on fine-tuned domain-specific languages for efficient communication. This is compounded by often similar vocabulary, which carries different meanings in different disciplines. A prominent example is the notion of “multiscale” itself, which, depending on the scientific community considered, can be associated with “broad and continuous Fourier spectra” (e.g., in turbulence theory), with the presence of large (asymptotic) scale separations [e.g., in (stiff) chemistry or geophysical fluid dynamics], or with non-asymptotic metastabilities (e.g., in molecular dynamics). Progress in interdisciplinary projects is often hampered by subtle misunderstandings arising from such discipline-dependent “concept overloading.” The authors who have contributed to the present special collection of papers have taken particular care to use precise and unambiguous language that will hopefully help make their contributions efficiently accessible to the readers of the Journal of Mathematical Physics.