Related papers: Multiscale functions, Scale dynamics and Applicati…
Multiscale thermodynamics is a theory of relations among levels of investigation of complex systems. It includes the classical equilibrium thermodynamics as a special case but it is applicable to both static and time evolving processes in…
The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since…
The dynamics of any spherical cosmology with a scalar field (`scalaron') coupling to gravity is described by the nonlinear second-order differential equations for two metric functions and the scalaron depending on the `time' parameter. The…
This paper introduces a novel data driven framework for constructing accurate and general equivariant models of multiscale phenomena which does not rely on specific assumptions about the underlying physics. This framework is illustrated…
Scaling properties of time series are usually studied in terms of the scaling laws of empirical moments, which are the time average estimates of moments of the dynamic variable. Nonlinearities in the scaling function of empirical moments…
We extend the phase field crystal model to accommodate exact atomic configurations and vacancies by requiring the order parameter to be non-negative. The resulting theory dictates the number of atoms and describes the motion of each of…
Different methods are used to determine the scaling exponents associated with a time series describing a complex dynamical process, such as those observed in geophysical systems. Many of these methods are based on the numerical evaluation…
The Herglotz problem is a generalization of the fundamental problem of the calculus of variations. In this paper, we consider a class of non-differentiable functions, where the dynamics is described by a scale derivative. Necessary…
In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their…
Conventional weak-coupling Rayleigh-Schr\"odinger perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale analysis, a powerful and sophisticated…
Scaling ideas and renormalization group approaches proved crucial for a deep understanding and classification of critical phenomena in thermal equilibrium. Over the past decades, these powerful conceptual and mathematical tools were…
We adopt the so--called \emph{occupation number representation}, originally used in quantum mechanics and recently adopted in the description of several classical systems, in the analysis of the dynamics of some models of closed ecosystems.…
Field equations with time and coordinates derivatives of noninteger order are derived from stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a…
A central notion of physics is the rate of change. While mathematically the concept of derivative represents an idealization of the linear growth, power law types of non-linearities even in noiseless physical signals cause derivative…
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…
There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both…
Modeling of fluid flows requires corresponding adequate and effective approaches that would account for multiscale nature of the considered physics. Despite the tremendous growth of computational power in the past decades, modeling of fluid…
The rich and diverse dynamics of particle-based systems ultimately originates from the coupling of their degrees of freedom via internal interactions. To arrive at a tractable approximation of such many-body problems, coarse-graining is…
We derive backward and forward fractional Schr\"odinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and…
The notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order. This type of equation appears in the description of chaotic dynamics, wave propagation in fractal media, and…