Entropy evolution of the bimodal field interacting with an effective two-level atom via the Raman transition in Kerr medium

For a general class of two-mode, simple analytic expressions are derived for the evolution of the field quantum entropy in the bimodal field interacting with an effective two-level atom via the Raman transition, with an additional Kerr-like medium. The effect of a Kerr-like medium on the entropy is analyzed. It is shown that the addition of the Kerr medium has an important effect on the properties of the entropy and the entanglement. The results show that the effect of the Kerr medium changes the quasi-period of the field entropy evolution and entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.     

Influence of intrinsic decoherence on entanglement degree in the atom–field coupling system

In this paper, we use the quantum mutual entropy to measure the degree of entanglement in the time development of a two-level particle (atom or trapped ion). We find an exact solution of the Milburn equation for the system. The exact solution is then used to discuss the influence of intrinsic decoherence on degree of entanglement. The exact results are employed to perform a careful investigation of the temporal evolution of the entropy. It is shown that the degree of entanglement is very sensitive to the changes of the intrinsic decoherence. The results show that the effect of the intrinsic decoherence decreases the quasiperiod of the entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

Complexity principle of extremality in evolution of living organisms by information-theoretic entropy

We consider evolution from a multiorgan (multistage) organism, which has a number of identical organs (e.g. a trilobite with many pairs of legs), to another organism, which has one organ modified (specialized) into a different part of the body (e.g. claws of a crab) at the expense of reduction in the number of non-modified organs. We observe that in early stages of evolution multiple organs (pairs of legs) may be created, and that extra organs may rapidly be reduced during later stages. We ask: Why do extra organs evolve during early stages of evolution? To answer the question we construct and then analyze a simple although basic model of evolution based on information-theoretic entropy. We show that an extremality principle is valid in which the increase in number of identical organs is led by the gradient of information entropy increasing with the number of these organs. On the other hand, the reduction in number of these organs, observed for later stages of evolution, results from catastrophes between submanifolds of evolution, the surfaces on which modifications (specializations) of organs may occur. Our conclusion is that modification (specialization) of organs, while in principle consistent with the entropy principle of extremality, may lead evolution to a region of catastrophes, and that these catastrophes may explain extinction of some species. The same mathematical model is applied for explanation of parallel evolution of animals and for some evolution problems of flowers.     

A Geometric Theory of Growth Mechanics

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. The time dependence of the metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and “shape”. We show that the time dependency of the material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange–d’Alembert principle with Rayleigh’s dissipation functions to derive the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of the deformation gradient, F=F _e F _g, into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory.     

Phase-field models for fluid mixtures

The paper investigates the modelling of phase transitions in multiphase fluid mixtures. The order parameter is identified with the set of concentrations and is a phase field in that it varies smoothly in the space region. This in turn requires that the continuity equations be regarded as constraints on the pertinent fields. The phase field is viewed as an internal variable whose evolution is subject to thermodynamic requirements. The second law allows for an extra entropy flux which proves to be proportional to the time derivative of the order parameter. Previous papers on the subject are revisited. It follows that their recourse to external mass supplies or to ad-hoc entropy fluxes can be avoided. The analogy of the phase-field model, with that of mixtures with mass–density gradients and extra entropy flux, is emphasized.     

Multiplicity of stationary solutions to the Euler-Poisson equations

Consider the system of Euler-Poisson as a model for the time evolution of gaseous stars through the self-induced gravitational force. We study the existence, uniqueness and multiplicity of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy a priori. These results generalize the previous works on the irrotational or the rotational gaseous stars around an axis, and then they hold in more general physical settings. Under the assumption of radial symmetry, the monotonicity properties of the radius of the gas with respect to either the strength of the velocity field or the center density are also given which yield the uniqueness under some circumstances.     

Travelling Waves with a Singularity in Magnetic Nanowires

We model the evolution of the magnetization in an infinite cylinder by harmonic map heat flow with an additional external field. Using variational methods, we prove the existence of corotationally symmetric travelling wave solutions with a moving vortex. We moreover show that for weak and strong fields the travelling waves connect the original state anti-parallel to the external magnetic field with the totally reversed state in direction of the external field. Our results match numeric simulations. For thicker wires several groups have found a reversal mode where a domain wall with a corotational symmetry and a vortex is propagating through the wire.     

Physical entropy, information entropy and their evolution equations

Inspired by the evolution equation of nonequilibrium statistical physics entropy and the concise statistical formula of the entropy production rate, we develop a theory of the dynamic information entropy and build a nonlinear evolution equation of the information entropy density changing in time and state variable space. Its mathematical form and physical meaning are similar to the evolution equation of the physical entropy: The time rate of change of information entropy density originates together from drift, diffusion and production. The concise statistical formula of information entropy production rate is similar to that of physical entropy also. Furthermore, we study the similarity and difference between physical entropy and information entropy and the possible unification of the two statistical entropies, and discuss the relationship among the principle of entropy increase, the principle of equilibrium maximum entropy and the principle of maximum information entropy as well as the connection between them and the entropy evolution equation.     

Quantum entropy of Dirac fields in black holes

We use the brick-wall model to study the quantum entropy of the Dirac field in a static black hole with a global monopole or a cosmic string. We show that the entropy of the Dirac field contains a quadratically divergent term and two logarithmically divergent ones and it is not proportional to the entropy of the scalar field. The contribution of the logarithmic term to the entropy depends on the black-hole characteristics and is always negative. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 60–64, October, 2006.     

Entropies and Equilibria of Many-Particle Systems: An Essay on Recent Research

This essay is intended to present a fruitful collaboration which has developed among a group of people whose names are listed above: entropy methods have proved over the last years to be an efficient tool for the understanding of the qualitative properties of physically sound models, for accurate numerics and for a more mathematical understanding of nonlinear PDEs. The goal of this essay is to sketch the historical development of the concept of entropy in connection with PDEs of continuum mechanics, to present recent results which have been obtained by the members of the group and to emphasize the most striking achievements of this research. The presentation is by no way an exhaustive review of the methods and results involving the entropy, not even in the field of PDEs. Many other researchers in and outside Europe have contributed to the development of this field, including – but not only – in collaboration with some of the people of the group. However, it can be claimed that this group had a leading role over the recent years and this essay is intended to explain how this occurred.     

Entropies and Equilibria of Many-Particle Systems: An Essay on Recent Research

This essay is intended to present a fruitful collaboration which has developed among a group of people whose names are listed above: entropy methods have proved over the last years to be an efficient tool for the understanding of the qualitative properties of physically sound models, for accurate numerics and for a more mathematical understanding of nonlinear PDEs. The goal of this essay is to sketch the historical development of the concept of entropy in connection with PDEs of continuum mechanics, to present recent results which have been obtained by the members of the group and to emphasize the most striking achievements of this research. The presentation is by no way an exhaustive review of the methods and results involving the entropy, not even in the field of PDEs. Many other researchers in and outside Europe have contributed to the development of this field, including – but not only – in collaboration with some of the people of the group. However, it can be claimed that this group had a leading role over the recent years and this essay is intended to explain how this occurred.     

Application of quorum response and information entropy to animal collective motion modeling

“Quorum response” is a type of social interaction in which an individual's chance of choosing an option is a nonlinear function of the number of other individuals already committing to it. This interaction has been widely used to characterize collective decision‐making in animal groups. Here, we first implement it in 1D and 2D models of collective animal movement, and find that the resulting group motion shows the characteristic behaviors which were observed in previous experimental and modeling studies. Further, the analytic form of quorum response renders us an opportunity to propose a mean field theory in 1D with globally interacting particles, so we can estimate the average time period between changes in the group direction (mean switching time). We find that the theoretical results provide an upper bound to the simulation results when the interaction radius grows from local to global. Information entropy, a concept widely used to quantify the uncertainty of a random variable, is introduced here as a new order parameter to study the evolution of systems of two cases in 2D models. The explicitly formulated probability of a particle's dynamic state in the framework of quorum response makes information entropy directly computable. We find that, besides the global order, information entropy can also capture the structural features of local order of the system which previous order parameters such as alignment cannot. © 2016 Wiley Periodicals, Inc. Complexity 21: 584–592, 2016     

交叉熵——计算语言学消歧的一种工具

歧义问题的描述和消除问题是制约计算语言学发展的瓶颈问题.将交叉熵引入计算语言学消岐领域.采用语句的真实语义作为交叉熵的训练集的先验信息,将机器翻译的语义作为测试集后验信息,计算两者的交叉熵,并以交叉熵指导对歧义的辨识和消除.实例表明,该方法简洁有效,易于计算机自适应实现,交叉熵不失为计算语言学消岐的一种较为有效的工具.     

First integrals for charged perfect fluid distributions

We study the evolution of shear-free spherically symmetric charged fluids in general relativity. We find a new parametric class of solutions to the Einstein-Maxwell system of field equations. Our charged results are a generalisation of earlier treatments for neutral relativistic fluids. We regain the first integrals found previously for uncharged matter as a special case. In addition an explicit first integral is found which is necessarily charged.     

Fractional dynamics of a system with particles subjected to impacts

This paper studies the dynamics of a system composed of a collection of particles that exhibit collisions between them. Several entropy measures and different impact conditions of the particles are tested. The results reveal a Power Law evolution both of the system energy and the entropy measures, typical in systems having fractional dynamics.     

On weak-strong uniqueness property for full compressible magnetohydrodynamics flows

This paper is devoted to the study of the weak-strong uniqueness property for full compressible magnetohydrodynamics flows. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and an additional equation which describes the evolution of the magnetic field. Using the relative entropy inequality, we prove that a weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists.