scholarly journals Nonequilibrium Time Reversibility with Maps and Walks

Entropy ◽  
2022 ◽  
Vol 24 (1) ◽  
pp. 78
Author(s):  
William Graham Hoover ◽  
Carol Griswold Hoover ◽  
Edward Ronald Smith
Keyword(s):  
Piecewise Linear ◽  
Second Law Of Thermodynamics ◽  
Model Systems ◽  
Computational Time ◽  
Nonequilibrium Systems ◽  
Time Reversibility ◽  
One Dimensional ◽  
Fractal Properties ◽  
Dynamical Simulations ◽  
The One

Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt’s and Zermélo’s paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the irreversible Second Law of Thermodynamics (Loschmidt’s) as well as periodic in the time (Zermélo’s, illustrating Poincaré recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals’ information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here.

scholarly journals Anomalous Diffusion with an Apparently Negative Diffusion Coefficient in a One-Dimensional Quantum Molecular Chain Model

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 506
Author(s):  
Sho Nakade ◽  
Kazuki Kanki ◽  
Satoshi Tanaka ◽  
Tomio Petrosky
Keyword(s):  
Diffusion Coefficient ◽  
Diffusion Constant ◽  
Mean Square Displacement ◽  
Second Law Of Thermodynamics ◽  
Molecular Chain ◽  
Chain Model ◽  
Phase Mixing ◽  
One Dimensional ◽  
The One ◽  
Negative Diffusion

An interesting anomaly in the diffusion process with an apparently negative diffusion coefficient defined through the mean-square displacement in a one-dimensional quantum molecular chain model is shown. Nevertheless, the system satisfies the H-theorem so that the second law of thermodynamics is satisfied. The reason why the “diffusion constant” becomes negative is due to the effect of the phase mixing process, which is a characteristic result of the one-dimensionality of the system. We illustrate the situation where this negative “diffusion constant” appears.

scholarly journals Comparison of numerical schemes of river flood routing with an inertial approximation of the Saint Venant equations

RBRH ◽  
2018 ◽  
Vol 23 (0) ◽  
Author(s):  
Alice César Fassoni-Andrade ◽  
Fernando Mainardi Fan ◽  
Walter Collischonn ◽  
Artur César Fassoni ◽  
Rodrigo Cauduro Dias de Paiva
Keyword(s):  
Numerical Stability ◽  
Flood Routing ◽  
Computational Time ◽  
Numerical Schemes ◽  
Time Step ◽  
Simple Formulation ◽  
One Dimensional ◽  
Saint Venant Equations ◽  
Original Scheme ◽  
The One

ABSTRACT The one-dimensional flow routing inertial model, formulated as an explicit solution, has advantages over other explicit models used in hydrological models that simplify the Saint-Venant equations. The main advantage is a simple formulation with good results. However, the inertial model is restricted to a small time step to avoid numerical instability. This paper proposes six numerical schemes that modify the one-dimensional inertial model in order to increase the numerical stability of the solution. The proposed numerical schemes were compared to the original scheme in four situations of river’s slope (normal, low, high and very high) and in two situations where the river is subject to downstream effects (dam backwater and tides). The results are discussed in terms of stability, peak flow, processing time, volume conservation error and RMSE (Root Mean Square Error). In general, the schemes showed improvement relative to each type of application. In particular, the numerical scheme here called Prog Q(k+1)xQ(k+1) stood out presenting advantages with greater numerical stability in relation to the original scheme. However, this scheme was not successful in the tide simulation situation. In addition, it was observed that the inclusion of the hydraulic radius calculation without simplification in the numerical schemes improved the results without increasing the computational time.

Comparison Between ODT and DNS for Ignition Occurrence in Turbulent Premixed Jet Combustion: Safety-Relevant Applications

2017 ◽  
Vol 231 (10) ◽  
pp. 1709-1735 ◽  
Author(s):  
Abouelmagd Abdelsamie ◽  
David O. Lignell ◽  
Dominique Thévenin
Keyword(s):  
Turbulent Flows ◽  
Reference Data ◽  
Computational Time ◽  
Gas Mixture ◽  
One Dimensional ◽  
Physicochemical Processes ◽  
Numerical Tool ◽  
The One ◽  
Very High ◽  
Turbulent Premixed

Abstract This work investigates the ability of the one-dimensional turbulence model (ODT) to detect, in a predictive manner, occurrence of successful ignition or misfire in a reacting gas mixture subjected to turbulence. Since ODT is computationally very efficient, this significantly aids in the analysis of safety-relevant applications. ODT delivers fast predictions, while still capturing most relevant physicochemical processes controlling ignition. However, ODT contains some empirical parameters that must be set by comparison with reliable reference data. In order to determine these parameters and check the accuracy of resulting ODT predictions, they are compared in this work with reference data from direct numerical simulation (DNS). DNS is recognized as the most accurate numerical tool to investigate ignition in turbulent flows. However, it requires very high computational times, so that it cannot be used for practical safety predictions. It is demonstrated in this article that, thanks to validation and comparison with DNS, ODT realizations can be used to predict correctly the occurrence of ignition in turbulent premixed flames while saving more than 90% of the required computational time, memory and disk space.

scholarly journals A New Diffusion Scheme for Numerical Models Based on Full Irreversibility

2009 ◽  
Vol 24 (2) ◽  
pp. 595-600 ◽  
Author(s):  
C. Liu ◽  
Y. Liu ◽  
H. Xu
Keyword(s):  
Numerical Models ◽  
Weather Prediction ◽  
Forecast Accuracy ◽  
Second Law Of Thermodynamics ◽  
New Physics ◽  
Numerical Weather Prediction Model ◽  
One Dimensional ◽  
Diffusion Scheme ◽  
The One ◽  
Mean Square Errors

Abstract In this work, the forecast accuracy of a numerical weather prediction model is improved by emulating physical dissipation as suggested by the second law of thermodynamics, which controls the irreversible evolutionary direction of a many-body system like the atmosphere. The ability of the new physics-based scheme to improve model accuracy is demonstrated via the case of the one-dimensional viscous Burgers equation and the one-dimensional diffusion equation, as well as a series of numerical simulations of the well-known 1998 successive torrential rains along the Yangtze River valley and 365 continuous 24-h simulations during 2005–06 with decreased root-mean-square errors and improved forecasts in all of the simulations.

GENERALIZATIONS OF THE CHUA EQUATIONS

1992 ◽  
Vol 02 (04) ◽  
pp. 889-909 ◽  
Author(s):  
RAY BROWN
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Type Ii ◽  
Two Dimensional ◽  
Chua’S Circuit ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Chua's Circuit ◽  
The One ◽  
Block Approach

In this paper we present two generalizations of the equations governing Chua’s circuit. In order to obtain the first generalization we simplify Chua’s equations by replacing the piecewise-linear term with a signum function. The resulting simplified system produces a double scroll similar to the one observed experimentally in Chua’s circuit. What is significant about this simplified system is that it can be reduced to what we shall call a two-dimensional single scroll, and from the two-dimensional single scroll we are able to derive a one-dimensional map. This entire derivation is carried out analytically, in contrast to the one-dimensional map analysis that has been carried out for the Lorenz equations which is based on axioms. After we have carried out our analysis for this simplified version of Chua’s equations, we use these equations as a guide to the construction of the first generalization to be presented in this paper. We call this a type-I generalization of Chua’s equations. The generalization consists in using a two-dimensional autonomous flow as a component in a three-dimensional autonomous flow in such a way that the resulting equations will have double scroll attractors similar to those observed experimentally in Chua’s circuit. The value of this generalization is that: (1) it provides a building block approach to the construction of chaotic circuits from simpler two-dimensional components which are not chaotic by themselves. In so doing it provides an insight into how chaotic systems can be built up from simple nonchaotic parts; (2) it illustrates a precise relationship between three-dimensional flows and one-dimensional maps. Of particular significance in this regard is a recent paper of Misiurewicz [1993], which analytically connects the two-dimensional single scroll to the class of unimodal maps, thus providing a framework within which a theory linking unimodal maps to three-dimensional flows may be possible. The second generalization is suggested by considering three-dimensional flows whose only nonlinearities are sigmoid, sgn, or piecewise-linear functions. Clearly, such flows are a generalization of the Chua equations. We call these equations type-II generalization Chua equations. The significance of this direction of investigation is that attractors similar to the Lorenz and Rössler attractors can be produced from type-II generalized Chua equations in a building block approach using only piecewise-linear vector fields. As a result we have a method of producing the Lorenz and Rössler dynamics in a circuit without the use of multipliers. This suggests that the type-II generalized Chua equations are in some sense fundamental in that the dynamics of the three most important autonomous three-dimensional differential equations producing chaos are seen as variations of a single class of equations whose nonlinearities are generalizations of the Chua diode.

BIFURCATION ANALYSIS OF A PWL CHAOTIC CIRCUIT BASED ON HYSTERESIS THROUGH A ONE-DIMENSIONAL MAP

2001 ◽  
Vol 11 (07) ◽  
pp. 1911-1927 ◽  
Author(s):  
FEDERICO BIZZARRI ◽  
MARCO STORACE ◽  
LAURA GARDINI ◽  
RENZO LUPINI
Keyword(s):  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Chaotic Circuit ◽  
One Dimensional ◽  
Local And Global Bifurcations ◽  
The One ◽  
Planar Flows ◽  
Analytical Expressions ◽  
Linear Nature

Bifurcations in the dynamics of a chaotic circuit based on hysteresis are evaluated. Owing to the piecewise-linear nature of the nonlinear elements of the circuit, such bifurcations are discussed by resorting to a suitable one-dimensional map. As the ordinary differential equations governing the circuit are piecewise linear, the analytical expressions of their solutions can be derived in each linear region. Consequently, the proposed results have been obtained not by resorting to numerical integration, but by properly connecting pieces of planar flows. The bifurcation analysis is carried out by varying one of the three dimensionless parameters that the system of normalized circuit equations depends on. Local and global bifurcations, regular and chaotic asymptotic behaviors are pointed out by analyzing both the one-dimensional map and the three-dimensional flow induced by the circuit dynamics.

scholarly journals Anomalous Diffusion With an Apparently Negative Diffusion Coefficient in a One-Dimensional Quantum Molecular Chain Model

2021 ◽  
Author(s):  
Sho Nakade ◽  
Kazuki Kanki ◽  
Satoshi Tanaka ◽  
Tomio Petrosky
Keyword(s):  
Diffusion Coefficient ◽  
Diffusion Constant ◽  
Mean Square Displacement ◽  
Second Law Of Thermodynamics ◽  
Molecular Chain ◽  
Chain Model ◽  
Phase Mixing ◽  
One Dimensional ◽  
The One ◽  
Negative Diffusion

An interesting anomaly of the diffusion process with an apparently negative diffusion coefficient defined through the mean-square displacement in a one-dimensional quantum molecular chain model is shown. Nevertheless, the system satisfies the H-theorem, so that the second law of thermodynamics is satisfied. The reason why the “diffusion constant” becomes negative is due to the effect of the phase mixing process, which is a characteristic result of the one-dimensionality of the system. We illustrate the situation where this negative “diffusion constant” appears.

scholarly journals NUMERICAL SIMULATION OF CHARGED FULLERENE SPECTRUM

2019 ◽  
Vol 24 (2) ◽  
pp. 263-275
Author(s):  
Rafael Arutyunyan ◽  
Yuri Obukhov ◽  
Petr Vabishchevich
Keyword(s):  
Spectral Problem ◽  
Electron Spectrum ◽  
Computational Algorithm ◽  
Piecewise Linear ◽  
Spherically Symmetric ◽  
Charged Sphere ◽  
One Dimensional ◽  
Spectral Problems ◽  
The One ◽  
The Mathematical Model

The mathematical model of the electron spectrum of a charged fullerene is constructed on the basis of the potential of a charged sphere and the spherically symmetric potential of an uncharged fullerene. The electron spectrum is defined as the solution of the spectral problem for the one-dimensional Schr\"odinger equation. For the numerical solution of the spectral problem, piecewise-linear finite elements are used. The computational algorithm was tested on the analytical solution of the problem of the spectrum of the hydrogen atom. For solution of matrix spectral problems, a free library for solving spectral problems of SLEPc is used. The results of calculations of the electron spectrum of a charged fullerene C60 are presented.

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