Entropic Measure for Localized Energy Configurations: Kinks, Bounces, and Bubbles
Marcelo Gleiser, Nikitas Stamatopoulos
备注：
5 pages, 2 PDF figures, uses RevTex4. v2: Modified the normalization condition in the entropy calculation to be more general and the figures were replaced to reflect that. Additional comments were added for clarity. v3: Minor rewordings, version to be published in Physics Letters B
We construct a configurational entropy measure in functional space. We apply it to several nonlinear scalar field models featuring solutions with spatially-localized energy, including solitons and bounces in one spatial dimension, and critical bubbles in three spatial dimensions, typical of first-order phase transitions. Such field models are of widespread interest in many areas of physics, from high energy and cosmology to condensed matter. Using a variational approach, we show that the higher the energy of a trial function that approximates the actual solution, the higher its relative configurational entropy, defined as the absolute difference between the configurational entropy of the actual solution and of the trial function. Furthermore, we show that when different trial functions have degenerate energies, the configurational entropy can be used to select the best fit to the actual solution. The configurational entropy relates the dynamical and informational content of physical models with localized energy configurations.
We consider a class of topological defects in $(1,1)$-dimensions with a deformed $ϕ^4$ kink structure whose stability analysis leads to a Schrödinger-like equation with a zero-mode and at least one vibrational (shape) mode. We are interested in the dynamics of kink-antikink collisions, focusing on the structure of two-bounce windows. For small deformation and for one or two vibrational modes, the observed two-bounce windows are explained by the standard mechanism of a resonant effect between the first vibrational and the translational modes. With the increasing of the deformation, the effect o…
We revisit the derivation of the Hawking-Moss transition rate. Using the static coordinates, we show that the Euclidean action is entirely determined by the contribution of the entropy of de Sitter space which is proportional to the surface area of the horizon. This holographic feature is common to any static spacetime with a horizon on which the shift vector vanishes.
We study kink-antikink collisions in a particular case of the double sine-Gordon model depending on only one parameter $r$. The scattering process of large kink-antikink shows the changing of the topological sector. For some parameter intervals we observed two connected effects: the production of up to five antikink-kink pairs and up to three solitary oscillations. The scattering process for small kink-antikink has several possibilities: the changing of the topological sector, one-bounce collision, two-bounce collision, or formation of a bion state. In particular, we observed for small values …
We study collisions of coherent structures in higher-order field-theoretic models, such as the $ϕ^8$, $ϕ^{10}$ and $ϕ^{12}$ ones. The main distinguishing feature, of the example models considered herein, is that the collision arises due to the long-range interacting algebraic tails of these solitary waves. We extend the approach to suitably initialize the relevant kinks, in the additional presence of finite initial speed, in order to minimize the dispersive wave radiation potentially created by their slow spatial decay. We find that, when suitably initialized, these models still feature the mu…
We investigate the existence and properties of kink-like solitons in a class of models with two interacting scalar fields. In particular, we focus on models that display both double and single-kink solutions, treatable analytically using the Bogomol'nyi--Prasad--Sommerfield bound (BPS). Such models are of interest in applications that include Skyrmions and various superstring-motivated theories. Exploring a region of parameter space where the energy for very different spatially-bound configurations is degenerate, we show that a newly-proposed momentum-space entropic measure called Configuratio…
We calculate new corrections to the Brane-world dynamics, lying in a 5D Schwarzschild-De Sitter black hole, generalizing the result of Nojiri, Odintsov and Ogushi (NOO) in Ref.\cite{Nojiri:2002vu}, The NOO entropy effect is based on the Logharitmic correction to the bulk entropy firstly calculated by Mukherji and Pal in Ref.\cite{Mukherji:2002de}. We calculate higher order contributions to the brane worldsheet. The extra terms obtained lead to interesting implications in brane-cosmology. In particular, new entropic terms rapidly disappear in the late Universe while exploding in the very Early …
In this work we consider a model where the potential has two topological sectors connecting three adjacent minima, as occurs with the $ϕ^6$ model. In each topological sector, the potential is symmetric around the local maximum. For $ϕ>0$ there is a linear map between the model and the $λϕ^4$ model. For $ϕ<0$ the potential is reflected. Linear stability analysis of kink and antikink lead to discrete and continuum modes related by a linear coordinate transformation to those known analytically for the $λϕ^4$ model. Fixing one topological sector, the structure of antikink-kink scattering is relate…
We study the collision of a kink and an antikink in the double sine-Gordon model with and without the excited vibrational mode. In the latter case, we find that there is a limited range of the parameters where the resonance windows exist, despite the existence of a vibrational mode. Still, when the vibrational mode is initially excited, its energy can turn into translational energy after the collision. This creates one-bounce as well as a rich structure of higher-bounce resonance windows that depend on the wobbling phase being in or out of phase at the collision and the wobbling amplitude bein…
We study kink-antikink collisions in the one-dimensional non-integrable scalar phi^6 model. Although the single-kink solutions for this model do not possess an internal vibrational mode, our simulations reveal a resonant scattering structure, thereby providing a counterexample to the standard belief that existence of such a mode is a necessary condition for multi-bounce resonances in general kink-antikink collisions. We investigate the two-bounce windows in detail, and present evidence that this structure is caused by existence of bound states in the spectrum of small oscillations about a comb…
Coleman tunneling in a general scalar potential with two non-degenerate minima is known to have an approximation in terms of a piecewise linear triangular-shaped potential with sharp 'kinks' at the place of the local minima. This approximate potential has a regime where the existence of the bounce solution needs the scalar field to 'wait' for some amount of Euclidean time at one of the 'kinks'. We discuss under which conditions a kink approximation of locally smooth 'cap' regions provides a good estimate for the bounce action.
We propose a gravitational model with a Brans-Dicke-type scalar field having, in the would-be action, a "wrong-sign" kinetic term and a quartic interaction term. In a cosmological context, we obtain, depending on the boundary conditions, either the Friedmann solution or a kink-bounce solution. The expanding-universe Friedmann solution has a big bang curvature singularity, whereas the kink-bounce solution has a nonsingular bouncing behavior of the cosmic scale factor. The bounce occurs precisely at the moment when the scalar field of the kink-type configuration goes through zero, making for a v…
We study tachyon kinks with and without electromagnetic fields in the context of boundary string field theory. For the case of pure tachyon only an array of kink-antikink is obtained. In the presence of electromagnetic coupling, all possible static codimension-one soliton solutions such as array of kink-antikink, single topological BPS kink, bounce, half kink, as well as nonBPS topological kink are found, and their properties including the interpretation as branes are analyzed in detail. Spectrum of the obtained kinks coincides with that of Dirac-Born-Infeld type effective theory.
In this paper we examine the scattering processes among the members of a rich family of kinks which arise in a (1+1)-dimensional relativistic two scalar field theory. These kinks carry two different topological charges that determine the mutual interactions between the basic energy lumps (extended particles) described by these topological defects. Processes like topological charge exchange, kink-antikink bound state formation or kink repulsion emerge depending on the charges of the scattered particles. Two-bounce resonant windows have been found in the antikink-kink scattering processes, but n…
The observed temperature fluctuations in the cosmic microwave background can be traced back to primordial curvature modes that are sourced by adiabatic and/or entropic matter perturbations. In this paper, we explore the entropic mechanism in the context of non-singular bouncing cosmologies. We show that curvature modes are naturally generated during `graceful exit,' i.e., when the smoothing slow contraction phase ends and the universe enters the bounce stage. Here, the key role is played by the kinetic energy components that come to dominate the energy density and drive the evolution towards t…
In this work, we study kink collisions in a scalar field model with scalar-kinetic coupling. This model supports kink/antikink solutions with inner structure in the energy density. The collision of two such kinks is simulated by using the Fourier spectral method. We numerically calculate how the critical velocity and the widths of the first three two bounce windows vary with the model parameters. After that, we report some interesting collision results including two-bion escape final states, kink-bion-antikink intermediate states and kink or antikink intertwined final states. These results sho…
We present a more detailed numerical investigation of the head-on collision of a two-kink/two-antikink system. We identified the escape of oscillon-like configurations as a pair of kinks of the standard $ϕ^4$ model moving apart from each other. New pieces of evidence support that the lump-like defects can emerge from the two-kinks interaction to form metastable configurations. Moreover, these configurations signalize the windows of escape that have a fractal structure similar to the $n$-bounce sequence when the kinks of $ϕ^4$ interact. As the last piece of the numerical experiment, we show tha…
This paper investigates a model containing $ϕ^4$ kinks interacting with fermions. The fermion back-reaction is included in the equations of motion, which affects the kink-antikink collisions. We show that the fermion field generates a force that can be either attractive or repulsive. Moreover, we investigate three different scenarios, which exhibit a wide variety of behaviors including the usual scenarios observed in the $ϕ^4$ model as well as the formation of two oscillons, reflection without contact, one-bounce resonance windows, and the creation of kink-antikink pairs. We also find evidence…
In this Letter, we describe how a spectrum of entropic perturbations generated during a period of slow contraction can source a nearly scale-invariant spectrum of curvature perturbations on length scales larger than the Hubble radius during the transition from slow contraction to a classical non-singular bounce (the `graceful exit' phase). The sourcing occurs naturally through higher-order scalar field kinetic terms common to classical (non-singular) bounce mechanisms. We present a concrete example in which, by the end of the graceful exit phase, the initial entropic fluctuations have become n…
We consider the (1+1)-dimensional Lorentz-symmetric field-theoretic model with logarithmic potential having a Mexican-hat form with two local minima similar to that of the quartic Higgs potential in conventional electroweak theory with spontaneous symmetry breaking and mass generation. We demonstrate that this model allows topological solutions -- kinks. We analyze the kink excitation spectrum, and show that it does not contain any vibrational modes. We also study the scattering dynamics of kinks for a wide range of initial velocities. The critical value of the initial velocity occurs in kink-…
We consider the decay of "false kinks," that is, kinks formed in a scalar field theory with a pair of degenerate symmetry-breaking false vacua in 1+1 dimensions. The true vacuum is symmetric. A second scalar field and a peculiar potential are added in order for the kink to be classically stable. We find an expression for the decay rate of a false kink. As with any tunneling event, the rate is proportional to $\exp(-S_E)$ where $S_E$ is the Euclidean action of the bounce describing the tunneling event. This factor varies wildly depending on the parameters of the model. Of interest is the fact t…
We study Born-Infeld type tachyonic effective action of unstable D2-brane with a runaway potential and find rich structure of static regular solitonic solutions. There exists only periodic array of tachyon kink-antikinks in pure tachyonic theory, however, in the presence of electromagnetic fields, solutions include periodic arrays, topological tachyon kinks, half kink, and bounces. Computed tension of each kink or single unit of the periodic array has $T_{1}=\sqrt{2}πT_{2}$ or that with a multiplicative factor depending on electric field. When both electric and magnetic fields are turned on, f…
In this paper the scattering between the non-topological kinks arising in a family of two-component scalar field theory models is analyzed. A winding charge is carried by these defects. As a consequence, two different classes of kink scattering processes emerge: (1) collisions between kinks that carry the same winding number and (2) scattering events between kinks with opposite winding number. The variety of scattering channels is very rich and it strongly depends on the collision velocity and the model parameter. For the first type of events, four distinct scattering channels are found: \text…
We study the behaviour of fermions localized on moving kinks as these collide with either antikinks or spacetime boundaries. We numerically solve for the evolution of the scalar kinks and the bound (i.e. localized) fermion modes, and calculate the number of fermions transfered to the antikink and boundary in terms of Bogoliubov coefficients. Interpreting the boundary as the brane on which we live, this models the ability of fermions on branes incoming from the bulk to ``stick'' on the world brane, even when the incoming branes bounce back into the bulk.
In this thesis, we study interactions between topological defects in two-dimensional spacetimes. These defects are called kinks. They are solutions of scalar field theories with localized energy which propagate without losing its shape. In order to understand the resonance phenomenon exhibited by those models, we built a toy model where the kink's vibrational mode turns into a quasinormal mode. This causes the suppression of resonance windows and, consequently, its fractal structure is lost. Considering a higher order polynomial as the scalar field potential, we find kinks with long-range tail…
Recent studies have emphasized the important role that a shape deformability of scalar-field models pertaining to the same class with the standard $ϕ^4$ field, can play in controlling the production of a specific type of breathing bound states so-called oscillons. In the context of cosmology, the built-in mechanism of oscillons suggests that they can affect the standard picture of scalar ultra-light dark matter. In the present work kink scatterings are investigated in a parametrized model of bistable system admitting the classical $ϕ^4$ field as an asymptotic limit, with focus on the formation…
The so-called configurational entropy (CE) framework has proved to be an efficient instrument to study nonlinear scalar field models featuring solutions with spatially-localized energy, since its proposal by Gleiser and Stamapoulos. Therefore, in this work, we apply this new physical quantity in order to investigate the properties of degenerate Bloch branes. We show that it is possible to construct a configurational entropy measure in functional space from the field configurations, where a complete set of exact solutions for the model studied displays both double and single-kink configurations…
Effective-potential classification of kink-antikink collision channels in the $ϕ^8$ scalar field theory
SDU-ANU Joint Science College, Shandong University（山东大学-澳国立大学联合学院）；Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shandong University（山东大学空间科学学院）
We search for exact tachyon kink solutions of DBI type effective action describing an unstable D-brane with worldvolume gauge field turned in both the flat and a curved background. There are various kinds of solutions in the presence of electromagnetic fields in the flat space, such as periodic arrays, topological tachyon kinks, half kinks, and bounces. We identify a BPS object, D($p$-1)F1 bound state, which describes a thick brane with string flux density. The curved background of interest is the ten-dimensional lift of the Salam-Sezgin vacuum and, in the asymptotic limit, it approaches ${\rm…
In this work we group three research topics apparently disconnected, namely solitons, Lorentz symmetry breaking and entropy. Following a recent work [Phys. Lett. B 713 (2012) 304], we show that it is possible to construct in the context of travelling wave solutions a configurational entropy measure in functional space, from the field configurations. Thus, we investigate the existence and properties of travelling solitons in Lorentz and CPT breaking scenarios for a class of models with two interacting scalar fields. Here, we obtain a complete set of exact solutions for the model studied which d…
In cyclic universe models based on a single scalar field (e.g., the radion determining the distance between branes in M-theory), virtually the entire universe makes it through the ekpyrotic smoothing and flattening phase, bounces, and enters a new epoch of expansion and cooling. This stable evolution cannot occur, however, if scale-invariant curvature perturbations are produced by the entropic mechanism because it requires two scalar fields (e.g., the radion and the Calabi-Yau dilaton) evolving along an unstable classical trajectory. In fact, we show here that an overwhelming fraction of the u…
