Modern Mathematical Physics

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 $\sigma$ 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 $\lambda$ 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.

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.

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.