Conservation Laws and Boundary Value Problems in Far-from-Equilibrium Dynamics

Far-from-equilibrium dynamics controls a broad range of processes in nature and technology, from celestial events to atoms. Examples include supernovae, climate change, turbulence, nanofabrication and fusion. Far-from-equilibrium dynamics is a challenge to study in theory, simulations, and data science. Analytically, one needs to solve the conservation laws in the bulk augmented with singular boundary value problem at a freely evolving discontinuity and with ill-posed initial value problem. Their numerical simulations encounter tight requirements on the method accuracy, computational precision and the span of spatial and temporal scales. In data science, these processes are challenging to lucidly capture.

Far-from-equilibrium dynamics, interfaces and inter-facial mixing couple kinetic to continuous scales. These processes are inhomogeneous, anisotropic, non-local, and statistically unsteady. Their dynamics often involve sharply and rapidly changing fields, and may also include accelerations and shocks, diffusion and radiation, among other effects. At macro scales, their scaling and spectral laws differ from those in canonical turbulence. At micro scales, their dynamics depart dramatically from the standard Gibbs ensemble averages and the quasi-static Boltzmann equation. At the same time, these processes may lead to self-organization and order. This opens novel perspectives for capturing symmetries of non-equilibrium dynamics, exploring invariant forms of interfacial mixing, and enabling solution of boundary value and initial value problems. It can further advance methods of analytical, numerical and data modeling of realistic complexity. Significant success has recently been achieved in understanding far-from-equilibrium dynamics for theoretical analysis, numerical simulations, and data analysis. It is now an exciting time to explore the fundamentals of these complex processes, and to apply this knowledge to address challenges in modern mathematics, science and technology.

This Research Topic aims to explore and assess the state of the art in far-from-equilibrium dynamics and to chart future research directions. It offers the unique opportunity to bring together researchers from different areas of mathematical sciences and motivate discussions on analytical approaches and state-of-the-art numerical simulations, along with advanced data analysis techniques. The scope of the Research Topic is shaped by invited lectures from the 2025 Joint Mathematics Meeting in the United States of America and the 2023 International Congress on Industrial and Applied Mathematics in Japan.

Contributions are cordially invited from both experienced and early career-stage researchers, spanning a broad range of disciplines in mathematical sciences, physical sciences, engineering and technology. Original Research, Perspective, and Review articles are particularly welcome.

Article types and fees

This Research Topic accepts the following article types, unless otherwise specified in the Research Topic description:

• Brief Research Report
• Curriculum, Instruction, and Pedagogy
• Data Report
• Editorial
• FAIR² Data
• FAIR² DATA Direct Submission
• General Commentary
• Hypothesis and Theory
• Methods
• ... View all formats

Articles that are accepted for publication by our external editors following rigorous peer review incur a publishing fee charged to Authors, institutions, or funders.

Article types

This Research Topic accepts the following article types, unless otherwise specified in the Research Topic description:

• Brief Research Report
• Curriculum, Instruction, and Pedagogy
• Data Report
• Editorial
• FAIR² Data
• FAIR² DATA Direct Submission
• General Commentary
• Hypothesis and Theory
• Methods
• Mini Review
• Opinion
• Original Research
• Perspective
• Review
• Technology and Code

Keywords: far-from-equilibrium dynamics, conservation laws, boundary value problems, interface dynamics, interfacial mixing, self-similar dynamics, scale-dependent dynamics, nonlinear processes, non-local processes, unstable and statistically unsteady processes

Important note: All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.