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Non-Linear Lattice

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (30 November 2015) | Viewed by 72302

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Ignazio Licata

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Guest Editor
1. ISEM Institute for Scientific Methodology, Via Ugo La Malfa n. 153, 90146 Palermo, Italy
2. School of Advanced International Studies on Applied Theoretical and Non Linear Methodologies of Physics, 70121 Bari, Italy
Interests: foundation of quantum theories; quantum cosmology; de sitter holographic models; dissipative quantum field theories; physics of emergence and organization; fisher information; sub- and super-Turing computation models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Sauro Succi

E-Mail Website
Guest Editor
Center for Life Nano Science @Sapienza, Italian Institute of Technology, Viale Regina Elena, 295, I-00161 Roma, Italy
Interests: statistical and computational physics

Special Issue Information

Dear Colleague,

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of Non-Linear Lattices, a theme among the most refined and interdisciplinary of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of Non-Linear Lattices, from the theoretical ones to more applicable ones. The Non-Linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete time.

Prof. Dr. Ignazio Licata
Dr. Sauro Succi
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Plastic Deformations in Crystals
  • Sin-Gordon Equations
  • Fermi-Pasta-Ulam Problem
  • Toda Lattice
  • Boltzmann Lattice
  • nonlinear waves and solitons
  • Ergodicity and integrability problems
  • Small Molecules with non linear local modes
  • Protein Models
  • DNA Models
  • Neuronal Lattices Models
  • Computational Issues
  • Cellular Automata
  • Historical Aspects in Non Linear Science
  • Lattice gas
  • Lattice Boltzmann
  • Lattice gravity
  • Quantum field theory
  • Ultracold lattice fluids

Published Papers (13 papers)

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Research

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3634 KiB  
Article
Long-Range Electron Transport Donor-Acceptor in Nonlinear Lattices
by Alexander P. Chetverikov, Werner Ebeling and Manuel G. Velarde
Entropy 2016, 18(3), 92; https://doi.org/10.3390/e18030092 - 11 Mar 2016
Cited by 5 | Viewed by 4853
Abstract
We study here several simple models of the electron transfer (ET) in a one-dimensional nonlinear lattice between a donor and an acceptor and propose a new fast mechanism of electron surfing on soliton-like excitations along the lattice. The nonlinear lattice is modeled as [...] Read more.
We study here several simple models of the electron transfer (ET) in a one-dimensional nonlinear lattice between a donor and an acceptor and propose a new fast mechanism of electron surfing on soliton-like excitations along the lattice. The nonlinear lattice is modeled as a classical one-dimensional Morse chain and the dynamics of the electrons are considered in the tight-binding approximation. This model is applied to the processes along a covalent bridge connecting donors and acceptors. First, it is shown that the electron forms bound states with the solitonic excitations in the lattice. These so-called solectrons may move with supersonic speed. In a heated system, the electron transfer between a donor and an acceptor is modeled as a diffusion-like process. We study in detail the role of thermal factors on the electron transfer. Then, we develop a simple model based on the classical Smoluchowski–Chandrasekhar picture of diffusion-controlled reactions as stochastic processes with emitters and absorbers. Acceptors are modeled by an absorbing boundary. Finally, we compare the new ET mechanisms described here with known ET data. We conclude that electron surfing on solitons could be a special fast way for ET over quite long distances. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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3739 KiB  
Article
Three-Dimensional Lattice Boltzmann Simulation of Liquid Water Transport in Porous Layer of PEMFC
by Bo Han, Meng Ni and Hua Meng
Entropy 2016, 18(1), 17; https://doi.org/10.3390/e18010017 - 31 Dec 2015
Cited by 23 | Viewed by 7197
Abstract
A three-dimensional two-phase lattice Boltzmann model (LBM) is implemented and validated for qualitative study of the fundamental phenomena of liquid water transport in the porous layer of a proton exchange membrane fuel cell (PEMFC). In the present study, the three-dimensional microstructures of a [...] Read more.
A three-dimensional two-phase lattice Boltzmann model (LBM) is implemented and validated for qualitative study of the fundamental phenomena of liquid water transport in the porous layer of a proton exchange membrane fuel cell (PEMFC). In the present study, the three-dimensional microstructures of a porous layer are numerically reconstructed by a random generation method. The LBM simulations focus on the effects of the porous layer porosity and boundary liquid saturation on liquid water transport in porous materials. Numerical results confirm that liquid water transport is strongly affected by the microstructures in a porous layer, and the transport process prefers the large pores as its main pathway. The preferential transport phenomenon is more profound with a decreased porous layer porosity and/or boundary liquid saturation. In the transport process, the breakup of a liquid water stream can occur under certain conditions, leading to the formation of liquid droplets inside the porous layer. This phenomenon is related to the connecting bridge or neck resistance dictated by the surface tension, and happens more frequently with a smaller porous layer porosity. Results indicate that an optimized design of porous layer porosity and the combination of various pore sizes may improve both the liquid water removal and gaseous reactant transport in the porous layer of a PEMFC. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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6468 KiB  
Article
A Lattice Gas Automata Model for the Coupled Heat Transfer and Chemical Reaction of Gas Flow Around and Through a Porous Circular Cylinder
by Hongsheng Chen, Zhong Zheng, Zhiwei Chen and Xiaotao T. Bi
Entropy 2016, 18(1), 2; https://doi.org/10.3390/e18010002 - 22 Dec 2015
Cited by 12 | Viewed by 5902
Abstract
Coupled heat transfer and chemical reaction of fluid flow in complex boundaries are explored by introducing two additional properties, i.e. particle type and energy state into the Lattice gas automata (LGA) Frisch–Hasslacher–Pomeau (FHP-II) model. A mix-redistribute of energy and type of particles is [...] Read more.
Coupled heat transfer and chemical reaction of fluid flow in complex boundaries are explored by introducing two additional properties, i.e. particle type and energy state into the Lattice gas automata (LGA) Frisch–Hasslacher–Pomeau (FHP-II) model. A mix-redistribute of energy and type of particles is also applied on top of collision rules to ensure randomness while maintaining the conservation of mass, momentum and energy. Simulations of heat transfer and heterogeneous reaction of gas flow passing a circular porous cylinder in a channel are presented. The effects of porosity of cylinder, gas inlet velocity, and reaction probability on the reaction process are further analyzed with respect to the characteristics of solid morphology, product concentration, and temperature profile. Numerical results indicate that the reaction rate increases with increasing reaction probability as well as gas inlet velocity. Cylinders with a higher value of porosity and more homogeneous structure also react with gas particles faster. These results agree well with the basic theories of gas–solid reactions, indicating the present model provides a method for describing gas–solid reactions in complex boundaries at mesoscopic level. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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1983 KiB  
Article
Entropy-Assisted Computing of Low-Dissipative Systems
by Ilya V. Karlin, Fabian Bösch, Shyam S. Chikatamarla and Sauro Succi
Entropy 2015, 17(12), 8099-8110; https://doi.org/10.3390/e17127867 - 08 Dec 2015
Cited by 12 | Viewed by 4905
Abstract
Entropy feedback is reviewed and highlighted as the guiding principle to reach extremely low dissipation. This principle is illustrated through turbulent flow simulations using the entropic lattice Boltzmann scheme. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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2578 KiB  
Article
From Lattice Boltzmann Method to Lattice Boltzmann Flux Solver
by Yan Wang, Liming Yang and Chang Shu
Entropy 2015, 17(11), 7713-7735; https://doi.org/10.3390/e17117713 - 13 Nov 2015
Cited by 44 | Viewed by 7536
Abstract
Based on the lattice Boltzmann method (LBM), the lattice Boltzmann flux solver (LBFS), which combines the advantages of conventional Navier–Stokes solvers and lattice Boltzmann solvers, was proposed recently. Specifically, LBFS applies the finite volume method to solve the macroscopic governing equations which provide [...] Read more.
Based on the lattice Boltzmann method (LBM), the lattice Boltzmann flux solver (LBFS), which combines the advantages of conventional Navier–Stokes solvers and lattice Boltzmann solvers, was proposed recently. Specifically, LBFS applies the finite volume method to solve the macroscopic governing equations which provide solutions for macroscopic flow variables at cell centers. In the meantime, numerical fluxes at each cell interface are evaluated by local reconstruction of LBM solution. In other words, in LBFS, LBM is only locally applied at the cell interface for one streaming step. This is quite different from the conventional LBM, which is globally applied in the whole flow domain. This paper shows three different versions of LBFS respectively for isothermal, thermal and compressible flows and their relationships with the standard LBM. In particular, the performance of isothermal LBFS in terms of accuracy, efficiency and stability is investigated by comparing it with the standard LBM. The thermal LBFS is simplified by using the D2Q4 lattice velocity model and its performance is examined by its application to simulate natural convection with high Rayleigh numbers. It is demonstrated that the compressible LBFS can be effectively used to simulate both inviscid and viscous flows by incorporating non-equilibrium effects into the process for inviscid flux reconstruction. Several numerical examples, including lid-driven cavity flow, natural convection in a square cavity at Rayleigh numbers of 107 and 108 and transonic flow around a staggered-biplane configuration, are tested on structured or unstructured grids to examine the performance of three LBFS versions. Good agreements have been achieved with the published data, which validates the capability of LBFS in simulating a variety of flow problems. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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233 KiB  
Article
A Truncation Scheme for the BBGKY2 Equation
by Gregor Chliamovitch, Orestis Malaspinas and Bastien Chopard
Entropy 2015, 17(11), 7522-7529; https://doi.org/10.3390/e17117522 - 30 Oct 2015
Cited by 4 | Viewed by 4457
Abstract
In recent years, the maximum entropy principle has been applied to a wide range of different fields, often successfully. While these works are usually focussed on cross-disciplinary applications, the point of this letter is instead to reconsider a fundamental point of kinetic theory. [...] Read more.
In recent years, the maximum entropy principle has been applied to a wide range of different fields, often successfully. While these works are usually focussed on cross-disciplinary applications, the point of this letter is instead to reconsider a fundamental point of kinetic theory. Namely, we shall re-examine the Stosszahlansatz leading to the irreversible Boltzmann equation at the light of the MaxEnt principle. We assert that this way of thinking allows to move one step further than the factorization hypothesis and provides a coherent—though implicit—closure scheme for the two-particle distribution function. Such higher-order dependences are believed to open the way to a deeper understanding of fluctuating phenomena. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
1468 KiB  
Article
Extension of the Improved Bounce-Back Scheme for Electrokinetic Flow in the Lattice Boltzmann Method
by Qing Chen, Hongping Zhou, Xuesong Jiang, Linyun Xu, Qing Li and Yu Ru
Entropy 2015, 17(11), 7406-7419; https://doi.org/10.3390/e17117406 - 28 Oct 2015
Cited by 7 | Viewed by 5463
Abstract
In this paper, an improved bounce-back boundary treatment for fluid systems in the lattice Boltzmann method [Yin, X.; Zhang J. J. Comput. Phys. 2012, 231, 4295–4303] is extended to handle the electrokinetic flows with complex boundary shapes and conditions. Several numerical [...] Read more.
In this paper, an improved bounce-back boundary treatment for fluid systems in the lattice Boltzmann method [Yin, X.; Zhang J. J. Comput. Phys. 2012, 231, 4295–4303] is extended to handle the electrokinetic flows with complex boundary shapes and conditions. Several numerical simulations are performed to validate the electric boundary treatment. Simulations are presented to demonstrate the accuracy and capability of this method in dealing with complex surface potential situations, and simulated results are compared with analytical predictions with excellent agreement. This method could be useful for electrokinetic simulations with complex boundaries, and can also be readily extended to other phenomena and processes. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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2375 KiB  
Article
Two-Dimensional Lattice Boltzmann for Reactive Rayleigh–Bénard and Bénard–Poiseuille Regimes
by Suemi Rodríguez-Romo and Oscar Ibañez-Orozco
Entropy 2015, 17(10), 6698-6711; https://doi.org/10.3390/e17106698 - 29 Sep 2015
Viewed by 5492
Abstract
We perform a computer simulation of the reaction-diffusion and convection that takes place in Rayleigh–Bénard and Bénard–Poiseuille regimes. The lattice Boltzmann equation (LBE) is used along with the Boussinesq approximation to solve the non-linear coupled differential equations that govern the systems’ thermo-hydrodynamics. Another [...] Read more.
We perform a computer simulation of the reaction-diffusion and convection that takes place in Rayleigh–Bénard and Bénard–Poiseuille regimes. The lattice Boltzmann equation (LBE) is used along with the Boussinesq approximation to solve the non-linear coupled differential equations that govern the systems’ thermo-hydrodynamics. Another LBE, is introduced to calculate the evolution concentration of the chemical species involved in the chemical reactions. The simulations are conducted at low Reynolds numbers and in terms of steady state between the first and second thermo-hydrodynamics instability. The results presented here (with no chemical reactions) are in good agreement with those reported in the scientific literature which gives us high expectations about the reliability of the chemical kinetics simulation. Some examples are provided. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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780 KiB  
Article
Short-Lived Lattice Quasiparticles for Strongly Interacting Fluids
by Miller Mendoza Jimenez and Sauro Succi
Entropy 2015, 17(9), 6169-6178; https://doi.org/10.3390/e17096169 - 03 Sep 2015
Cited by 2 | Viewed by 4448
Abstract
It is shown that lattice kinetic theory based on short-lived quasiparticles proves very effective in simulating the complex dynamics of strongly interacting fluids (SIF). In particular, it is pointed out that the shear viscosity of lattice fluids is the sum of two contributions, [...] Read more.
It is shown that lattice kinetic theory based on short-lived quasiparticles proves very effective in simulating the complex dynamics of strongly interacting fluids (SIF). In particular, it is pointed out that the shear viscosity of lattice fluids is the sum of two contributions, one due to the usual interactions between particles (collision viscosity) and the other due to the interaction with the discrete lattice (propagation viscosity). Since the latter is negative, the sum may turn out to be orders of magnitude smaller than each of the two contributions separately, thus providing a mechanism to access SIF regimes at ordinary values of the collisional viscosity. This concept, as applied to quantum superfluids in one-dimensional optical lattices, is shown to reproduce shear viscosities consistent with the AdS-CFT holographic bound on the viscosity/entropy ratio. This shows that lattice kinetic theory continues to hold for strongly coupled hydrodynamic regimes where continuum kinetic theory may no longer be applicable. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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332 KiB  
Article
The Effect of a Long-Range Correlated-Hopping Interaction on Bariev Spin Chains
by Tao Yang, Fa-Kai Wen, Kun Hao, Li-Ke Cao and Rui-Hong Yue
Entropy 2015, 17(9), 6044-6055; https://doi.org/10.3390/e17096044 - 28 Aug 2015
Cited by 1 | Viewed by 4813
Abstract
We introduce a long-range particle and spin interaction into the standard Bariev model and show that this interaction is equivalent to a phase shift in the kinetic term of the Hamiltonian. When the particles circle around the chain and across the boundary, the [...] Read more.
We introduce a long-range particle and spin interaction into the standard Bariev model and show that this interaction is equivalent to a phase shift in the kinetic term of the Hamiltonian. When the particles circle around the chain and across the boundary, the accumulated phase shift acts as a twist boundary condition with respect to the normal periodic boundary condition. This boundary phase term depends on the total number of particles in the system and also the number of particles in different spin states, which relates to the spin fluctuations in the system. The model is solved exactly via a unitary transformation by the coordinate Bethe ansatz. We calculate the Bethe equations and work out the energy spectrum with varying number of particles and spins. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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233 KiB  
Article
Exact Solutions of Non-Linear Lattice Equations by an Improved Exp-Function Method
by Sheng Zhang, Jiahong Li and Yingying Zhou
Entropy 2015, 17(5), 3182-3193; https://doi.org/10.3390/e17053182 - 13 May 2015
Cited by 9 | Viewed by 4765
Abstract
In this paper, the exp-function method is improved to construct exact solutions of non-linear lattice equations by modifying its exponential function ansätz. The improved method has two advantages. One is that it can solve non-linear lattice equations with variable coefficients, and the other [...] Read more.
In this paper, the exp-function method is improved to construct exact solutions of non-linear lattice equations by modifying its exponential function ansätz. The improved method has two advantages. One is that it can solve non-linear lattice equations with variable coefficients, and the other is that it is not necessary to balance the highest order derivative with the highest order nonlinear term in the procedure of determining the exponential function ansätz. To show the advantages of this improved method, a variable-coefficient mKdV lattice equation is considered. As a result, new exact solutions, which include kink-type solutions and bell-kink-type solutions, are obtained. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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Review

Jump to: Research

1886 KiB  
Review
Nonlinear Phenomena of Ultracold Atomic Gases in Optical Lattices: Emergence of Novel Features in Extended States
by Gentaro Watanabe, B. Prasanna Venkatesh and Raka Dasgupta
Entropy 2016, 18(4), 118; https://doi.org/10.3390/e18040118 - 31 Mar 2016
Cited by 15 | Viewed by 5023
Abstract
The system of a cold atomic gas in an optical lattice is governed by two factors: nonlinearity originating from the interparticle interaction, and the periodicity of the system set by the lattice. The high level of controllability associated with such an arrangement allows [...] Read more.
The system of a cold atomic gas in an optical lattice is governed by two factors: nonlinearity originating from the interparticle interaction, and the periodicity of the system set by the lattice. The high level of controllability associated with such an arrangement allows for the study of the competition and interplay between these two, and gives rise to a whole range of interesting and rich nonlinear effects. This review covers the basic idea and overview of such nonlinear phenomena, especially those corresponding to extended states. This includes “swallowtail” loop structures of the energy band, Bloch states with multiple periodicity, and those in “nonlinear lattices”, i.e., systems with the nonlinear interaction term itself being a periodic function in space. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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20200 KiB  
Review
Hydrodynamic Force Evaluation by Momentum Exchange Method in Lattice Boltzmann Simulations
by Binghai Wen, Chaoying Zhang and Haiping Fang
Entropy 2015, 17(12), 8240-8266; https://doi.org/10.3390/e17127876 - 17 Dec 2015
Cited by 22 | Viewed by 6666
Abstract
As a native scheme to evaluate hydrodynamic force in the lattice Boltzmann method, the momentum exchange method has some excellent features, such as simplicity, accuracy, high efficiency and easy parallelization. Especially, it is independent of boundary geometry, preventing from solving the Navier–Stokes equations [...] Read more.
As a native scheme to evaluate hydrodynamic force in the lattice Boltzmann method, the momentum exchange method has some excellent features, such as simplicity, accuracy, high efficiency and easy parallelization. Especially, it is independent of boundary geometry, preventing from solving the Navier–Stokes equations on complex boundary geometries in the boundary-integral methods. We review the origination and main developments of the momentum exchange method in lattice Boltzmann simulations. Then several practical techniques to fill newborn fluid nodes are discussed for the simulations of fluid-structure interactions. Finally, some representative applications show the wide applicability of the momentum exchange method, such as movements of rigid particles, interactions of deformation particles, particle suspensions in turbulent flow and multiphase flow, etc. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
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