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Recent Advances in Non-Equilibrium Statistical Mechanics and Its Application

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: closed (30 November 2015) | Viewed by 57304

Special Issue Editor

Department of Physics, Université Libre de Bruxelles (U.L.B.), Bvd du Triomphe, Campus de la Plaine C.P. 231, 1050 Brussels, Belgium
Interests: thermodynamics of irreversible processes; entropy information theory; thermodynamic field theories; metric geometry in thermodynamics; non-linear dynamics; hydrodynamic fluctuations; transport processes in tokamak-plasmas

Special Issue Information

Dear Colleagues,

It is well-known that many natural systems today still remain beyond the scope of currently-known macroscopic thermodynamic methods. Biophysics or physics of the life sciences, for example, will have to deal with fundamental problems in non-equilibrium statistical mechanics. One also finds concepts and models of non-equilibrium physics, in computer science, social science, economy, or in the field of complex systems. However, also well within “classical” physics, major problems remain. A good example is turbulence, the great-unsolved problem of classical physics. Therefore, non-equilibrium statistical mechanics, even today, does not yet exist as a complete and systematic theory, at least, certainly not at the same level as its equilibrium counterpart. However, in recent years, research on non-equilibrium statistical mechanics has made significant progress, in particular in the field of thermodynamics of irreversible processes, viewed as a thermodynamical field theory, of the entropy information theory or the stochastic theory.

The aim of this Special Issue is to encourage scientists to present original and recent developments on non-equilibrium statistical mechanics (stochastic theory, kinetic theory, entropy information theory, etc.) applied to dynamical systems.

One of the objectives of the issue is, therefore, to promote a cross-fertilization among scientists working in a wide range of disciplines ranging from bio-medical dynamical systems to plasmas.

Dynamical and chaotic systems should be treated by Non-equilibrium Statistical Mechanics, e.g., by Stochastic Modeling (Fokker-Planck and/or Master equations, etc.), Kinetic Theory (Boltzmann-equation, Vlasov-Equation, etc.), Entropy Information Theory (Shannon, Rényi, Tsallis entropies, etc.), and Thermodynamics of Irreversible Processes (entropy production, thermodynamically field theories, etc.).

Applications can include mathematical models applied to biomedical systems and plasmas. Concerning plasmas, kinetic theory/non-equilibrium statistical mechanics of weakly turbulent plasmas may be proposed, should be focused on

(i) understanding resonant and non-resonant transport processes of the beam-plasma system, characterizing transitions for increasing beam and/or fluctuation strength.

(ii) understanding the role of sources and collisions in a dissipative system with multiple kinetic resonances.

(iii) complex behaviors, turbulence and self-organization: investigating self-organization of nonlinear systems under the effect of coherent nonlinear interactions vs random perturbations.

Prof. Dr. Giorgio Sonnino
Guest Editor

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

  • stochastic analysis method (Fokker-Planck, Langevin, etc.)
  • time series analysis
  • fluctuation phenomena, random processes, noise, and Brownian motion
  • perturbation and fractional calculus method
  • computational methods in statistical physics and nonlinear dynamics
  • entropy and other measures of information
  • non-equilibrium and irreversible thermodynamics
  • statistical mechanics, thermodynamics, models and pathways
  • self-organized systems, time delay systems, and nonlocal theories and models
  • transport processes in non-equilibrium systems
  • plasma turbulence
  • control of chaos, applications of chaos, high-dimensional and low-dimensional chaos

Published Papers (11 papers)

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Research

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415 KiB  
Article
Chemical Reactions Using a Non-Equilibrium Wigner Function Approach
Entropy 2016, 18(10), 369; https://doi.org/10.3390/e18100369 - 19 Oct 2016
Cited by 3 | Viewed by 3702
Abstract
A three-dimensional model of binary chemical reactions is studied. We consider an ab initio quantum two-particle system subjected to an attractive interaction potential and to a heat bath at thermal equilibrium at absolute temperature T > 0 . Under the sole action of [...] Read more.
A three-dimensional model of binary chemical reactions is studied. We consider an ab initio quantum two-particle system subjected to an attractive interaction potential and to a heat bath at thermal equilibrium at absolute temperature T > 0 . Under the sole action of the attraction potential, the two particles can either be bound or unbound to each other. While at T = 0 , there is no transition between both states, such a transition is possible when T > 0 (due to the heat bath) and plays a key role as k B T approaches the magnitude of the attractive potential. We focus on a quantum regime, typical of chemical reactions, such that: (a) the thermal wavelength is shorter than the range of the attractive potential (lower limit on T) and (b) ( 3 / 2 ) k B T does not exceed the magnitude of the attractive potential (upper limit on T). In this regime, we extend several methods previously applied to analyze the time duration of DNA thermal denaturation. The two-particle system is then described by a non-equilibrium Wigner function. Under Assumptions (a) and (b), and for sufficiently long times, defined by a characteristic time scale D that is subsequently estimated, the general dissipationless non-equilibrium equation for the Wigner function is approximated by a Smoluchowski-like equation displaying dissipation and quantum effects. A comparison with the standard chemical kinetic equations is made. The time τ required for the two particles to transition from the bound state to unbound configurations is studied by means of the mean first passage time formalism. An approximate formula for τ, in terms of D and exhibiting the Arrhenius exponential factor, is obtained. Recombination processes are also briefly studied within our framework and compared with previous well-known methods. Full article
340 KiB  
Article
Hydrodynamic Theories for Flows of Active Liquid Crystals and the Generalized Onsager Principle
Entropy 2016, 18(6), 202; https://doi.org/10.3390/e18060202 - 24 May 2016
Cited by 37 | Viewed by 4743
Abstract
We articulate and apply the generalized Onsager principle to derive transport equations for active liquid crystals in a fixed domain as well as in a free surface domain adjacent to a passive fluid matrix. The Onsager principle ensures fundamental variational structure of the [...] Read more.
We articulate and apply the generalized Onsager principle to derive transport equations for active liquid crystals in a fixed domain as well as in a free surface domain adjacent to a passive fluid matrix. The Onsager principle ensures fundamental variational structure of the models as well as dissipative properties of the passive component in the models, irrespective of the choice of scale (kinetic to continuum) and of the physical potentials. Many popular models for passive and active liquid crystals in a fixed domain subject to consistent boundary conditions at solid walls, as well as active liquid crystals in a free surface domain with consistent transport equations along the free boundaries, can be systematically derived from the generalized Onsager principle. The dynamical boundary conditions are shown to reduce to the static boundary conditions for passive liquid crystals used previously. Full article
2696 KiB  
Article
Analysis of the Chaotic Behavior of the Lower Hybrid Wave Propagation in Magnetised Plasma by Hamiltonian Theory
Entropy 2016, 18(5), 175; https://doi.org/10.3390/e18050175 - 07 May 2016
Cited by 2 | Viewed by 4622
Abstract
The Hamiltonian character of the ray tracing equations describing the propagation of the Lower Hybrid Wave (LHW) in a magnetic confined plasma device (tokamak) is investigated in order to study the evolution of the parallel wave number along the propagation path. The chaotic [...] Read more.
The Hamiltonian character of the ray tracing equations describing the propagation of the Lower Hybrid Wave (LHW) in a magnetic confined plasma device (tokamak) is investigated in order to study the evolution of the parallel wave number along the propagation path. The chaotic diffusion of the “time-averaged” parallel wave number at higher values (with respect to that launched by the antenna at the plasma edge) has been evaluated, in order to find an explanation of the filling of the spectral gap (Fisch, 1987) by “Hamiltonian chaos” in the Lower Hybrid Current Drive (LHCD) experiments (Fisch, 1978). The present work shows that the increase of the parallel wave number \(n_{\parallel}\) due to toroidal effects, in the case of the typical plasma parameters of the Frascati Tokamak Upgrade (FTU) experiment, is insufficient to explain the filling of the spectral gap, and the consequent current drive and another mechanism must come into play to justify the wave absorption by Landau damping. Analytical calculations have been supplemented by a numerical algorithm based on the symplectic integration of the ray equations implemented in a ray tracing code, in order to preserve exactly the symplectic character of a Hamiltonian flow. Full article
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4120 KiB  
Article
Mixed Diffusive-Convective Relaxation of a Warm Beam of Energetic Particles in Cold Plasma
Entropy 2016, 18(4), 143; https://doi.org/10.3390/e18040143 - 16 Apr 2016
Cited by 14 | Viewed by 3984
Abstract
This work addresses the features of fast particle transport in the bump-on-tail problem for varying the width of the fluctuation spectrum, in the view of possible applications to studies of energetic particle transport in fusion plasmas. Our analysis is built around the idea [...] Read more.
This work addresses the features of fast particle transport in the bump-on-tail problem for varying the width of the fluctuation spectrum, in the view of possible applications to studies of energetic particle transport in fusion plasmas. Our analysis is built around the idea that strongly-shaped beams do not relax through diffusion only and that there exists an intermediate time scale where the relaxations are convective (ballistic-like). We cast this idea in the form of a self-consistent nonlinear dynamical model, which extends the classic equations of the quasi-linear theory to “broad” beams with internal structure. We also present numerical simulation results of the relaxation of a broad beam of energetic particles in cold plasma. These generally demonstrate the mixed diffusive-convective features of supra-thermal particle transport essentially depending on nonlinear wave-particle interactions and phase-space structures. Taking into account the modes of the stable linear spectrum is crucial for the self-consistent evolution of the distribution function and the fluctuation intensity spectrum. Full article
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495 KiB  
Article
Phase Transitions in Equilibrium and Non-Equilibrium Models on Some Topologies
Entropy 2016, 18(3), 81; https://doi.org/10.3390/e18030081 - 03 Mar 2016
Cited by 8 | Viewed by 4970
Abstract
On some regular and non-regular topologies, we studied the critical properties of models that present up-down symmetry, like the equilibrium Ising model and the nonequilibrium majority vote model. These are investigated on networks, like Apollonian (AN), Barabási–Albert (BA), small-worlds (SW), Voronoi–Delaunay (VD) and [...] Read more.
On some regular and non-regular topologies, we studied the critical properties of models that present up-down symmetry, like the equilibrium Ising model and the nonequilibrium majority vote model. These are investigated on networks, like Apollonian (AN), Barabási–Albert (BA), small-worlds (SW), Voronoi–Delaunay (VD) and Erdös–Rényi (ER) random graphs. The review here is on phase transitions, critical points, exponents and universality classes that are compared to the results obtained for these models on regular square lattices (SL). Full article
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943 KiB  
Article
Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay
Entropy 2016, 18(3), 64; https://doi.org/10.3390/e18030064 - 27 Feb 2016
Cited by 24 | Viewed by 5321
Abstract
We consider the paradigmatic Brusselator model for the study of dissipative structures in far from equilibrium systems. In two dimensions, we show the occurrence of a self-replication phenomenon leading to the fragmentation of a single localized spot into four daughter spots. This instability [...] Read more.
We consider the paradigmatic Brusselator model for the study of dissipative structures in far from equilibrium systems. In two dimensions, we show the occurrence of a self-replication phenomenon leading to the fragmentation of a single localized spot into four daughter spots. This instability affects the new spots and leads to splitting behavior until the system reaches a hexagonal stationary pattern. This phenomenon occurs in the absence of delay feedback. In addition, we incorporate a time-delayed feedback loop in the Brusselator model. In one dimension, we show that the delay feedback induces extreme events in a chemical reaction diffusion system. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical distribution, which is often considered as a signature of the presence of rogue waves, appears for sufficiently strong feedback intensity. The generality of our analysis suggests that the feedback-induced instability leading to the spontaneous formation of rogue waves in a controllable way is a universal phenomenon. Full article
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219 KiB  
Article
Non-Extensive Entropic Distance Based on Diffusion: Restrictions on Parameters in Entropy Formulae
Entropy 2016, 18(2), 42; https://doi.org/10.3390/e18020042 - 27 Jan 2016
Cited by 3 | Viewed by 4152
Abstract
Based on a diffusion-like master equation we propose a formula using the Bregman divergence for measuring entropic distance in terms of different non-extensive entropy expressions. We obtain the non-extensivity parameter range for a universal approach to the stationary distribution by simple diffusive dynamics [...] Read more.
Based on a diffusion-like master equation we propose a formula using the Bregman divergence for measuring entropic distance in terms of different non-extensive entropy expressions. We obtain the non-extensivity parameter range for a universal approach to the stationary distribution by simple diffusive dynamics for the Tsallis and the Kaniadakis entropies, for the Hanel–Thurner generalization, and finally for a recently suggested log-log type entropy formula which belongs to diverging variance in the inverse temperature superstatistics. Full article
452 KiB  
Article
Interacting Brownian Swarms: Some Analytical Results
Entropy 2016, 18(1), 27; https://doi.org/10.3390/e18010027 - 14 Jan 2016
Cited by 4 | Viewed by 3966
Abstract
We consider the dynamics of swarms of scalar Brownian agents subject to local imitation mechanisms implemented using mutual rank-based interactions. For appropriate values of the underlying control parameters, the swarm propagates tightly and the distances separating successive agents are iid exponential random variables. [...] Read more.
We consider the dynamics of swarms of scalar Brownian agents subject to local imitation mechanisms implemented using mutual rank-based interactions. For appropriate values of the underlying control parameters, the swarm propagates tightly and the distances separating successive agents are iid exponential random variables. Implicitly, the implementation of rank-based mutual interactions, requires that agents have infinite interaction ranges. Using the probabilistic size of the swarm’s support, we analytically estimate the critical interaction range below that flocked swarms cannot survive. In the second part of the paper, we consider the interactions between two flocked swarms of Brownian agents with finite interaction ranges. Both swarms travel with different barycentric velocities, and agents from both swarms indifferently interact with each other. For appropriate initial configurations, both swarms eventually collide (i.e., all agents interact). Depending on the values of the control parameters, one of the following patterns emerges after collision: (i) Both swarms remain essentially flocked, or (ii) the swarms become ultimately quasi-free and recover their nominal barycentric speeds. We derive a set of analytical flocking conditions based on the generalized rank-based Brownian motion. An extensive set of numerical simulations corroborates our analytical findings. Full article
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4596 KiB  
Article
The Bogdanov–Takens Normal Form: A Minimal Model for Single Neuron Dynamics
Entropy 2015, 17(12), 7859-7874; https://doi.org/10.3390/e17127850 - 30 Nov 2015
Cited by 4 | Viewed by 5359
Abstract
Conductance-based (CB) models are a class of high dimensional dynamical systems derived from biophysical principles to describe in detail the electrical dynamics of single neurons. Despite the high dimensionality of these models, the dynamics observed for realistic parameter values is generically planar and [...] Read more.
Conductance-based (CB) models are a class of high dimensional dynamical systems derived from biophysical principles to describe in detail the electrical dynamics of single neurons. Despite the high dimensionality of these models, the dynamics observed for realistic parameter values is generically planar and can be minimally described by two equations. In this work, we derive the conditions to have a Bogdanov–Takens (BT) bifurcation in CB models, and we argue that it is plausible that these conditions are verified for experimentally-sensible values of the parameters. We show numerically that the cubic BT normal form, a two-variable dynamical system, exhibits all of the diversity of bifurcations generically observed in single neuron models. We show that the Morris–Lecar model is approximately equivalent to the cubic Bogdanov–Takens normal form for realistic values of parameters. Furthermore, we explicitly calculate the quadratic coefficient of the BT normal form for a generic CB model, obtaining that by constraining the theoretical I-V curve’s curvature to match experimental observations, the normal form appears to be naturally cubic. We propose the cubic BT normal form as a robust minimal model for single neuron dynamics that can be derived from biophysically-realistic CB models. Full article
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258 KiB  
Article
Minimum Dissipation Principle in Nonlinear Transport
Entropy 2015, 17(11), 7567-7583; https://doi.org/10.3390/e17117567 - 30 Oct 2015
Cited by 4 | Viewed by 4465
Abstract
We extend Onsager’s minimum dissipation principle to stationary states that are only subject to local equilibrium constraints, even when the transport coefficients depend on the thermodynamic forces. Crucial to this generalization is a decomposition of the thermodynamic forces into those that are held [...] Read more.
We extend Onsager’s minimum dissipation principle to stationary states that are only subject to local equilibrium constraints, even when the transport coefficients depend on the thermodynamic forces. Crucial to this generalization is a decomposition of the thermodynamic forces into those that are held fixed by the boundary conditions and the subspace that is orthogonal with respect to the metric defined by the transport coefficients. We are then able to apply Onsager and Machlup’s proof to the second set of forces. As an example, we consider two-dimensional nonlinear diffusion coupled to two reservoirs at different temperatures. Our extension differs from that of Bertini et al. in that we assume microscopic irreversibility, and we allow a nonlinear dependence of the fluxes on the forces. Full article
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Review

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930 KiB  
Review
Energy Flows in Low-Entropy Complex Systems
Entropy 2015, 17(12), 8007-8018; https://doi.org/10.3390/e17127857 - 04 Dec 2015
Cited by 14 | Viewed by 10842
Abstract
Nature’s many complex systems—physical, biological, and cultural—are islands of low-entropy order within increasingly disordered seas of surrounding, high-entropy chaos. Energy is a principal facilitator of the rising complexity of all such systems in the expanding Universe, including galaxies, stars, planets, life, society, and [...] Read more.
Nature’s many complex systems—physical, biological, and cultural—are islands of low-entropy order within increasingly disordered seas of surrounding, high-entropy chaos. Energy is a principal facilitator of the rising complexity of all such systems in the expanding Universe, including galaxies, stars, planets, life, society, and machines. A large amount of empirical evidence—relating neither entropy nor information, rather energy—suggests that an underlying simplicity guides the emergence and growth of complexity among many known, highly varied systems in the 14-billion-year-old Universe, from big bang to humankind. Energy flows are as centrally important to life and society as they are to stars and galaxies. In particular, the quantity energy rate density—the rate of energy flow per unit mass—can be used to explicate in a consistent, uniform, and unifying way a huge collection of diverse complex systems observed throughout Nature. Operationally, those systems able to utilize optimal amounts of energy tend to survive and those that cannot are non-randomly eliminated. Full article
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