Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives

There has been increasing research activity in recent years concerning the properties and the applications of nonlinear partial differential equations that are closely related to nonstandard entropic functionals, such as the Tsallis and Renyi entropies. It is well known that some fundamental partial differential equations of applied mathematics and of mathematical physics—such as the linear diffusion equation—are closely linked to the standard, logarithmic Boltzmann–Gibbs–Shannon–Jaynes entropic measure. This link can be extended to the realm of nonlinear partial differential equations via the aforementioned nonstandard (or generalized) entropic functionals. In particular, nonlinear diffusion and Fokker–Planck equations endowed with power-law nonlinearities in the diffusion term (Laplacian term) admit exact time-dependent solutions exhibiting the form of Tsallis maximum entropy distributions, where the Tsallis measure is optimized under a small number of appropriate constraints. The connection between nonlinear Fokker–Planck equations and nonstandard entropies proved to be a remarkably fertile field of research, leading to a set of ideas and techniques that has been successfully applied to the analysis of a variegated family of systems or processes in physics, biology, and other areas. As examples we can mention applications to the behavior of granular media and to the study of the motility of biological microorganisms. Most of the above developments concern purely classical (as opposed to quantum mechanical) concepts. However, new results arising in recent years have made it clear that the connection between nonstandard entropies and nonlinear partial differential equations can also be extended to new nonlinear wave equations inspired on quantum mechanics. Nonlinear versions of the celebrated Klein–Gordon and Dirac equations have been discovered that admit exact time-dependent soliton-like solutions having the forms of the so-called q-plane waves. These q-plane waves are power-law generalizations of the complex exponential plane wave solutions of the linear Klein–Gordon and Dirac equations. The q-plane waves are complex-valued counterparts of the real-valued functions arising from the generalized Tsallis or Renyi maximum entropy principles. The aforementioned nonlinear extensions of the relativistic wave equations exhibit interesting physico-mathematical properties that have been explored in the recent research literature. Non-relativistic complex wave equations have also been incorporated to these research efforts, in terms of a parameterized family of power-law nonlinear Schrödinger equations. The field of research discussed above is currently experiencing rapid development and is giving rise to new lines of enquiry, in many cases of a multi-disciplinary character. It thus seems necessary to devote a special issue of the Journal Entropy to these subjects. It would constitute a very useful, timely, and unique contribution to the current scientific literature. It should stimulate further research in the field, and we can expect it to be highly cited. A brief outline of the present ten contributions follows, and makes for exciting reading.