Modern Mathematical Physics doi: 10.3390/mmphys2020005

Authors: Jorge A. González Salvador Jiménez Alberto J. Bellorín Leonardo Reyes

Long-range topological objects can exist in many physical systems, and they can tunnel through very wide barriers. Thus, the propagation of long-range kink-like objects through disordered media can be extremely enhanced. When the potential is asymmetric, the long-range kink-like excitations can enter a regime of superpropagation, where, essentially, they can move through almost any disordered medium. We believe these phenomena can find applications in macroscopic quantum technologies (including robust qubits), energy devices for energy harvesting and storage, and high-Tc superconductivity in hydrides. We expect that many of these results can be generalized to other topological objects, e.g., fluxons, domain walls, skyrmions, topological defects, stripes, textures, dislocations in crystals, strings, monopoles, instantons, vortices, and spiral waves.

Modern Mathematical Physics doi: 10.3390/mmphys2020004

Authors: Kazuma Morimoto Hiroshi Kobayashi Hiroyuki Shima

Massive-scale, ultra-high-resolution numerical simulations for climate change prediction provide data of exceptional accuracy and reliability. However, this comes at the cost of enormous computational resources, and the underlying processes often remain a “black box”. In contrast to these sophisticated methods, we theoretically analyzed the water vapor feedback effect using a highly simplified model that focuses exclusively on the most critical physical factors governing climate change. Specifically, we formulated a two-layer box model by dividing the entire atmosphere into layers of equal optical thickness. Using this model, we quantitatively verified the extent to which the water vapor feedback effect—a key driver of global warming—can be theoretically reproduced.

Modern Mathematical Physics doi: 10.3390/mmphys2010003

Authors: Aritro Mukherjee

Quantum stochastic processes are widely used in describing open quantum systems and in the context of quantum foundations. Physically relevant quantum stochastic processes driven by multiplicative colored noise are generically non-Markovian and analytically intractable. Further, their Markovian limits are generically inequivalent when using either the Ito or Stratonovich conventions for the same quantum stochastic processes. We introduce a quantum noise homogenization scheme that temporally coarse-grains non-Markovian, colored-noise-driven quantum stochastic processes and connects them to their effective white-noise (Markovian) limits. Our approach uses a novel phase-space augmentation that maps the non-Markovian dynamics into a higher-dimensional Markovian system and then applies a controlled perturbative coarse-graining scheme in the characteristic time scales of the noise. This allows an explicit analytical algorithm to derive effective Markovian generators with renormalized coefficients and enables various physical constraints, such as causality, to be imposed on them. We thus resolve the Ito–Stratonovich ambiguity for multiplicative colored-noise-driven quantum stochastic processes, wherein we show that their consistent Markovian limit corresponds to the Stratonovich convention with renormalized coefficients as well as correction terms in Ito’s convention. By assuming their Markovian limit unravels causal, completely positive and trace-preserving dynamics, we further characterize a physically relevant family of non-Markovian quantum stochastic processes driven by multiplicative colored noise.

Modern Mathematical Physics doi: 10.3390/mmphys2010002

Authors: Panos D. Karageorge George N. Makrakis

We present a construction of the anisotropic Gaussian semi-classical Schrödinger propagator, emblematic of a class of Fourier integral operators of quadratic phase kernels related to the Schrödinger equation. We deduce a set of algebraic relations of the variational matrices, solutions of the variational system pertaining to single Gaussian wave packet semi-classical time evolution, some already known in the literature, representing the symplectic and other invariances of the dynamics, which are subsequently utilized in order to derive the Van Vleck formula from the semi-classical Schrödinger propagator.

Modern Mathematical Physics doi: 10.3390/mmphys2010001

Authors: Mesfin Asfaw Taye

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.

Modern Mathematical Physics doi: 10.3390/mmphys1030011

Authors: Yoonseok (John) Chae

We survey the recent developments on quantum invariants of 3-manifolds and links: Z^ and FL. They are q-series invariants originated from mathematical physics, inspired by the categorification of a numerical quantum invariant—the Witten–Reshetikhin–Turaev (WRT) invariant—of 3-manifolds. They exhibit rich features, for example, quantum modularity, infinite-dimensional Verma module structures, and knot–quiver correspondence. Furthermore, they have connections to the 3d-3d correspondence and other topological invariants. We also provide a review of an extension of the above series invariants to Lie superalgebras.

Modern Mathematical Physics doi: 10.3390/mmphys1030010

Authors: Bonan Wang Chenxi Zheng Shaoqiang Tang

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.

Modern Mathematical Physics doi: 10.3390/mmphys1030009

Authors: Chris Vuille Andrei Ludu

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.

Modern Mathematical Physics doi: 10.3390/mmphys1020008

Authors: Gro Hovhannisyan

The integrable versions of SIR epidemic models are introduced. The exact solutions of these models are derived. The advantage of these models is the possibility of full analysis of obtained solutions and the simplicity of explicit formulas for the important metrics of spread of disease. The effectiveness of these formulas is illustrated by applications to the spread of COVID-19.

Modern Mathematical Physics doi: 10.3390/mmphys1020007

Authors: Jun Yan

The Fermi condensation flows in the sine-Gordon–Thirring model with two impurities coupling are investigated in this paper; these matter flows can be induced by the Ricci flow perturbation in the two-dimensional string σ model. The Ricci flow perturbation equations are derived according to the Gauss–Codazzi equations, and the two-loop asymptotic perturbation solution of the cigar soliton is reduced by using a small parameter expansion method. Moreover, the thermodynamic quantities on the cigar soliton background are obtained by using the variational functional integrals method. Subsequently, the Fermi condensation flows varying with the momentum scale λ are analyzed and discussed. We find that the energy density, the correlation function, and the condensation fluctuations will decrease, but the entropy will increase monotonically. The Fermi condensed matter can maintain thermodynamic stability under the Ricci flow perturbation.

Modern Mathematical Physics doi: 10.3390/mmphys1020006

Authors: Andras Kovacs

Bose–Einstein condensation is an intensely studied quantum phenomenon that emerges at low temperatures. While preceding Bose–Einstein condensation models do not consider what statistics apply above the condensation temperature, we show that neglecting this question leads to inconsistencies. A mathematically rigorous calculation of Bose–Einstein condensation temperature requires evaluating the thermodynamic balance between coherent and incoherent particle populations. The first part of this work develops such an improved Bose–Einstein condensation temperature calculation, for both three-dimensional and two-dimensional scenarios. The progress over preceding Bose–Einstein condensation models is particularly apparent in the two-dimensional case, where preceding models run into mathematical divergence. In the Discussion section, we compare our mathematical model against experimental superconductivity data. A remarkable match is found between experimental data and the calculated Bose–Einstein condensation temperature formulas. Our mathematical model therefore appears applicable to superconductivity, and may facilitate a rational search for higher-temperature superconductors.

Modern Mathematical Physics doi: 10.3390/mmphys1010005

Authors: Călin-Ioan Gheorghiu

This paper concerns accurate spectral collocation solutions, more precisely Chebyshev collocation (ChC), to some third-order nonlinear and singular boundary value problems on unbounded domains. The problems model some draining or coating fluid flows. We use exclusively ChC, in the form of Chebfun, avoid any obsolete shooting-type method, and provide reliable information about the convergence and accuracy of the method, including the order of Newton’s method involved in solving the nonlinear algebraic systems. As a complete novelty, we combine a graphical representation of the convergence of the Newton method with a numerical estimate of its order of convergence for a more realistic value. We treat five challenging examples, some of which have only been solved by approximate methods. The found numerical results are judged in the context of existing ones; at least from a qualitative point of view, they look reasonable.

Modern Mathematical Physics doi: 10.3390/mmphys1010004

Authors: Alexander D. Popov

Non-relativistic quantum mechanics was originally formulated to describe particles. Using ideas from the geometric quantization approach, we show how the concept of antiparticles can and should be introduced in the non-relativistic case without appealing to quantum field theory. We discuss this in detail using the example of the one-dimensional harmonic oscillator.

Modern Mathematical Physics doi: 10.3390/mmphys1010003

Authors: Jian-Xin Lu

In this paper, we address how to implement T-duality to the closed string tree cylinder amplitude between a Dp brane and a Dp′ brane with p−p′=2n. To achieve this, we compute the closed string tree cylinder amplitude, for the first time, between these two D-branes with common longitudinal and transverse circle compactifications. We then show explicitly how to perform a T-duality for this amplitude along either a longitudinal or a transverse compactified direction to both branes. At the decompactification limit, we show that either the compactified cylinder amplitude or the T dual compactified cylinder gives the known non-compactified one as expected.

Modern Mathematical Physics doi: 10.3390/mmphys1010002

Authors: Chang-Pu Sun Murray Batchelor Yupeng Wang

It is our great pleasure to be involved in the launch of Modern Mathematical Physics (MMP) [...]

Modern Mathematical Physics doi: 10.3390/mmphys1010001

Authors: Lin Li

On 6 June 2024, we had the privilege of visiting Professor Chang-Pu Sun at the China Academy of Engineering Physics [...]