Shannon Information Entropy in Heavy-ion Collisions

The general idea of information entropy provided by C.E. Shannon "hangs over everything we do" and can be applied to a great variety of problems once the connection between a distribution and the quantities of interest is found. The Shannon information entropy essentially quantify the information of a quantity with its specific distribution, for which the information entropy based methods have been deeply developed in many scientific areas including physics. The dynamical properties of heavy-ion collisions (HICs) process make it difficult and complex to study the nuclear matter and its evolution, for which Shannon information entropy theory can provide new methods and observables to understand the physical phenomena both theoretically and experimentally. To better understand the processes of HICs, the main characteristics of typical models, including the quantum molecular dynamics models, thermodynamics models, and statistical models, etc, are briefly introduced. The typical applications of Shannon information theory in HICs are collected, which cover the chaotic behavior in branching process of hadron collisions, the liquid-gas phase transition in HICs, and the isobaric difference scaling phenomenon for intermediate mass fragments produced in HICs of neutron-rich systems. Even though the present applications in heavy-ion collision physics are still relatively simple, it would shed light on key questions we are seeking for. It is suggested to further develop the information entropy methods in nuclear reactions models, as well as to develop new analysis methods to study the properties of nuclear matters in HICs, especially the evolution of dynamics system.Comment: 56 pages, 19 figures.Version accepted by Progress in Particle and Nuclear Physics. To be published in 201

Investigating the quark flavor dependence of the chiral magnetic effect with a multiphase transport model

Because the properties of the QCD phase transition and the chiral magnetic effect (CME) depend on the number of quark flavors ($N_{f}$) and quark mass, relativistic heavy-ion collisions provide a natural environment to investigate the flavor features if quark deconfinement occurs. We introduce an initial two-flavor or three-flavor dipole charge separation into a multiphase transport (AMPT) model to investigate the flavor dependence of the CME. By taking advantage of the recent ALICE data of charge azimuthal correlations with identified hadrons, we attempt to disentangle two-flavor and three-flavor CME scenarios in Pb+Pb collisions at 2.76 TeV. We find that the experimental data show a certain potential to distinguish the two scenarios, therefore we further suggest to collect more data to clarify the possible flavor dependence in future experiments.Comment: 12 pages, 4 figures; final published versio

Cycle lengths and minimum degree of graphs

There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let $G$ be a graph with minimum degree at least $k+1$. We prove that if $G$ is bipartite, then there are $k$ cycles in $G$ whose lengths form an arithmetic progression with common difference two. For general graph $G$, we show that $G$ contains $\lfloor k/2\rfloor$ cycles with consecutive even lengths and $k-3$ cycles whose lengths form an arithmetic progression with common difference one or two. In addition, if $G$ is 2-connected and non-bipartite, then $G$ contains $\lfloor k/2\rfloor$ cycles with consecutive odd lengths. Thomassen (1983) made two conjectures on cycle lengths modulo a fixed integer $k$: (1) every graph with minimum degree at least $k+1$ contains cycles of all even lengths modulo $k$; (2) every 2-connected non-bipartite graph with minimum degree at least $k+1$ contains cycles of all lengths modulo $k$. These two conjectures, if true, are best possible. Our results confirm both conjectures when $k$ is even. And when $k$ is odd, we show that minimum degree at least $k+4$ suffices. This improves all previous results in this direction. Moreover, our results derive new upper bounds of the chromatic number in terms of the longest sequence of cycles with consecutive (even or odd) lengths

Pairing-energy coefficients of neutron-rich fragments in spallation reactions

The ratio of pairing-energy coefficient to temperature ($a_{p}/T$) of neutron-rich fragments produced in spallation reactions has been investigated by adopting an isobaric yield ratio method deduced in the framework of a modified Fisher model. A series of spallation reactions, 0.5$A$ and 1$A$ GeV $^{208}$Pb + $p$, 1$A$ GeV $^{238}$U + $p$, 0.5$A$ GeV $^{136}$Xe + $d$, 0.2$A$, 0.5$A$ and 1$A$ GeV $^{136}$Xe + $p$, and $^{56}$Fe + $p$ with incident energy ranging from 0.3$A$ to 1.5$A$ GeV, has been analysed. An obvious odd-even staggering is shown in the fragments with small neutron excess ($I\equiv N - Z$), and in the relatively small-$A$ fragments which have large $I$. The values of $a_{p}/T$ for the fragments, with $I$ from 0 to 36, have been found to be in a range from -4 to 4, and most values of $a_{p}/T$ fall in the range from -1 to 1. It is suggested that a small pairing-energy coefficient should be considered in predicting the cross sections of fragments in spallation reactions. It is also concluded that the method proposed in this article is not good for fragments with $A/A_{s} >$ 85\% (where $A_{s}$ is the mass number of the spallation system).Comment: 8 pages, 6 figures, to appear on Chinese Physics

Coarse-graining Langevin dynamics using reduced-order techniques

This paper considers the reduction of the Langevin equation arising from bio-molecular models. To facilitate the construction and implementation of the reduced models, the problem is formulated as a reduced-order modeling problem. The reduced models can then be directly obtained from a Galerkin projection to appropriately defined Krylov subspaces. The equivalence to a moment-matching procedure, previously implemented in , 2), is proved. A particular emphasis is placed on the reduction of the stochastic noise, which is absent in many order-reduction problems. In particular, for order less than six we can show the reduced model obtained from the subspace projection automatically satisfies the fluctuation-dissipation theorem. Details for the implementations, including a bi-orthogonalization procedure and the minimization of the number of matrix multiplications, will be discussed as well

Phase Behavior in Thin Films of Confined Colloid-Polymer Mixtures

Using self-consistent field and density-functional theories, we first investigate colloidal self-assembling of colloid/polymer films confined between two soft surfaces grafted by polymers. With the increase of colloidal concentrations, the film undergoes a series of transitions from disordered liquid $\to$ sparse square $\to$ hexagonal (or mixed square-hexagonal) $\to$ dense square $\to$ cylindric structures in plane, which results from the competition between the entropic elasticity of polymer brushes and the steric packing effect of colloidal particles. A phase diagram displays the stable regions of different in-layer ordering structures as the colloidal concentration is varied and layering transitions as the polymer-grafted density is decreased. Our results show a new control mechanism to stabilize the ordering of structures within the films

From Generalized Langevin Equations to Brownian Dynamics and Embedded Brownian Dynamics

We present the reduction of generalized Langevin equations to a coordinate-only stochastic model, which in its exact form, involves a forcing term with memory and a general Gaussian noise. It will be shown that a similar fluctuation-dissipation theorem still holds at this level. We study the approximation by the typical Brownian dynamics as a first approximation. Our numerical test indicates how the intrinsic frequency of the kernel function influences the accuracy of this approximation. In the case when such an approximate is inadequate, further approximations can be derived by embedding the nonlocal model into an extended dynamics without memory. By imposing noises in the auxiliary variables, we show how the second fluctuation-dissipation theorem is still exactly satisfied.Comment: 28 pages, 8 figure

Attractor Bifurcation of Three-Dimensional Double-Diffusive Convection

In this article, we present a bifurcation analysis on the double-diffusive convection. Two pattern selections, rectangles and squares, are investigated. It is proved that there are two different types of attractor bifurcations depending on the thermal and salinity Rayleigh numbers for each pattern. The analysis is based on a newly developed attractor bifurcation theory, together with eigen-analysis and the center manifold reductions

Improved Thermometer from Intermediate Mass Fragments in Heavy-Ion Collisions with Isobaric Yield Ratio Difference

\item[Background] Temperature is an important parameter in studying many important questions in heavy-ion collisions. A thermometer based on the isobaric yield ratio (IYR) has been proposed [Ma \textit{et al.}, Phys. Rev. C \textbf{86}, 054611 (2012) and Ma \textit{et al.}, \textit{ibid.}, Phys. Rev. C \textbf{88}, 014609 (2013)]. \item[Purpose] An improved thermometer ($T_{IB}$) is proposed based on the difference between IYRs. $T_{IB}$ obtained from isobars in different reactions will be compared. \item[Methods] The yields of three isobars are employed in $T_{IB}$. The residual free energy of the three isobars are replaced by that of the binding energy. No secondary decay modification for odd $A$ fragment is used in $T_{IB}$. \item[Results] The measured fragment yields in the 140$A$ MeV $^{40, 48}$Ca + $^{9}$Be ($^{181}$Ta) and $^{58, 64}$Ni + $^9$Be ($^{181}$Ta), the 1$A$ GeV $^{124, 136}$Xe + Pb, and the $^{112,124}$Sn + $^{112,124}$Sn reactions have been analyzed to obtain $T_{IB}$ from IMFs. $T_{IB}$ from most of the fragments in the $^{40, 48}$Ca and $^{58, 64}$Ni reactions is in the range of 0.6 MeV $< T_{IB} <$ 3.5 MeV. $T_{IB}$ from most of the fragments in the $^{124}$Xe and $^{112,124}$Sn reactions is in the range of 0.5 MeV $< T_{IB} <$ 2.5 MeV, while the range is 0.5 MeV $< T_{IB} <$ 4 MeV from most of the fragments in the $^{136}$Xe reaction. In general, for most of the fragments $T_{IB}$ in the $^{40, 48}$Ca and $^{58, 64}$Ni reactions are very similar (except in the very neutron-rich fragments), and $T_{IB}$ from IMFs in the $^{124, 136}$Xe and $^{112,124}$Sn reactions is also similar. A slightly dependence of $T_{IB}$ on $A$ is found. \item[Conclusions] Using the binding energy of the nucleus, $T_{IB}$ can be obtained without the knowledge of the free energies of fragments. In the investigated reactions, $T_{IB}$ from most of the IMFs is low.Comment: 7 pages, 9 figures. To appear on Physical Review

Bifurcation and Stability of Two-Dimensional Double-Diffusive Convection

In this article, we present a bifurcation and stability analysis on the double-diffusive convection. The main objective is to study 1) the mechanism of the saddle-node bifurcation and hysteresis for the problem, 2) the formation, stability and transitions of the typical convection structures, and 3) the stability of solutions. It is proved in particular that there are two different types of transitions: continuous and jump, which are determined explicitly using some physical relevant nondimensional parameters. It is also proved that the jump transition always leads to the existence of a saddle-node bifurcation and hysteresis phenomena