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Register a new userClassical evolution of fractal measures on the lattice
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We consider the classical evolution of a lattice of non-linear coupled oscillators for a special case of initial conditions resembling the equilibrium state of a macroscopic thermal system at the critical point. The displacements of the oscillators define initially a fractal measure on the lattice associated with the scaling properties of the order parameter fluctuations in the corresponding critical system. Assuming a sudden symmetry breaking (quench), leading to a change in the equilibrium position of each oscillator, we investigate in some detail the deformation of the initial fractal geometry as time evolves. In particular we show that traces of the critical fractal measure can sustain for large times and we extract the properties of the chain which determine the associated time-scales. Our analysis applies generally to critical systems for which, after a slow developing phase where equilibrium conditions are justified, a rapid evolution, induced by a sudden symmetry breaking, emerges in time scales much shorter than the corresponding relaxation or observation time. In particular, it can be used in the fireball evolution in a heavy-ion collision experiment, where the QCD critical point emerges, or in the study of evolving fractals of astrophysical and cosmological scales, and may lead to determination of the initial critical properties of the Universe through observations in the symmetry broken phase.
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We investigate the classical evolution of a $phi^4$ scalar field theory, using in the initial state random field configurations possessing a fractal measure expressed by a non-integer mass dimension. These configurations resemble the equilibrium state of a critical scalar condensate. The measures of the initial fractal behavior vary in time following the mean field motion. We show that the remnants of the original fractal geometry survive and leave an imprint in the system time averaged observables, even for large times compared to the approximate oscillation period of the mean field, determined by the model parameters. This behavior becomes more transparent in the evolution of a deterministic Cantor-like scalar field configuration. We extend our study to the case of two interacting scalar fields, and we find qualitatively similar results. Therefore, our analysis indicates that the geometrical properties of a critical system initially at equilibrium could sustain for several periods of the field oscillations in the phase of non-equilibrium evolution.
Anomalous short- and long-time self-diffusion of non-overlapping fractal particles on a percolation cluster with spreading dimension $1.67(2)$ is studied by dynamic Monte Carlo simulations. As reported in Phys. Rev. Lett. 115, 097801 (2015), the disordered phase formed by these particles is that of an unconfined, homogeneous and monodisperse fluid in fractal space. During particle diffusion in thermodynamic equilibrium, the mean squared chemical displacement increases as a nonlinear power of time, with an exponent of $0.96(1)$ at short times and $0.63(1)$ at long times. At finite packing fractions the steric hindrance among nearest neighbor particles leads to a sub-diffusive regime that separates short-time anomalous diffusion from long-time anomalous diffusion. Particle localization is observed over eight decades in time for packing fractions of $sim 60%$ and higher.
Infarction- or ischaemia-induced cardiac fibrosis can be arrythmogenic. We use mathematcal models for diffuse fibrosis ($mathcal{DF}$), interstitial fibrosis ($mathcal{IF}$), patchy fibrosis ($mathcal{PF}$), and compact fibrosis ($mathcal{CF}$) to study patterns of fibrotic cardiac tissue that have been generated by new mathematical algorithms. We show that the fractal dimension $mathbb{D}$, the lacunarity $mathcal{L}$, and the Betti numbers $beta_0$ and $beta_1$ of such patterns are textit{fibrotic-tissue markers} that can be used to characterise the arrhythmogenicity of different types of cardiac fibrosis. We hypothesize, and then demonstrate by extensive textit{in silico} studies of detailed mathematical models for cardiac tissue, that the arrhytmogenicity of fibrotic tissue is high when $beta_0$ is large and the lacunarity parameter $b$ is small.
In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrodinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.
Phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of four. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from that of the square-lattice Ising model. An exponential decay is observed in the density matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.
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