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Phase Transitions
Published in Jeffrey Olafsen, Sturge’s Statistical and Thermal Physics, 2019
In Chapter 4, we found that in a simple noninteracting system the relative fluctuations of any macroscopic quantity about its mean are extremely small, of order N−1/2 where N is the number of particles. We then developed thermodynamics and its statistical underpinning on the assumption that the systems with which we are dealing have well-defined average properties, with small fluctuations about the mean. In this chapter, we have found that this assumption ceases to hold at a critical point, where fluctuations become so large that one can no longer talk in terms of average properties. While some thermodynamic quantities remain well defined (e.g., temperature, since it is defined by the heat bath and not by the system under study), other quantities such as the density do not, and a new approach is needed. We have come full circle, to the point where a reexamination of the very basis of the theory is required. This has been the subject of active research over the past thirty years, and there is now a well established theory of critical phenomena based on the renormalization group. This theory is beyond the scope of this book.30
A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation
Published in Applicable Analysis, 2019
Zongbiao Zhang, Meng Li, Zhongchi Wang
The classical (integer) complex Ginzburg–Landau type equation, which is proposed to describe a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity and Bose-Einstein condensation to liquid crystals and strings in field theory, is a kind of important nonlinear evolution equation in the physics community [1]. During the last few years, the interest to fractional equations has been growing continually because of numerous applications. For example, Tarasov et al. [2,3] derived the fractional generalization of Ginzburg–Landau equation from the variational Euler-Lagrange equation for fractal media. Since the fractals can be realized in nature as a fractal process or fractal media, there are wide range of applications for the fractional Ginzburg–Landau equation (FGLE) in physical phenomena, such as the dynamical processes in a medium with fractal dispersion [2], a fairly general class of critical phenomena when the organization of the system near the phase transition point is influenced by a competing nonlocal ordering [4] and a network of diffusively Hindmarsh-Rose (HR) neurons with long-range synaptic coupling [5].
Effective field study of the magnetism and superconductivity in idealised Ising-type X@Y60 endohedral fullerene system
Published in Philosophical Magazine, 2019
We should also mention that the nanomaterials with core–shell structure have been utilised for applications in diverse fields such as drug delivery [29], ultra-high density magnetic recording media [30–32], sensors [33], environmental remediation [34], nonlinear optics [35], bio-molecular motors [36] and permanent magnets [37]. Therefore, nanomaterials with core–shell have been studied in various experimental and theoretical investigations [38–45]. The magnetic properties of nanostructured core–shell systems have been also studied by the Ising model with different methods such as effective-field theory (EFT), mean-field approximation (MFA), and Monte Carlo simulations (MCS) [46–60]. Ising model is a well-known and widely used pioneering theoretical model on the magnetic systems and plays an important role in the deeper understanding of phase transitions and critical phenomena. In order to simulate the behaviours of many complex systems, such as magnetic nanoparticle systems the MCS is a strong numerical approach. Because of long calculation times originating from the exhausting sampling averaging procedures, it needs a large computer opportunity. Hence, this method has a disadvantage. Moreover, due to its mathematical simplicity, the EFT is considered to be quite superior to conventional MFT since considers partially the spin–spin correlations as a result it gives more accurate results than the MFT in the calculations. On the other hand, the experimental systems have been modelled and investigated by using Ising models with various methods [61–66]. The magnetic and hysteresis behaviours in the experimental works are reported in quantitatively good agreement with Ising model.
General power laws of the causalities in the causal Bayesian networks
Published in International Journal of General Systems, 2023
Boyuan Li, Xiaoyang Li, Zhaoxing Tian, Xia Lu, Rui Kang
The critical phenomena and phase transitions are the typical features of the real complex systems, and usually indicate the dynamic self-organization and self-adaptation under the complicated interactions. Further, the power law has been found to effectively represent the critical phenomena and phase transitions, which has been studied in biosystems like the human brain (Kitzbichler et al. 2009; Srivastava, Sahni, and Satsangi 2016), artificial systems like the power grids (Yan et al. 2020; Peng 2018), and even social systems such as population migration and social media (Gao, Ding, and Ying 2006; Xu et al. 2019; Zhang et al. 2023; Ellouze, Mechti, and Belguith 2023).