Irreversible Dynamical Systems Research Articles

Functional Description of C*-Algebras Associated with Group Graded Systems

The well known pure algebraic concept of group grading arises naturally in considering the crossed products, especially in the context of irreversible dynamical systems. In the paper some general aspects concerning group graded systems and related algebras are considered. In particular, a functional description of a C*-algebra associated with an Abelian group graded system is presented.

On C⁎-algebras associated to product systems

Let P be a unital subsemigroup of a group G. We propose an approach to C⁎-algebras associated to product systems over P. We call the C⁎-algebra of a given product system E its covariance algebra and denote it by A×EP, where A is the coefficient C⁎-algebra. We prove that our construction does not depend on the embedding P↪G and that a representation of A×EP is faithful on the fixed-point algebra for the canonical coaction of G if and only if it is faithful on A. We compare this with other constructions in the setting of irreversible dynamical systems, such as Cuntz–Nica–Pimsner algebras, Fowler's Cuntz–Pimsner algebra, semigroup C⁎-algebras of Xin Li and Exel's crossed products by interaction groups.

Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty

A major conceptual step forward in understanding the logical architecture of living systems was advanced by von Neumann with his universal constructor, a physical device capable of self-reproduction. A necessary condition for a universal constructor to exist is that the laws of physics permit physical universality, such that any transformation (consistent with the laws of physics and availability of resources) can be caused to occur. While physical universality has been demonstrated in simple cellular automata models, so far these have not displayed a requisite feature of life—namely open-ended evolution—the explanation of which was also a prime motivator in von Neumann’s formulation of a universal constructor. Current examples of physical universality rely on reversible dynamical laws, whereas it is well-known that living processes are dissipative. Here we show that physical universality and open-ended dynamics should both be possible in irreversible dynamical systems if one entertains the possibility of state-dependent laws. We demonstrate with simple toy models how the accessibility of state space can yield open-ended trajectories, defined as trajectories that do not repeat within the expected Poincaré recurrence time and are not reproducible by an isolated system. We discuss implications for physical universality, or an approximation to it, as a foundational framework for developing a physics for life.

Group-Graded Systems and Algebras

In the paper, we discuss some problems concerning the structural properties of crossed products. While expansions of C*-algebras under group actions have been studied rather extensively, known difficulties in the transition to irreversible dynamical systems require the development of new methods. The first step in this direction is to study actions of inverse semigroups, whose properties are closest to those of groups. The main tool to achieve the goal is the concept of grading. The detection of the grading structure in the corresponding constructions seems to be very promising.

Representation of irreversible systems in a metric thermodynamic phase space

This paper studies geometric properties of a class of irreversible dynamical systems, referred to in the literature as metriplectic systems. This class of systems, related to generalized (or dissipative) Hamiltonian systems, are generated by a conserved component and a dissipative component and appear, for example, in non-equilibrium thermodynamics. In non-equilibrium thermodynamics, the two potentials generating the dynamics are interpreted as generalized energy and generalized entropy, respectively. Stability and stabilization results for metriplectic systems have been presented in the literature, however, some aspects are still poorly understood, in particular the existence of dynamical invariants such as periodic orbits. In this note, we study the properties of metriplectic systems by considering a lift from the n-dimensional state space to a (2n+1)-dimensional contact space, following an approach introduced in recent years to study irreversible control systems. This lift leads to a deeper geometric characterization of metriplectic systems in the extended space. An example is provided to illustrate the approach proposed in this paper.

Reversible extensions of irreversible dynamical systems: the C*-method

A construction of a reversible extension of irreversible dynamical systems is presented. It is based on calculating the maximal ideal spaces of the C*-algebras generated by these systems and the corresponding reversible extensions of endomorphisms. Connections between the objects that arise and dynamical systems of Smale horseshoe and other types are revealed.Bibliography: 20 titles.

Plus-operators in Krein spaces and dichotomous behavior of irreversible dynamical systems with discrete time

We study dichotomous behavior of solutions to a non-autonomous linear difference equation in a Hilbert space. The evolution operator of this equation is not continuously invertible and the corresponding unstable subspace is of infinite dimension in general. We formulate a condition ensuring the dichotomy in terms of a sequence of indefinite metrics in the Hilbert space. We also construct an example of a difference equation in which dichotomous behavior of solutions is not compatible with the signature of the indefinite metric.

Fluctuation relations and positivity of the entropy production in irreversible dynamical systems

We give a generalization of fluctuation identities and inequalities for the entropy production when the microscopic dynamics preserves the phase space volume but is not assumed to be dynamically reversible. We find that general first properties such as the strict positivity of the entropy production or the validity of an H-theorem are unaffected by the irreversibility of the dynamics. Fluctuation identities describing the relation between entropy production and its time-reversal must, however, be changed and now involve the time-reversed dynamics.

An explicit approach to the Λ-operator and the H-theorem in the Prigogine theory of irreversibility

Abstract A reformulation of the Prigogine theory of irreversible dynamical systems is given. In contrast to the previous works a stochastic integral with respect operator valued martingales is employed here. This approach provides a mathematically rigorous and simple construction of the Λ operator for dynamical flows. It has an explicit random character. We derive also the Boltzmann H-theorem explicitly from the properties of the semigroup only. Finally, the form of the Boltzmann kinetic equation for the irreversible dynamics is found.

On Irreversible Dynamical Systems

Journal of Mathematics and PhysicsVolume 2, Issue 1-4 p. 73-87 Article On Irreversible Dynamical Systems Joseph Lipka, Joseph LipkaSearch for more papers by this author Joseph Lipka, Joseph LipkaSearch for more papers by this author First published: October 1923 https://doi.org/10.1002/sapm19232173AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume2, Issue1-4October 1923Pages 73-87 RelatedInformation