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Entropy
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Non-Hamiltonian Dynamics, Open Systems and Entropy
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Non-Hamiltonian Dynamics, Open Systems and Entropy

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

Deadline for manuscript submissions: closed (30 April 2022) | Viewed by 7917

Special Issue Editors

Dr. Massimo Materassi

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Guest Editor
Istituto dei Sistemi Complessi del Consiglio Nazionale delle Ricerche, CNR-ISC, via Madonna del Piano 10, 50019 Sesto Fiorentino (Firenze), Italy
Interests: dissipative processes; metriplectic dynamics; information theory and dynamical systems; space weather
Special Issues, Collections and Topics in MDPI journals
Dr. Giuseppe Consolini

E-Mail Website
Guest Editor
National Institute for Astrophysics-Institute for Space Astrophysics and Planetology (INAF-IAPS), 00133 Rome, Italy
Interests: complexity and turbulence in space plasmas; dynamical systems and information theory approaches to Sun-Earth relationships and Earth’s magnetospheric dynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In Hamiltonian systems, the dynamics, expressed in terms of Poisson brackets, results in a purely algebraic construction, a matter of differential geometry and topology. Algebrization of dynamics is also the simplest path to quantization, as stated by Dirac’s isomorphism, mapping the classical Poisson bracket algebra of Hamiltonian systems into the algebra of commutation brackets of quantum observables.

Classical systems with dissipation and open quantum systems are non-Hamiltonian systems, and the problem of their algebrization is currently under the spotlight. Classical systems in which dissipation coexists with a Hamiltonian structure are algebrized as metriplectic systems, or in the equivalent scheme named GENERIC, more focused on tensor operators. Open quantum systems are described with the celebrated Lindblad equations, showing striking analogies with the classical metriplectic formalism.

An excellent point about non-Hamiltonian dynamics is the role apparently played by entropy-like quantities: in classical metriplectic systems, the entropy of the medium draining mechanical energy via dissipation generates the irreversible part of dynamics. In quantum open systems, entanglement plays the role of coupling the system with the environment, giving rise to its classical properties, in a suitable macroscopic limit.

In this Special Issue, contributions will be collected on the unifying role of entropy-like quantities in algebrized dynamics of non-Hamiltonian systems, both classical and quantum. In particular, the objective is that of investigating the general relationship between irreversibility and classical behaviour, and the appearance of information-like quantities in the dynamics.

Dr. Massimo Materassi
Dr. Giuseppe Consolini
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

  • dissipation
  • irreversibility
  • non-Hamiltonian systems
  • open quantum systems
  • metriplectic dynamics
  • GENERIC formalism
  • non-equilibrium thermodynamics
  • Lindblad equations
  • entanglement and quantum entropy
  • emergence of thermodynamics

Published Papers (4 papers)

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Research

20 pages, 969 KiB  
Article
Electron Kinetic Entropy across Quasi-Perpendicular Shocks
by Martin Lindberg, Andris Vaivads, Savvas Raptis, Per-Arne Lindqvist, Barbara L. Giles and Daniel Jonathan Gershman
Entropy 2022, 24(6), 745; https://doi.org/10.3390/e24060745 - 24 May 2022
Cited by 4 | Viewed by 1531
Abstract
We use Magnetospheric Multiscale (MMS) data to study electron kinetic entropy per particle Se across Earth’s quasi-perpendicular bow shock. We have selected 22 shock crossings covering a wide range of shock conditions. Measured distribution functions are calibrated and corrected for spacecraft potential, [...] Read more.
We use Magnetospheric Multiscale (MMS) data to study electron kinetic entropy per particle Se across Earth’s quasi-perpendicular bow shock. We have selected 22 shock crossings covering a wide range of shock conditions. Measured distribution functions are calibrated and corrected for spacecraft potential, secondary electron contamination, lack of measurements at the lowest energies and electron density measurements based on plasma frequency measurements. All crossings display an increase in electron kinetic entropy across the shock ΔSe being positive or zero within their error margin. There is a strong dependence of ΔSe on the change in electron temperature, ΔTe, and the upstream electron plasma beta, βe. Shocks with large ΔTe have large ΔSe. Shocks with smaller βe are associated with larger ΔSe. We use the values of ΔSe, ΔTe and density change Δne to determine the effective adiabatic index of electrons for each shock crossing. The average effective adiabatic index is ⟨γe⟩=1.64±0.07. Full article
(This article belongs to the Special Issue Non-Hamiltonian Dynamics, Open Systems and Entropy)
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14 pages, 348 KiB  
Article
Metriplectic Structure of a Radiation–Matter-Interaction Toy Model
by Massimo Materassi, Giulia Marcucci and Claudio Conti
Entropy 2022, 24(4), 506; https://doi.org/10.3390/e24040506 - 04 Apr 2022
Viewed by 1512
Abstract
A dynamical system defined by a metriplectic structure is a dissipative model characterized by a specific pair of tensors, which defines a Leibniz bracket; and a free energy, formed by a “Hamiltonian” and an entropy, playing the role of dynamics generator. Generally, these [...] Read more.
A dynamical system defined by a metriplectic structure is a dissipative model characterized by a specific pair of tensors, which defines a Leibniz bracket; and a free energy, formed by a “Hamiltonian” and an entropy, playing the role of dynamics generator. Generally, these tensors are a Poisson bracket tensor, describing the Hamiltonian part of the dynamics, and a symmetric metric tensor, that models purely dissipative dynamics. In this paper, the metriplectic system describing a simplified two-photon absorption by a two-level atom is disclosed. The Hamiltonian component is sufficient to describe the free electromagnetic radiation. The metric component encodes the radiation–matter coupling, driving the system to an asymptotically stable state in which the excited level of the atom is populated due to absorption, and the radiation has disappeared. First, a description of the system is used, based on the real–imaginary decomposition of the electromagnetic field phasor; then, the whole metriplectic system is re-written in terms of the phase–amplitude pair, named Madelung variables. This work is intended as a first result to pave the way for applying the metriplectic formalism to many other irreversible processes in nonlinear optics. Full article
(This article belongs to the Special Issue Non-Hamiltonian Dynamics, Open Systems and Entropy)
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19 pages, 338 KiB  
Article
Para-Hamiltonian form for General Autonomous ODE Systems: Introductory Results
by Artur Kobus and Jan L. Cieśliński
Entropy 2022, 24(3), 338; https://doi.org/10.3390/e24030338 - 26 Feb 2022
Cited by 1 | Viewed by 1628
Abstract
We propose a new tool to deal with autonomous ODE systems for which the solution to the Hamiltonian inverse problem is not available in the usual, classical sense. Our approach allows a class of formally conserved quantities to be constructed for dynamical systems [...] Read more.
We propose a new tool to deal with autonomous ODE systems for which the solution to the Hamiltonian inverse problem is not available in the usual, classical sense. Our approach allows a class of formally conserved quantities to be constructed for dynamical systems showing dissipative behavior and other, more general, phenomena. The only ingredients of this new framework are Hamiltonian geometric mechanics (to sustain certain desirable properties) and the direct reformulation of the notion of the derivative along the phase curve. This seemingly odd and inconsistent marriage of apparently remote ideas leads to the existence of the generator of motion for every autonomous ODE system. Having constructed the generator, we obtained the Lie invariance of the symplectic form ω for free. Various examples are presented, ranging from mathematics, classical mechanics, and thermodynamics, to chemical kinetics and population dynamics in biology. Applications of these ideas to geometric integration techniques of numerical analysis are suggested. Full article
(This article belongs to the Special Issue Non-Hamiltonian Dynamics, Open Systems and Entropy)
44 pages, 449 KiB  
Article
General Non-Markovian Quantum Dynamics
by Vasily E. Tarasov
Entropy 2021, 23(8), 1006; https://doi.org/10.3390/e23081006 - 31 Jul 2021
Cited by 21 | Viewed by 2250
Abstract
A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. [...] Read more.
A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested. Full article
(This article belongs to the Special Issue Non-Hamiltonian Dynamics, Open Systems and Entropy)
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