A New Criterion Beyond Divergence for Determining the Dissipation of a System: Dissipative Power

Divergence is usually used to determine the dissipation of a dynamical system, but some researchers have noticed that it can lead to elusive contradictions. In this article, a criterion, dissipative power, beyond divergence for judging the dissipation of a system is presented, which is based on the knowledge of classical mechanics and a novel dynamic structure by Ao. Moreover, the relationship between the dissipative power and potential function (or called Lyapunov function) is derived, which reveals a very interesting, important, and apparently new feature in dynamical systems: to classify dynamics into dissipative or conservative according to the change of “energy function” or “Hamiltonian,” not according to the change of phase space volume. We start with two simple examples corresponding to two types of attractors in planar dynamical systems: fixed points and limit cycles. In judging the dissipation by divergence, these two systems have both the elusive contradictions pointed by researchers and new ones noticed by us. Then, we analyze and compare these two criteria in these two examples, further consider the planar linear systems with the coefficient matrices being the four types of Jordan’s normal form, and find that the dissipative power works when divergence exhibits contradiction. Moreover, we also consider another nonlinear system to analyze and compare these two criteria. Finally, the obtained relationship between the dissipative power and the Lyapunov function provides a reasonable way to explain why some researchers think that the Lyapunov function does not coexist with the limit cycle. Those results may provide a deeper understanding of the dissipation of dynamical systems.

How to design a focusing guide: The large moderator case1

As continuously shaped super-mirrors are becoming available, the conceptual design of focusing guides should explore a wider range of possibilities to accomplish an efficient neutron beam extraction. Starting from a desired phase-space volume at the sample position and using an upstream ray-tracing approach, the acceptance diagram of any focusing guide can be calculated at the moderator position. To ensure high brilliance transfer and homogeneous coverage, the acceptance diagram should be fully included in the neutron source emission phase-space volume. Following this idea, the guide system can be scaled into dimensionless geometric figures that convey performance limits for a desired cross-section reduction. Moreover, if we impose a monotonic increase of the reflection angle with divergence angle at the sample position, the shape of the mirror is analytically determined. This approach was applied in the design of a focusing guide for SNAP instrument at SNS, at ORNL, USA. The results of McStas simulations are presented with different options included. Our approach facilitates finding an optimal solution for connecting multiple guide pieces to avoid excessive losses and ensure a homogeneous phase space coverage.

Discrete Competitive Lotka–Volterra Model with Controllable Phase Volume

The simulation of population dynamics and social processes is of great interest in nonlinear systems. Recently, many scholars have paid attention to the possible applications of population dynamics models, such as the competitive Lotka–Volterra equation, in economic, demographic and social sciences. It was found that these models can describe some complex behavioral phenomena such as marital behavior, the stable marriage problem and other demographic processes, possessing chaotic dynamics under certain conditions. However, the introduction of external factors directly into the continuous system can influence its dynamic properties and requires a reformulation of the whole model. Nowadays most of the simulations are performed on digital computers. Thus, it is possible to use special numerical techniques and discrete effects to introduce additional features to the digital models of continuous systems. In this paper we propose a discrete model with controllable phase-space volume based on the competitive Lotka–Volterra equations. This model is obtained through the application of semi-implicit numerical methods with controllable symmetry to the continuous competitive Lotka–Volterra model. The proposed model provides almost linear control of the phase-space volume and, consequently, the quantitative characteristics of simulated behavior, by shifting the symmetry of the underlying finite-difference scheme. We explicitly show the possibility of introducing almost arbitrary law to control the phase-space volume and entropy of the system. The proposed approach is verified through bifurcation, time domain and phase-space volume analysis. Several possible applications of the developed model to the social and demographic problems’ simulation are discussed. The developed discrete model can be broadly used in modern behavioral, demographic and social studies.

Prospects for the study of event-by-event fluctuations at MPD/NICA project

One of the main physics goals of the Multi Purpose Detector (MPD) is to investigate hot and dense baryonic matter in heavy ion collisions at NICA energies to search for the possible critical end point (CEP). Since the location of CEP is not clear the entire accessible region of the QCD phase diagram needs to be explored by scanning the full range of available beam energies. In case of CEP existence it can be observed by abnormal fluctuations of various quantities such as net-proton multiplicity. This task requires excellent particle identification (PID) capability over as large as possible phase space volume. The identification of charged hadrons is achieved at the momenta of 0:1 – 3 GeV/c. The results of hadron identification and preliminary possibility estimation of the study of event-by-event fluctuations at MPD will be presented.

Generalized higher-order Snyder model

In this paper we consider the higher-order generalization of the Snyder algebra. We find the modified inner product for this algebra. We also discuss the one-dimensional hydrogen atom eigenvalue problem. We find the invariant phase space volume for the generalized Snyder algebra. Finally we use the density of states in three dimension to discuss the Maxwellian distribution and cosmological constant.

A Family of Hierarchical Symplectic Maps for N-body Simulations with Post-Newtonian Corrections

Symplectic maps are well known for preserving the phase-space volume in Hamiltonian dynamics and are particularly suited for problems that require long integration times, such as the [Formula: see text]-body problem. However, when combined with a varying time-step scheme, they end up losing its symplecticity and become numerically inefficient. We address this problem by using a recursive Hamiltonian splitting based on the time-symmetric value of the individual time-steps required by the particles in the system. We present a family of 48 quasi-symplectic maps with different orders of convergence (2nd-, 4th- & 6th-order) and three time-stepping schemes: i) 16 using constant time-steps, ii) 16 using shared adaptive time-steps, and iii) 16 using hierarchical (individual) time-steps. All maps include post-Newtonian corrections up to order 3.5PN. We describe the method and present some details of the implementation.

An Entropy for Groups of Intermediate Growth

One of the few accepted dynamical foundations of nonadditive (“nonextensive”) statistical mechanics is that the choice of the appropriate entropy functional describing a system with many degrees of freedom should reflect the rate of growth of its configuration or phase space volume. We present an example of a group, as a metric space, that may be used as the phase space of a system whose ergodic behavior is statistically described by the recently proposed δ-entropy. This entropy is a one-parameter variation of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from the power-law entropies that have been recently studied. We use the first Grigorchuk group for our purposes. We comment on the connections of the above construction with the conjectured evolution of the underlying system in phase space.