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Register a new userClassical evolution of fractal measures generated by a scalar field on the lattice
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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.
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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.
We give a class of exact solutions of quartic scalar field theories. These solutions prove to be interesting as are characterized by the production of mass contributions arising from the nonlinear terms while maintaining a wave-like behavior. So, a quartic massless equation has a nonlinear wave solution with a dispersion relation of a massive wave and a quartic scalar theory gets its mass term renormalized in the dispersion relation through a term depending on the coupling and an integration constant. When spontaneous breaking of symmetry is considered, such wave-like solutions show how a mass term with the wrong sign and the nonlinearity give rise to a proper dispersion relation. These latter solutions do not change the sign maintaining the property of the selected value of the equilibrium state. Then, we use these solutions to obtain a quantum field theory for the case of a quartic massless field. We get the propagator from a first order correction showing that is consistent in the limit of a very large coupling. The spectrum of a massless quartic scalar field theory is then provided. From this we can conclude that, for an infinite countable number of exact classical solutions, there exist an infinite number of equivalent quantum field theories that are trivial in the limit of the coupling going to infinity.
We give two characterizations, one for the class of generalized Young measures generated by $mathcal A$-free measures, and one for the class generated by $mathcal B$-gradient measures $mathcal Bu$. Here, $mathcal A$ and $mathcal B$ are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The characterization places the class of generalized $mathcal A$-free Young measures in duality with the class of $mathcal A$-quasiconvex integrands by means of a well-known Hahn--Banach separation property. A similar statement holds for generalized $mathcal B$-gradient Young measures. Concerning applications, we discuss several examples that showcase the rigidity or the failure of $mathrm{L}^1$-compensated compactness when concentration of mass is allowed. These include the failure of $mathrm{L}^1$-estimates for elliptic systems and the failure of $mathrm{L}^1$-rigidity for the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $Omega$, the inclusions [ mathrm{L}^1(Omega) cap ker mathcal A hookrightarrow mathcal M(Omega) cap ker mathcal A, ] [ {mathcal B uin mathrm{C}^infty(Omega)} hookrightarrow {mathcal B uin mathcal M(Omega)}, ] are dense with respect to area-functional convergence of measures
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.
It has been expected that astronomical observations to detect the orbital angular momenta of electromagnetic waves may give us a new insight into astrophysics. Previous works pointed out the possibility that a rotating black hole can produce orbital angular momenta of electromagnetic waves through gravitational scattering, and the spin parameter of the black hole can be measured by observing them. However, the mechanism how the orbital angular momentum of the electromagnetic wave is generated by the gravitational scattering has not been clarified sufficiently. In this paper, in order to understand it from a point of view of gravitational lensing effects, we consider an emitter which radiates a spherical wave of the real massless scalar field and study the deformation of the scalar wave by the gravitational scattering due to a black hole by invoking the geometrical optics approximation. We show that the frame dragging caused by the rotating black hole is not a necessary condition for generating the orbital angular momentum of the scalar wave. However, its components parallel to the direction cosines of images appear only if the black hole is rotating.
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