The baker map is investigated by
two different theories of irreversibility by Prigogine and his
colleagues, namely, the <mml:math alttext="$\Lambda$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Λ</mml:mi></mml:math>-transformation and complex
spectral theories, and their structures are compared. In both
theories, the evolution operator <mml:math alttext="$U^\dag$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>U</mml:mi><mml:mo>†</mml:mo></mml:msup></mml:math> of observables (the
Koopman operator) is found to acquire dissipativity by
restricting observables to an appropriate subspace <mml:math alttext="$\Phi$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Φ</mml:mi></mml:math> of the
Hilbert space <mml:math alttext="$L^2$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> of square integrable functions. Consequently,
its spectral set contains an annulus in the unit disc. However,
the two theories are not equivalent. In the
<mml:math alttext="$\Lambda$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Λ</mml:mi></mml:math>-transformation theory, a bijective map <mml:math alttext="$\Lambda^{\dag -1}:\Phi\rightarrow L^2$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>Λ</mml:mi><mml:mrow><mml:mo>†</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>:</mml:mo><mml:mi>Φ</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> is looked for and the evolution operator
<mml:math alttext="$U$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math> of densities (the Frobenius-Perron operator) is transformed
to a dissipative operator <mml:math alttext="$W=\Lambda U\Lambda^{-1}$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mi>Λ</mml:mi><mml:mi>U</mml:mi><mml:msup><mml:mi>Λ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>. In the
complex spectral theory, the class of densities is restricted
further so that most values in the interior of the annulus are
removed from the spectrum, and the relaxation of expectation
values is described in terms of a few point spectra in the
annulus (Pollicott-Ruelle resonances) and faster decaying terms.

S. Tasaki

On Prigogine's approaches to irreversibility: a case study by the
baker map

Discrete chaotic dynamics (DCD) of living and thinking systems
 are presented in a form of networks of
interacting agents with the abilities of energy and information
exchange.  Special dynamical principles followed by the systems of
basic discrete time and space difference equations are introduced.
 Emergent,  self-organized
behavior of complex living and thinking systems is presented by the
different patterns generated by the DCD algorithms. Artificial
life and  brain systems based on DCD principles and mathematical
models are proposed.

V.  Gontar

The dynamics of living and thinking systems, biological networks,
and the laws of physics

We present the modification of the Prigogine-Herman kinetic
equation related to the network traffic. We discuss a solution of
this equation for homogeneous time-independent situations and for
the lognormal desired speed distribution function, obtained from
the traffic measurements. This solution clearly shows two modes
corresponding to individual flow patterns (low
concentration mode) and to collective flow patterns
(traffic jam mode). For situations with low concentration, there
is almost a linear dependence of the information flow versus the
concentration and the higher the average speed the lower the
concentration at which the optimum flow takes place. When
approaching the critical concentration, there are no essential
differences in the flow for different average speeds, whereas for
the individual flow regions there are dramatic differences.

I.  Antoniou

Victor V. Ivanov

Valery V. Ivanov

Yu. L. Kalinovsky

P. V. Zrelov

On a kinetic model of the internet traffic

This paper discusses how the concepts of nonlinearity, discrete
dynamics, chaos, and emergent order that Ilya Prigogine advanced
correspond to similar ideas involved in the adaptive,
evolutionary branch of economics.

Richard H. Day

Physics and the foundations of economic science: comments in
memory of Ilya Prigogine

First, I will recount the substance of several important
conversations I had with Ilya Prigogine over the years. There is
no doubt in my mind that Professor Prigogine firmly believed in
the underlying stochasticity of the universe. Second, I will
summarize my curiosity about the principle of detailed balance.
In my opinion, so far it has always been put in by hand. Third, I
will advance my own theory of microscopic irreversibility, that
is, irreversibility at the quantum level. This involves new
concepts of probability preference and probability violation. A
manifestation of this irreversibility may be seen during emission
of bound-free photons during field-matter interactions. Entropy
increase is tied to radiation increase, and the notion of heat
bath becomes light bath.

Karl  Gustafson

Microscopic irreversibility

It is widely accepted that consciousness or, in other words,
mental activity is in some way correlated to the behavior of the
brain or, in other words, material brain activity. Since quantum
theory is the most fundamental theory of matter that is currently
available, it is a legitimate question to ask whether quantum
theory can help us to understand consciousness. Several
approaches answering this question affirmatively, proposed in
recent decades, will be surveyed. It will be pointed out that
they make different epistemological assumptions, refer to
different neurophysiological levels of description, and adopt
quantum theory in different ways. For each of the approaches
discussed, these imply both problematic and promising features
which will be indicated.

Harald  Atmanspacher

Quantum theory and consciousness: an overview with selected
examples

Ioannis   Antoniou

Vladimir  Gontar

In memory of Ilya Prigogine

We discuss the stochastic limit approach to superfluidity.
Starting from the usual Hamiltonian describing the interaction
between the Bose condensate and the normal phase, we prove the
existence of superfluidity in the stochastic limit and deduce a
nonlinear (quadratic) kinetic equation describing the evolution
of the density of superfluid liquid.

L. Accardi

S. V. Kozyrev

Superfluidity in the stochastic limit

We describe bi-Hamiltonian structure and Lax-pair formulation with the
spectral parameter of the generalized fermionic Toda lattice
hierarchy as well as its bosonic and fermionic symmetries for
different (including periodic) boundary conditions.
Its two reductions<mml:math alttext="$N=4$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> and <mml:math alttext="$N=2$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> supersymmetric Toda lattice
hierarchiesin different (including canonical) bases are
investigated. Its <mml:math alttext="$r$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math>-matrix description, monodromy matrix, and
spectral curves are discussed.

V. V. Gribanov

V. G. Kadyshevsky

A. S. Sorin

Generalized fermionic discrete Toda hierarchy

We provide here a qualitative summary of the main ideas from
econophysics and finance theory, starting with a thorough
criticism of the standard ideas taught in typical economics
textbooks. The emphasis is on the Galilean or physicists'
approach to market dynamics, as opposed to the standard
nonempirical postulatory one.

Joseph L. McCauley

Cornelia M. K&#252;ffner

Economic system dynamics

About this journal

Discrete Dynamics in Nature and Society publishes research that links basic and applied research relating to discrete dynamics of complex systems encountered in the natural and social sciences.

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Chief Editor, Dr&nbsp;Renna, is an associate professor at the University of Basilicata, Italy. His research interests include manufacturing systems, production planning and enterprise networks.&nbsp;

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Discrete Dynamics in Nature and Society

This Issue is now open for submissions.
Papers are published upon acceptance, regardless of the Special Issue publication date.

Discrete Dynamics in Nature and Society

In memory of Ilya Prigogine

In memory of Ilya Prigogine

Articles

On Prigogine's approaches to irreversibility: a case study by the baker map

The dynamics of living and thinking systems, biological networks, and the laws of physics

On a kinetic model of the internet traffic

Physics and the foundations of economic science: comments in memory of Ilya Prigogine

Microscopic irreversibility

Quantum theory and consciousness: an overview with selected examples

In memory of Ilya Prigogine

Superfluidity in the stochastic limit

Generalized fermionic discrete Toda hierarchy

Economic system dynamics