Modern Mathematical Physics

We develop a balance-first framework for nonequilibrium thermodynamics in which entropy production follows directly from macroscopic conservation laws, without assuming the Gibbs relation, local equilibrium, or a predefined temperature field. Entropy flux and production emerge naturally from the flux–force structure, with non-negativity ensured by dissipative admissibility. Substituting microscopic currents from the Fokker–Planck equation recovers the canonical entropy production laws for both overdamped and underdamped Brownian motions, which demonstrates that macroscopic and stochastic descriptions share identical production and extraction identities. The framework further quantifies thermodynamic distances and establishes rigorous bounds linking current, activity, and dissipation, providing a unifying route from continuum balance to stochastic irreversibility. Thus, entropy appears as an emergent, gauge-invariant quantity that connects transport, distance, and control in nonequilibrium systems.

The nonlinear Schrödinger equation is a classical nonlinear evolution equation with wide applications. This paper explores the asymptotic behavior of solutions to the nonlinear Schrödinger equation with non-zero boundary conditions in the presence of a pair of second-order discrete spectra. We analyze the Riemann–Hilbert problem in the inverse scattering transform by the Deift–Zhou nonlinear steepest descent method. Then we propose a proper deformation to deal with the growing time term and give the conditions for the series in the process of deformation by the Laurent expansion. Finally, we provide the characterization of the interactions between the solitary waves corresponding to second-order discrete spectra and the coherent oscillations produced by the perturbation. Numerical verifications are also performed.

In this paper we present a brief review of extended general relativity in four dimensions and solve versions of the extended equations for the case of static spherical symmetry in various contexts, for a previously studied Lagrangian. The exterior vacuum yields a Schwarzschild solution with an additional scalar field potential that falls off logarithmically, the latter essentially an inverse square force. That is probably not adequate as a dark matter force, but might contribute. When a constant density field of ions holds sway in the exterior, a solution identical to the cosmological constant extension of Schwarzschild occurs, together with a scalar field potential declining as $r^{-3/2}$, however it is not asymptotically flat. An inverse square declining distribution of ionic material, according to perturbation theory, results in an additional linear gravity potential that would provide further attraction in the gravity term. A limited exact solution in the same case yields a cubic equation with a Schwarzschild solution, corresponding to $A=0$, and two MOND-like possible potentials, one vanishing at infinity, but a better solution must be found. The approximate solution is complex (one of many) and the system requires further study. Ionic matter is ubiquitous in the universe and provides a source for the scalar field, which suggests that the extended Einstein equations could be of utility in the dark matter problem, provided such an electromagnetic scalar force could be found and differentiated from the usual, far stronger electromagnetic forces. Further, it’s possible that the strong photon flux outside stars might have an influence, and is under current investigation. These calculations show that extending the concept of curvature and working in four dimensions with larger operators may bring new tools to the study of physics and unified field theories.