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From the perspective of mathematical epidemiology, COVID-19 epidemics emerge due to instabilities in epidemiological systems. It is shown that the COVID-19 outbreaks follow highly specified paths in epidemiological state spaces. These paths are described by phase portraits that can be readily computed from epidemiological models defined in terms of nonlinear dynamical systems. The paths are predicted by order parameters and amplitude equations that are well known in nonlinear physics and synergetics to exist at instability points. The approach is illustrated for SIR, SEIR and SEIAR models and epidemic outbreaks in China, Italy and West Africa. Identifying such COVID-19 order parameters may help in forecasting COVID-19 epidemics and predicting the impacts of intervention measures.

The density functional theory for the freezing hard spheres is studied. We use a variety of the hard sphere direct correlation functions (DCFs) such as the one introduced by Roth et al. [J. Phys. Condens. Matter14, 12063 (2002)]; we call it RELK DCF, a new hard sphere DCF developed here by a combination of the RELK and the Percus–Yevick DCFs, and finally the generalized mean spherical approximation (GMSA). The structure factor, the freezing, and order parameters are calculated using these DCFs. The structure factor obtained by the new DCF is in good agreement with the Monte Carlo simulation. The best result for the freezing parameters in comparison with the Monte Carlo simulations is obtained by using our new expression for the DCF. Finally we obtain the Helmholtz free energy of the hard sphere FCC crystals using modified weighted density approximation (MWDA), and again the best results are obtained by using the new expression for the hard sphere DCF.

A survey of recent studies of biaxial liquid crystals (LCs), whose nematic and/or smectic-A phases do not possess optical uniaxiality (viz., more than one optical axis exists), is given in this review. In particular, we emphasize on how Nuclear Magnetic Resonance (NMR) spectroscopy can help to advance the understanding of phase biaxiality in general, and to examine recent debates on the existence of biaxial nematic phase reported in low molecular mass bent-core or V-shaped mesogens. A general discussion of orientational order parameters is detailed, particularly in smectic-C (SmC) and biaxial nematic phases. How these orientational order parameters can be determined by various techniques such as NMR, IR absorbance and Raman scattering studies, will be mentioned. Recent X-ray observations of SmC clusters in the nematic phase of V-shaped mesogens are highlighted and contrasted with probable theory. Moreover, deuterium and carbon-13 NMR techniques are briefly reviewed, and their possible utilization to identify phase biaxiality in these biaxial LC systems is explored.

We study the two-component Ginzburg–Landau (GL) theory, in the presence of a self-consistent vortex line, to obtain the penetration depth (λ) and the effective healing length (ξ), in the asymptotic limit r → ∞. All these parameters versus T/T_c are analyzed for the materials MgB₂, V₃Si and LiFeAs in the interval 0.88 ≤ T/T_c ≤ 1.0, where the GL theory is assumed to be valid. We find that κ ≡ λ/ξ, which is another parameter not related to the GL parameter, is T-independent for V₃Si and LiFeAs, while is T-dependent for the compound MgB₂. This result suggests that even though all these three materials display two-gap superconductivity overall, near T_c superconductivity in V₃Si and LiFeAs seems to be different from the one in MgB₂. The use of this parameter, κ, as a new way to "study" the superconducting materials, under the presence of a single vortex, is valid for 0.88T_c≲ T ≤ T_c, namely, in the GL formalism.

The thermodynamics of solid–solid and solid–liquid transitions are investigated with an account of the number of molecular orientation. The variations of the positional and orientational orders with the reduced temperature are studied. It is found out that orientational order parameter is very sensitive to the number of allowed orientation. The reduced transition temperatures, volume changes and entropy changes of the phase transitions and theoretical phase diagrams are obtained. The entropy changes of melting transitions for different numbers of allowed orientation of the present model are compared with the theoretical results and some experimental data. The quantitative predictions of the model are compared with experimental results for plastic crystals and agreement between predictions of the model and the experimental results are approximately good. Also, different numbers of allowed orientation D correspond to different experimental results HI, HBr, H₂S for D = 2; HBr, CCl₄, HI for D = 4; C₂H₁₂ for D = 6; CH₄, PH₃ for D = 20.

A two-band and quartic interaction order Ginzburg–Landau model in the presence of a single vortex is studied in this work. Interactions of second (quadratic, with coupling parameter $\gamma$) and fourth (quartic, with coupling parameter $\tilde{\gamma }$) order between the two superconducting order parameters ($f_i$ with i = 1,2) are incorporated in a functional. Terms beyond quadratic gradient contributions are neglected in the corresponding minimized free energy. The solution of the system of coupled equations is solved by numerical methods to obtain the $f_i$-profiles, where our starting point was the calculation of the superconducting critical temperature $T_c$. With this at hand, we evaluate $f_i$ and the magnetic field along the z-axis, $B_0$, as function of $\gamma$, $\tilde{\gamma }$, the radial distance $r/{\lambda }_1(0)$ and the temperature $T$, for $T\approx T_c$. The self-consistent equations allow us to compute $\lambda$ (penetration depth) and the healing lengths of $f_i$ ($L_{h_i}$ with i = 1,2) as functions of T, $\gamma$ and $\tilde{\gamma }$. At the end, relevant discussions about type-1.5 superconductivity in the compounds we have studied are presented.

In this work, the ternary alloy (TA) of the form $A{B}_p{C}_{1-p}$ with spin-$\frac{3}{2}$, spin-2 and spin-$\frac{5}{2}$, respectively, is studied on the Bethe lattice in terms of exact recursion relations in the standard random approach. The bilinear interaction parameter $J_{AB}$ is assumed to be ferromagnetic between the nearest-neighbor spins with spin-$\frac{3}{2}$ and spin-2, while $J_{AC}$ is taken to be antiferromagnetic between spin-$\frac{3}{2}$ and spin-$\frac{5}{2}$. The possible phase diagrams are obtained from the thermal analysis of the order parameters for the given coordination numbers z = 3,4,5 and 6. This analysis has also revealed that the model gives both second- and first-order phase transitions in addition to the compensation temperatures.

From a dynamical systems perspective, COVID-19 infectious disease emerges via an instability in human populations. Accordingly, the human population free of COVID-19 infected individuals is an unstable state and the dynamics away from that unstable state is a bifurcation. Recent research has determined COVID-19 relevant bifurcation parameters for various countries in terms of basic reproduction ratios. However, little is known about the relevant order parameters (synergetics) of COVID-19 bifurcations and the corresponding time constants. Those order parameters describe directions in compartment model spaces in which infection dynamics initially evolves. The corresponding time constants describe the speed of the dynamics along those directions. COVID-19 order parameters and their time constants are derived within a standard SEIR dynamical systems framework and determined explicitly for two published studies on COVID-19 trajectories in Italy and China. The results suggest the existence of certain relationships between order parameters, time constants, and reproduction ratios. However, the examples from Italy and China also suggest that COVID-19 order parameters and time constants in general depend on regional differences and the stage of the local COVID-19 epidemic under consideration. These findings may help to improve the forecasting of COVID-19 outbreaks in new hotspots around the world.

Several recent modeling studies have attempted to understand the human immune reaction to SARS-CoV-2 infections. Such a model-based understanding provides a sound basis for fighting COVID-19 on the level of individual patients. However, in this context, a worked-out nonlinear physics analysis providing insights into underlying amplitude equations and potential COVID-19 order parameters has not been conducted so far. In order to conduct such an analysis, a three-variable virus dynamics model is considered that can account for the human immune reaction to SARS-CoV-2 infections. The model amplitudes equations are derived and the relevant order parameter is determined. In line with theoretical reasoning, it is demonstrated that the order parameter and its amplitude determine the initial stage of SARS-CoV-2 infections, in general, and the initial dynamics of immune reactions, in particular. Explicitly, this finding is demonstrated for data from four COVID-19 patients. For those patients it is also demonstrated that the remnant of the order parameter determines the final disease decline phase. In this context, a time-resolved eigenvalue analysis is conducted that reveals that the transition from the initial stage to the decline stage which is associated with a switch of the leading eigenvalue from a positive to a negative value. It is argued that the immune reaction essentially contributes to this switch. From a medical-physics point of view, this observation suggests that the immune reaction of COVID-19 patients can stabilize the virus-free fixed point of affected sites.