Poincare Cycle Research Articles

Nonlocal quantum computing theory and Poincare cycle in spherical states

Four new fundamental nonlocal quantum computing diagonal operator-state relations are derived which model the interaction between two adjacent atoms of an entangled atomic chain. Each atom possesses four eigen-states. These relations lead to four momentum-space cyclic transformations and are used as the computation states in one-dimensional cellular automaton. Four interacting half-observable periodic planar states appear with the same Poincare cycle. Due to the space-time rotational symmetry of these operator-state relations, a new type of periodic spherical state can be constructed consisting of eight finite space-time quadrants as the special quantum computing result.

Layout optimization of large-scale oil–gas gathering system based on combined optimization strategy

Layout optimization of large-scale oil–gas gathering system is a kind of NP-hard problem in the field of system optimization. It involves a large number of network nodes, coupled optimization variables, complex network structures and hydraulic constraints, which cause the great difficulty in constructing optimization models and solution methods. In this paper, a high-dimensional mixed integer nonlinear layout optimization mathematical model involving the pipeline network structure parameters and pipeline design parameters is established, which can be applied to large-scale oil gathering system and gas gathering system universally. A modified particle swarm optimization (MPSO) algorithm with global search ability is proposed. The convergence theorem of the stochastic optimization algorithm is established based on the Poincare cycle theory. Global convergence of MPSO algorithm is proved, and the performance of MPSO algorithm is analyzed by numerical experiments. Based on dimension reduction planning and modularization thought, the grid dissection set partition method is proposed, and the theoretical foundation and complexity of the grid dissection method are discussed. In order to reduce the dimension of the layout optimization problem, the concept of the fuzzy set of adjacent position, and a novel approach for the well-station connection mode optimization are put forward. Based on the MPSO algorithm, grid dissection set partition method and solution method by fuzzy set of adjacent position, a combined optimization strategy for layout optimization model is proposed. The reliability and practicality of the proposed layout optimization model and combined optimization strategy are verified by the successful application of a real-world large-scale oil field with 661 wells.

Ehrenfest urn model with interaction.

We studied the Ehrenfest urn model in which particles in the same urn interact with each other. Depending on the nature of interaction, the system undergoes a first- or second-order phase transition. The relaxation time to the equilibrium state, the Poincaré cycles of the equilibrium state and the most far-from-equilibrium state, and the duration time of the states during first-order phase transition are calculated. It was shown that the scaling behavior the Poincaré cycles could serve as an indication to the nature of phase transition, and the behavior of the ratio of duration time of the states could be strong evidence of the metastability during first-order phase transition.

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.

Irreversibility in a Reversible Lattice Gas

A simple lattice gas model, a microscopically reversible cellular automaton, is described and shown to exhibit thermodynamic irreversibility in processes similar to those in real gases. The model, which has no random elements, develops a long-lasting equilibrium state within a Poincare cycle. This state is an attractor resulting from the nonlinear nature of the collective particle collisions and motions. The results illustrate how the Second Law of Thermodynamics applies to real systems governed by reversible microscopic dynamics.

Variations on a theme by Mark Kac

Mark Kac's theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations, most of which are formulated in terms of 0–1 processes. Whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0, this condition is dropped in part of the variations; two others refer to the variance of the recurrence time and two to the Poincare cycle of a dynamical system. Most variations consist in inequalities and formal identities for the mean first-arrival time and subsequent recurrence times for the given state.

Example of the description of dissipative processes in terms of reversible dynamic equations and some comments on the fluctuation-dissipation theorem

The dynamics of a macroscopic oscillator which is interacting with a heat reservoir, which also consists of oscillators, is analyzed. This problem, which can be solved exactly in its general form in both the classical and quantum-mechanic cases, is used as an example for a study of the transition from a purely dynamic description to a statistical description. Since the system of linear oscillators is not ergodic, an averaging procedure must be regarded as taking an average over the time or over repeated measurements on a unique dynamic trajectory. Depending on the nature of the quadratic form of the potential energy, the oscillations of a macroscopic oscillator can decay in various ways, including exponentially, in the initial stage of the evolution. After a Poincare cycle, the system returns to its initial state, and the damping of the oscillations gives way to a growth. The reversibility of the motion means that the Green's function of the system of oscillators is of odd parity in the time. Equilibrium fluctuations of a macroscopic oscillator are examined. In the classical case the Callen-Welton fluctuation-dissipation theorem can be formulated as follows: The derivative of the coordinate correlation function is proportional to the Green's function of the macroscopic oscillator. In a description in terms of frequencies, the odd parity of the Green's function gives rise to an imaginary part of the Fourier transform of this function in the fluctuation-dissipation theorem. This result is a consequence of the reversibility of the motion in time. The fluctuation-dissipation theorem is proved for Hamiltonian systems without dissipation, but it also applies to systems with dissipation. The exact microscopic Green's function is replaced in this case by the Green's function of a simplified phenomenological description, which explicitly contains dissipative parameters. In the quantum-mechanical case, the results are analogous. The classical and quantum-mechanical versions of the Nyquist relation which follow from the fluctuation-dissipation theorem when the Green's function is approximated by an exponentially damped sinusoidal oscillation are discussed.

An apparent violation of second law of thermodynamics in biological systems

An apparent violation of second law of thermodynamics in a photosynthetic process and a Poincare cycle.

Irreversible Process in Nuclear Reactions

The time-dependent coupled equations of reaction processes are solved by means of the Green's functions of their stationary parts. It is pointed out that the yields of nuclear reac­ tions (time spectra) become the function of time and the characteristic parameters of the phase of compound nucleus having dimension of time: the number of degrees of freedom of motion, the Poincare cycle, the delay, the relaxation time and the lifetime. Under the strong correlation condition for the transition matrix elements, the solution of time-dependent coupled equations satisfies the master equation of irreversible process inside the compound nucleus given by the statistical mechanics. An example of the time spectrum for a two-step reaction is shown. § I. Introduction Existence of the irreversible process in nuclear reactions has been studied by means of the time-dependent perturbation, D the statistical treatments2), 3) and recently the pre-equilibrium modeL''·'' It has been shown experimentally that the reaction times of the direct interactions6' and the compound nucleus processes7' are about 10- 21 sec and 10-16 sec, respectively. It was also pointed out by a previous paper8 ' that the reaction steps (phase of compound nucleus) inside the compound nucleus were able to be lined sequentially starting from the simplest phase and ending in the most complex one of compound nucleus. The number of matrix elements for the transition from a state in the simple phase of compound nucleus to states in the complex one will be much larger than those from a state in the complex one to states in the simple one, because the level density of the former is much smaller than the latter. Then the time­ dependent coupled equations of each phase8' will be reduced to the master equation after the statistical averaging. 91 The nuclear reaction inside compound nuCleus starts from the simple phase and goes on step by step in the direction of more complex phases and finally reaches the statistical equilibrium of compound nucleus. It is our purpose to solve the time-dependent coupled equations among the phases of compound nucleus and to show the existence of an irreversible process in nuclear reactions. When a wave packet with the shortest time duration comes into a target nucleus, the fastest outgoing wave packet will be the direct reflection from the nuclear surface with zero delay; then many outgoing wave packets going through the n-phases of compound nucleus will come out of it in order of the

Sequential Series for Nuclear Reactions

A new time-dependent treatment of nuclear reactions is given, in which the wave function of compound nucleus is expanded by a sequential series of the reaction processes. The wave functions of the sequential series form another complete set of compound nucleus at the limit Δt →0. It is pointed out that the wave function is characterized by the quantities: the number of degrees of freedom of motion n, the period of the motion (Poincaré cycle) tn, the decay time tnµ and the relaxation time τn to the equilibrium of compound nucleus, instead of the usual quantum number λ, the energy eigenvalued Eλ and the total width Γλ of resonance levels, respectively. The transition matrix matrix elements and the yields of nuclear reactions also become the functions of time given by the Fourier transform of the usual ones. The Poincaré cycles of compound nuclei are compared with the observed correlations among resonance levels, which are about 10-17 ∼10-16 sec for medium and heavy nuclei and about 10-20 sec for the intermediate resonances.

Poincar� cycles and ergodic behavior of a linear diatomic chain

The ergodic behavior of a linear diatomic chain is shown to be analogous to that of a linear monatomic chain. Starting with the expressions for the time-relaxed correlation functions between any two particles in the chain, we show that the existence of Poincare cycles is not inconsistent with the development of an equilibrium state. Also, we show that those dynamical variables that are ergodic for the linear monoatomic chain remain ergodic in the diatomic chain. It is shown that the autocorrelation functions for particles with equal or different masses decay in time ast −1/2 .

ON THE NATURE OF THE SPECTRUM OF THE TIME CORRELATION FUNCTION MATRIX FOR STOCHASTIC PROCESSES

If a stochastic process has a finite Poincare recursion time,' the time correlation function matrix a(t) a(t + r) is an almost periodic function of r. Here, a is a generalized vector whose components represent the departure of the observable macroscopic properties of the system from their equilibrium or steady-state values. The almost periodic nature of a(t)au(t + r) is commonly taken to imply that the ensemble average of the time correlation function matrix R(t,r) a(t)W (t + r) is also almost periodic and thus has a discrete spectrum.' This leads to statements implying that in order to obtain irreversible behavior and an absolutely integrable spectrum, the recurrence time must be made infinite by taking the limit of an infinitely large system. In this paper it is found that R(t,r) vanishes irreversibly as T * oo, for all but a small class of very special (unphysical) stochastic models. The existence of finite recursion times and the exact use of deterministic mechanics do not change this property. The argument makes explicit an opinion of Uhlenbeck,3 and hopefully puts to rest the emphasis on the importance of Poincare cycles in statistical mechanics. For this study it is convenient to define certain probability distribution functions: p(a,t)da is the probability that the system be found at time t with its macroscopic state between a and a + dot; p(a,tla', t + r)dada' is the joint probability that the system be found at time t in a to a + du and at time t + T in a' to a' + da'; P(a,tl a',t + T)da' is the probability the system will be found between o' and a' + da' at time t + r under the condition that it be in a at time t. Then,

Dynamics of a Simple Many-Body System of Hard Rods

General formulas are given for the exact calculation of the nonequilibrium properties of the one-dimensional system of equal-mass hard rods both for a finite but large system and in the limit of infinite size. Only properties which depend upon labeling one or more of the particles are nontrivial in this system. Various results are obtained on Poincaré cycles, delocalization of a particle with time and electrical conductivity when one particle is charged.