Hamiltonian and Overdamped Complex Systems, Symmetry of Phase-Space Occupancy

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (31 March 2024) | Viewed by 8549

Special Issue Editors


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Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros s/n, 28040 Madrid, Spain

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Department of Physics and Astronomy, University of Catania and INFN-Section of Catania, 95123 Catania, Italy
Interests: complex systems; statistical mechanics; complex networks; agent-based models
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Guest Editor
Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey
Interests: statistical mechanics; nonlinear dynamics; complex systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Complexity naturally arises in nonlinear physics and elsewhere, for instance through processes of successive bifurcations generating complex spatiotemporal patterns, or in nontrivial configurations of the phase space of chaotic Hamiltonian systems possibly involving long-range interactions, where one typically can find a mixture of chaotic seas and regions with regular motion that can lead to statistical distributions with power-law and scale-free tails.

The same can occur in systems such as complex plasmas, superconductors or colloidal systems, which can be described by dissipative approaches including repulsive particles whose equation of motion, in the overdamped limit, takes the form of a first-order differential equations, where the velocity of the particles is proportional to the force over them. Additionally, in this case, for various kinds of repulsive potentials, anomalous distributions of the local density and of the velocities can be found, both analytically and numerically.

The phase space of strongly chaotic Hamiltonian systems presents translational symmetry, as expected within Boltzmann-Gibbs statistical mechanics, consistently exhibiting the celebrated exponential distribution of energies and the Maxwellian distribution of momenta. In contrast, alternative, anomalous, distributions arise whenever this basic symmetry is broken. Such is frequently the case when weak chaos is present, for instance if long-range interactions are involved, or when memory effects play an important role, overdamped complex systems constituting a paradigmatic example of this situation. The study of the symmetry of the phase-space occupancy is thus one of the crucial features to characterize the thermostatistical properties of many complex systems. 

The aim of this Special Issue is to stimulate further investigations along these directions, particularly in connection with frameworks such as q-generalized statistical mechanics, superstatistics, stochastic thermodynamics, and other modern statistical mechanical approaches, in both classical and quantum systems where nonlocal correlations are relevant.

We are soliciting contributions (both research and review articles) covering a broad range of topics related to complexity and symmetries in space and/or time, at micro-, meso-, and macroscopic scales, possibly revealing either grounding aspects or their applications.


Prof. Antonio Rodríguez
Prof. Dr. Alessandro Pluchino
Prof. Dr. Ugur Tirnakli
Guest Editors

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Published Papers (6 papers)

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Research

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12 pages, 373 KiB  
Article
About Stability of Nonlinear Stochastic Differential Equations with State-Dependent Delay
by Leonid Shaikhet
Symmetry 2022, 14(11), 2307; https://doi.org/10.3390/sym14112307 - 03 Nov 2022
Cited by 2 | Viewed by 927
Abstract
A nonlinear stage-structured population model with a state-dependent delay under stochastic perturbations is investigated. Delay-independent and delay-dependent conditions of stability in probability for two equilibria of the considered system are obtained via the general method of Lyapunov functionals construction and the method of [...] Read more.
A nonlinear stage-structured population model with a state-dependent delay under stochastic perturbations is investigated. Delay-independent and delay-dependent conditions of stability in probability for two equilibria of the considered system are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The model under consideration is not the aim of the work and was chosen only to demonstrate the proposed research method, which can be used for the study of other types of nonlinear systems with a state-dependent delay. Full article
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18 pages, 742 KiB  
Article
Exact Time-Dependent Solutions and Information Geometry of a Rocking Ratchet
by Eun-jin Kim and Rainer Hollerbach
Symmetry 2022, 14(2), 314; https://doi.org/10.3390/sym14020314 - 03 Feb 2022
Cited by 1 | Viewed by 986
Abstract
The noise-induced transport due to spatial symmetry-breaking is a key mechanism for the generation of a uni-directional motion by a Brownian motor. By utilising an asymmetric sawtooth periodic potential and three different types of periodic forcing G(t) (sinusoidal, square and [...] Read more.
The noise-induced transport due to spatial symmetry-breaking is a key mechanism for the generation of a uni-directional motion by a Brownian motor. By utilising an asymmetric sawtooth periodic potential and three different types of periodic forcing G(t) (sinusoidal, square and sawtooth waves) with period T and amplitude A, we investigate the performance (energetics, mean current, Stokes efficiency) of a rocking ratchet in light of thermodynamic quantities (entropy production) and the path-dependent information geometric measures. For each G(t), we calculate exact time-dependent probability density functions under different conditions by varying T, A and the strength of the stochastic noise D in an unprecedentedly wide range. Overall similar behaviours are found for different cases of G(t). In particular, in all cases, the current, Stokes efficiency and the information rate normalised by A and D exhibit one or multiple local maxima and minima as A increases. However, the dependence of the current and Stokes efficiency on A can be quite different, while the behaviour of the information rate normalised by A and D tends to resemble that of the Stokes efficiency. In comparison, the irreversibility measured by a normalised entropy production is independent of A. The results indicate the utility of the information geometry as a proxy of a motor efficiency. Full article
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13 pages, 323 KiB  
Article
Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction
by Angel Ricardo Plastino, Roseli S. Wedemann and Constantino Tsallis
Symmetry 2021, 13(9), 1621; https://doi.org/10.3390/sym13091621 - 03 Sep 2021
Cited by 5 | Viewed by 1660
Abstract
We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same [...] Read more.
We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those moving to the left. The dynamics of the system, effectively described by a non-linear, Fokker–Planck equation, exhibits peculiar features related to the way in which the drag force depends on velocity. The evolution equation satisfies an H-theorem involving the Sq nonadditive entropy, and admits particular, exact, time-dependent solutions closely related, but not identical, to the q-Gaussian densities. The departure from the canonical, q-Gaussian shape is related to the fact that in one spatial dimension, in contrast to what occurs in two or more spatial dimensions, the drag’s dependence on direction entails that its dependence on velocity is necessarily (and severely) non-linear. The results reported here provide further evidence of the deep connections between overdamped, many-body systems, non-linear Fokker–Planck equations, and the Sq-thermostatistics. Full article
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13 pages, 524 KiB  
Article
The Superconducting Critical Temperature
by Mike Guidry, Yang Sun and Lian-Ao Wu
Symmetry 2021, 13(5), 911; https://doi.org/10.3390/sym13050911 - 20 May 2021
Cited by 1 | Viewed by 1869
Abstract
Two principles govern the critical temperature for superconducting transitions: (1) intrinsic strength of the pair coupling and (2) the effect of the many-body environments on the efficiency of that coupling. Most discussions take into account only the former, but we argue that the [...] Read more.
Two principles govern the critical temperature for superconducting transitions: (1) intrinsic strength of the pair coupling and (2) the effect of the many-body environments on the efficiency of that coupling. Most discussions take into account only the former, but we argue that the properties of unconventional superconductors are governed more often by the latter, through dynamical symmetry relating to normal and superconducting states. Differentiating these effects is essential to charting a path to the highest-temperature superconductors. Full article
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Review

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65 pages, 773 KiB  
Review
Generalized Equations in Quantum Mechanics and Brownian Theory
by Pierre-Henri Chavanis
Symmetry 2023, 15(12), 2195; https://doi.org/10.3390/sym15122195 - 13 Dec 2023
Viewed by 685
Abstract
We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction [...] Read more.
We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction force, an effective thermal pressure, a pressure due to the self-interaction, and a quantum potential. These hydrodynamic equations have a form similar to the damped Euler equations obtained for self-interacting Brownian particles in the theory of simple liquids. In that case, the temperature is due to thermal motion and the pressure arises from spatial correlations between the particles. More generally, the correlations can be accounted for by using the dynamical density functional theory. We determine the excess free energy of Brownian particles that reproduces the standard quantum potential. We then consider a more general form of excess free energy functionals and propose a new class of generalized Schrödinger equations. For a certain form of excess free energy, we recover the generalized Schrödinger equation associated with the Tsallis entropy considered in a previous paper. Full article
33 pages, 11453 KiB  
Review
Nonextensive Footprints in Dissipative and Conservative Dynamical Systems
by Antonio Rodríguez, Alessandro Pluchino, Ugur Tirnakli, Andrea Rapisarda and Constantino Tsallis
Symmetry 2023, 15(2), 444; https://doi.org/10.3390/sym15020444 - 07 Feb 2023
Cited by 3 | Viewed by 1347
Abstract
Despite its centennial successes in describing physical systems at thermal equilibrium, Boltzmann–Gibbs (BG) statistical mechanics have exhibited, in the last several decades, several flaws in addressing out-of-equilibrium dynamics of many nonlinear complex systems. In such circumstances, it has been shown that an appropriate [...] Read more.
Despite its centennial successes in describing physical systems at thermal equilibrium, Boltzmann–Gibbs (BG) statistical mechanics have exhibited, in the last several decades, several flaws in addressing out-of-equilibrium dynamics of many nonlinear complex systems. In such circumstances, it has been shown that an appropriate generalization of the BG theory, known as nonextensive statistical mechanics and based on nonadditive entropies, is able to satisfactorily handle wide classes of anomalous emerging features and violations of standard equilibrium prescriptions, such as ergodicity, mixing, breakdown of the symmetry of homogeneous occupancy of phase space, and related features. In the present study, we review various important results of nonextensive statistical mechanics for dissipative and conservative dynamical systems. In particular, we discuss applications to both discrete-time systems with a few degrees of freedom and continuous-time ones with many degrees of freedom, as well as to asymptotically scale-free networks and systems with diverse dimensionalities and ranges of interactions, of either classical or quantum nature. Full article
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