# Complexity : microscopic and macroscopic aspects / Origine dynamique de la complexité

*Symposium organized **by **Ioannis Antoniou, Albert Goldbeter **and René Lefever from **8 to 14 July, 2002.*

**Participants**

Allen Peter (Cranfield University, Bedford, Royaume-Uni), Antoniou Ioannis (International Solvay Institutes, Bruxelles, Belgique), Balescu Radu (Université Libre de Bruxelles, Belgique), Belyi Viatcheslav (IZMIRAN, Troïtsk, Russie), Borckmans Pierre (Université Libre de Bruxelles, Belgique), Brandas Erkki (Uppsala University, Suède), Clavin Paul (IRPHE, Marseille), Coullet Pierre (Institut Non linéaire de Nice Sofia-Antipolis, Valbonne), Courbage Maurice (Université Paris VII, Paris), Driebe Dean (The University of Texas at Austin, USA), George Claude (Université Libre de Bruxelles, Belgique), Goldbeter Albert (Université Libre de Bruxelles, Belgique), De Haan Michel (Université Libre de Bruxelles, Belgique), Hasegawa Hiroshi (Ibaraki University, Mito, Japon), Karpov Evgueni (International Solvay Institutes, Bruxelles, Belgique ), Kondepudi Dilip (Wake Forest University, North Carolina, USA), Lefever René (Université Libre de Bruxelles, Belgique), Mayne Fernand (Université Libre de Bruxelles, Belgique), Misra Baidyanаth (International Solvay Institutes, New Delhi, Inde), Ordonez Gonzalo (The University of Texas at Austin, USA), Petrosky Torio (The University of Texas at Austin, USA), Pomeau Yves (CNRS, École normale Supérieure, Paris), Prigogine Ilyа (International Solvаy Institutes, Bruxelles, Belgique), Reich Linda (The University of Texas at Austin, USA), Sanglier Michèle (Université Libre de Bruxelles, Belgique), Sonnino Giorgio (European Commission Research, Bruxelles, Belgique), Tasaki Shuichi (Waseda University, Tokyo, Japon), Velarde Manuel G.(Universidad Complutense, Madrid, Espagne), Walgraef Daniel (Université Libre de Bruxelles, Belgique).

**Report **

Ilya Prigogine opened the Workshop on Tuesday 9 Ju1y with a review of the milestones which led him from the study of open systems far from equilibrium to the dynamical basis of Irreversibility in terms of Non-Integrable a la Poincaré Systems.

The complexity of Quantum Systems at the microscopic leve! was addressed by Erkki Brändas, who emphasized that the very origin of irre- versibility is intimately related to the intrinsic соmрlеxity generated by the interactions and correlations between the elementary particles constituting the macroscopic system. A variety of systems operate on an intermediate scale between the micro- and microscopic ones. Such mesoscopic systems exhibit a growing importance in the new quantum technology domain. The tools of modem physics extend into many new areas with the developments focused on a new level of description never before believed to be possible. Соmрlеxity as a paradigm ascribes particular emphasis to the appropriate level of formulation as well as the supposed boundary between classical and quantum physics. Recent developments in quantum technology involve basic quantum phenomena, stochastic resonances and information processing, ellipsoidal wavefunctions for scattering and signal processing, coherence-decoherence and dissipative systems, the phenomenon of high temperature cuprate superconductors.

Gonzalo Ordonez reviewed recent work with S. Kim on star-unitary transformations leading from dynamics to irreversibility and stochastic behaviour. This transformation avoids the use of coarse graining or other approximations. There is no loss of information since the transformation is invertible and it is obtained by an extension of the canonical (unitary) transformation that eliminates interactions. White unitary transformations can be constructed for integrable systems in the sense of Poincaré, for non-integrable systems there appear divergences in the perturbation expansion, due to resonances. This transformation is “star-unitary”. Starunitarity for non-integrable systems is an extension of unitarity for integrable systems. For a model of a classical harmonic oscillator coupled to a field, an exact non-unitarity transformation that maps Hamilton’s equations to the Langevin and Fokker-Planck equations was presented. The effects of Gaussian white noise in the Langevin equation are obtained by the non-distributive property of the non-unitary transformation with respect to products of dynamical variables.

Tomio Petmsky presented in the afternoon session an exciting application of the Star-unitary Transformation to obtain stochastic Maxwell equations for classical radiation damping. The description of classical motion of a charged particle with radiation damping leads to well-known difficulties, such as self-accelerated motion and non-causal emission of the field. An essential difficulty of the problem is that the system is not integrable because of the Poincaré resonance between the charged particle and the field. As a result, one cannot eliminate interactions between the particle and the field by any canonical transformation that is analytiс with respect to the coupling constant. However, a non-canonical star-untary transformation is obtained by an analytic continuation of the unitary transformation operator which removes Poincaré’s divergences due to the resonance. Bу applying this transformation to the radiation damping problem in a model that can be exactly solved through the transformation, results have been obtained that do not show anу of the difficulties mentioned above. These results are in excellent agreement with the numerical simulations. The transformation is non-distributive with respect to products of dynamical variables. This leads to an unexpected fluctuation in the field, in spite of the fast that there is no temperature in the emitted field. This fluctuation can be described in terms of a stochastic wave equation (or Maxwell equation) that contains a Langevin type of noise for the transformed normal mode of each field. This stochastic equation leads to a consistent line shape of the energy of the emitted field from the dressed unstable particle.

Evgueni Karpov continued this line of thought bу presenting results on microscopic entropy and non-locality in quantum mechanics. Recent analysis of the Liouville-von Neumann equation outside the Hilbert space has revealed that irreversibility is a direct consequence of microscopic dynamical processes associated with resonance singularities, and does not come from anthropomorphic principles often presented in the literature, such as our incomplete knowledge, coarse-graining approximations, or the so-called environmental approach. Bу an analytic extension of unitary transformation to a complex plane, microscopic entropy as a quantum observable for the Friedrichs model of the interaction of malter with the radiation field in the presence of resonances has been obtained. An important point is the description of еmission and absorption in terms of this entropy, whiсh in Heisenberg representation decays exponentially for positive times. This justifies the point of view of Planck on entropy relating entropy to dуnamics. Due to its non-local nature, this observable creates a consistent picture of the scattering of a wave packet of the field. In order to have scattering, one has to target the wave packet to the particle. This introduces a non-local pre-collisional correlation that lowers the value of entropy. During the particle excitation, there appears an еntropy flow from the field to the particle. This flow reduces the particle component of the entropy, in spite of the total entropy production always being positive. This implies that the excitation of the particle may be considered as a construction of a non-equilibrium structure due to an entropy flow. This was extended to the three dimensional Friedrichs model where the direction of the initial momentum of the wave packet has a big effect on the resulting picture. Of special interest is the case in which the initial condition corresponds to the non-local situation.

Another extension of Quantum Mechanics leading to Irreversibility was presented bу M. de Haan and C. D. George. This extension was based on self-enеrgy processes (already found in quantum electrodynamics), which enables the widening of the dynamics by the introduction of specific additional variables leading to the recognition of such processes. A projection technique inspired by the one introduced long ago by the

Brussels group defines a subdynamics, which is shown to contain entirely the starting description (it corresponds now to a restricted class of initial conditions, namely the reversible one). A new kinetic scheme is obtained for privileged components for which the original interaction leads to collision processes. This method is applied successively to various models and the presentation focused on the possible presence of temporal behaviours excluded in ordinary quantum mechanics. Among those differences, the possibility appears of a purеly exponential deсay for a particle, while usual quantum mechanics requires long time tails associated with the finite lower bound in the spectrum of the Hamiltonian.

Wednesday 10 July was devoted to Kinеtic theory, Markov Processes and Irreversibility. The first lecture was given by Yves Pomeau on Statistical Mechanics without equilibrium as is the case in gravitational interactions. Since the seminal paper on the subject by Prigogine and Severn, the central issue in the statistical mechanics of particles interaсting gravitationally has been to formulate a relevant kinetic theory and to build up predictions based on a thorough analysis of it. The example of three masses in a trap shows how time enters into various scaling laws for the asymptotic properties of such a system. The more complex many body problems, in the case of a self-gravitating cluster of many masses, were also considered. The kinetic theory of this system, at the Boltzmann order, puts on the same footing the resonant interaction and the close encounters, although the later are logarithmically dominant. This kinetic theоry predicts the formation of a core of deeрlу bound trajectories in a finite timе.

Recent progress in the theоry of anomalous transport in plasmas was the subject of Radu Balescu’s presentation. The problem of anomalous transport due to turbulence in plasmas is one of the major issues to be addressed in the quest for controlled fusion. An enormous amount of work has been devoted to this theme, by the way of laboratory experiments, numerical simulations and analytiсal theory. As regards the latter, mort existing works are based on oversimplified assumptions, which limit very seriously their range of validity. It is well known that the typical transport coefficient, i.e., the diffusion coefficient across the confining magnetic field is expressed as a time integral of the Lagrangian velocity correlation function of the particles, i.e., their correlation along the particle trajectories. This is, however, an extremely complex problem, as these trajectories cannot be calculated explicitly. The оnly available data is the corresponding Eulerian correlation, between particles at fixed positions and time. The relation between the two types of correlations embodies the full difficulty of the turbulence problem, both for neutral fluids and for plasmas. Usually, a rather crude approximation, the Corrsin approximation has been used in order to advance in the theoretical calculations. It is well known, however, that this approximation is limited to very weak levels of turbulence. In recent years, the members of Balescu’s group consisting of a Belgian-French-Romanian collaboration, has introduced a new concept, the Decorrelation Trajectory method. It departs radically from the Corrsin approximation and allows one to treat problems of string turbulence ; it has been validated by numerical simulations, aid recovers the known analytical results in appropriate limits. One of its main features is its taking into account the phenomenon of temporary trapping of the particles in the collective potential structures of the turbulent plasma : this introduces novel, unexpected features in the process of anomalous transport. The applications of the decorrelation trajectory method to various types of plasma turbulence were also discussed.

Dean Driebe reviewed the work of the Brussels-Austin group on fully chaotic maps with broken time symmetry. These maps provide simple pedagogical constructions of extended spectral decompositions in terms of resonances, which appear as eigenvalues. In this way probabitistic estimations of the rates of approach to equilibrium can be established.

Hiroshi Hasegawa opened the afternoon session with a presentation of his study of a system with long range correlations which shows fractional power scaling of excess heat production with respect to the period of an external transformation. The thermodynamics for a one-dimensional intermittent chaotic system was constructed and numeriсally confirmed the fractional power scaling of excess heat production. Hasegawa also commented on the scaling of the area of the hysteresis loop in magnetic inversion experiments.

The statistical and irreversible behaviours of pseudo-chaotic and space-time dynamical systems were discussed by Maurice Courbage, who first considered a family of area preserving maps of the two-dimensional torus with zero Lyapounov exponent as an example of dynamiсs between regular and chaotic mixing. He showed how to relate them to motions of some billiards and discussed their ergodic properties and the way thеу generate random walks sensitive to initial conditions according to some diophantine approximations of the parameters. Next, he considered the interesting issue of charаcterization of intrinsic space-time randomness and irreversibility of infinitely extended dynamicаl systems, like infinite systems of independent particles moving in a periodic field. This generalizes his previous works done together with B. Misra and I. Prigogine.

Ioannis Antoniou, the last speaker of the daу, addressed the inverse problem of Statistical Physics. Can Markov semigroups arise as exact projections of chaotic dynamics ? Hе showed in terms of positive dilations that this is indeed the case. For piecewise monotonic irreversible Markov maps he obtained, together with Karl Gustafson and Stanislav Shkarin, a formula which gives explicitly the underlying chaotic dуnamics, which were constructed for several concrete examples.

Shuiсhi Tasaki opened the first session of Thursday with the Quantum Zen° Effect and presented a possibility for Quantum State Control applicable for Quantum Computation. In the last decades, quantum computations are intensively studied and several new promising algorithms are proposed such as Shor’s factorization algorithm. Howеvеr, as firstly pointed out by Unruh, the decoherence in the individual quantum manipulations causes substantial deterioration of efficiency. Therefore, in order to realize quantum computers, *an *effective control suppressing the decoherence is necessary. As one of such methods, the so-called quantum bang-bang control was proposed by Lloyd. This is the method of suppressing the decoherence by strongly driving the system in target. On the other hand, as is known in quantum Zeno effect, string driving does cause decoherence. Therefore, it is interesting to look at the interrelation of the quantum bang-bang control and quantum Zeno effect. Tasaki explicitly considered this question for the trapped Be ion, where quantum Zeno effect was observed and 2-bit CNOT gate was realized. Hе showed that quantum bang-bang control is a constructive application of quantum Zeno effect.

Linda Reichl afterwards discussed quantum chaos and universality in open quantum systems. When the reaction region of a quantum scattering process has a classical counterpart that undergoes a transition to chaos, the quantum scattering process undergoes a transition to universal random matrix behaviour. She illustrated niсely this effect for scattering of electron matter waves in electron waveguides and also for atomic scattering in the presence of single mode laser fields.

The last lecture in the morning was given by Slava Belyi on fluctuation-dissipation relations for the non-local plasma. Using the Langevin approach and the time-space multiscale technique, he constructed a kinetic theory of the non-local fluctuations in the collisional plasma. First he derived a generalized version of the Callen Welton theorem and showed that not only the dissipation but also the derivatives of the dispersion determine the amplitude and the width of the spectral lines of the electrostatic field fluctuations, as well as the form factor. There appear significant differences with respect to the non-uniform plasma. In the kinetic regime the form factor is more sensitive to space gradient than the spectral function of the electrostatic field fluctuations. The inhomogeneity leads to asymmetry with respect to the inversion of the frequency sign. The differences in amplitude of peaks could become a new diagnostic tool for slow space variations in the plasma.

The next sessions, taking place on Friday and Saturday, were devoted to Prigogine’s concept of *dissipative structures. *They illustrated its importance in domains extending from reaction-diffusion systems and hydrodynamics up to biosystems and the social sciences.

Ti begin with, George Sonnino presented a theoretical contribution concerning the thermodynamic framework in which dissipative structures are defined. Hе showed that it is possible to formulate a covariant *thermodynamic field theory *(TFT) based on field equations whose solutions give the generalized relations between thermodynamic forces and flows. When the latter are related by a symmetric tensor, the thermodynamic space can be described by the Riemannian space. When the system is far from thermodynamic equilibrium, skew-symmetric components can appear in the expression that links thermodynamic forces and flows. The appropriate thermodynamic space is then the Weyl’s space. For systems close to equilibrium, the solutions of the field equations reduce to the Onsager-Casimir tensors and satisfy Prigogine’s *Minimum entropy production theorem *in the case of pure a—a or R—(3 processes. Moreover, the solutions of the field equations satisfy the Prigogine-Glansdorff General Evolution Criterion when the systems are far from thermodynamic equilibrium. This thermodynamic field theory predicts the possible existence of an interesting non-linear Hall effect in the case of materials simultaneously submitted to magnetic fields and current flows.

The next talk on dissipative structures was given by Dilip Kondepudi who discussed how the process of “chiral autocatalysis” may induce the generation and propagation of chiral asymmetry. He outlined the context in which these phenomena are of interest and summarized advances that significantly improve our understanding of the non-equilibrium symmetry breaking instabilities involved.

Albert Goldbeter reviewed the relationship of biochemical dissipative structures with circadian rhythms. These endogeneous temporal oscillations of a 24h period, exhibited by organisms ranging from сyanobacteria to mammals, are the prototype of biological rhythms. They represent an important class of temporal dissipative structures in living organisms. The experimental advances of the last decade, made with such key organisms as the fly *Drosophile (*and the fongus *Neurospora *have shed much light on the molecular mechanism of circadian rhythms. The general picture which emerges so far is that the molecular mechanism of circadian oscillations involves the negative autoregulation exerted by a protein on the expression of its gene. Deterministic models of such genetic regulatory processes, developed in collaboration with D. Gonze and J.C. Leloup, support the idea that circadian rhythms correspond to sustained lirait cycle oscillations around an instable steady state. The models further account for the entrainment of circadian rhythms by light-dark cycles, and for their phase-shifting by light pulses. In conditions where the system admits a coexistence between a stable steady state and a stable limit cycle, the models provide an explanation for the suppression of rhythmicity by a single light pulse, and for the subsequent restoration of oscillations by a second light pulse. The effect of molecular noise has been studied by considering the stochastic version of a model previously proposed for circadian oscillations of the PER and TIM proteins and their mRNAs in *Drrosophila. *The predictions of the stochastic approach compare well with those of the deterministic model ; they indicate furthermore that robust circadian oscillations can emerge even when the maximum numbers of mRNA and protein molecules involved in the oscillations are of the order of onlу a few tens or hundreds, rеspectively.

The next four talks by Pierre Borckmans, René Lefever, Pierre Coullet, Daniel Walgraef concerned examples of spatio-temporal dissipative structures of the reaction-diffusion type, i.e. involving dissipative processes in the absence of hydrodynamic couplings.

Chemical patterns emerging via a so-called Turing instability constitute the paradigm in this domain. As Pierre Borckmans reported in a talk, to which he associated the late, much regretted friend and collaborator Guy Dewel, our knowledge of these ordering behaviours has immensely progressed in the last decade. The invention of gel reactors plаyеd in this respect an important role in allowing for the systematic testing of many long standing theoretical predictions. The experimental and theoretical progresses deriving from this breakthrough were reviewed.

Within the same framework of ideas, Daniel Walgraef emphasized the progresses made by extending reaction-diffusion theory to material science, more particularlу to the studу of nanostructures in thin films. Such films have a wide variety of industrial applications. It is of capital economic and scientific interest to master the growth mechanisms which largely determine their properties and textures. Previous investigations based on computer simulations of molecular dynamics have provided a lot of information but, due to their atomistic nature, thеу *can *deal onlу with small systems, with linear dimension under the micron. This is why continuum, reaction-diffusion inspired approaches are of great interest in this context. They are indeed capable of simulating the growth of thin films of much larger dimensions. They have also been proposed to describe the spontaneous ordering of nanostructures or self-assembled quantum dots in multicomponents epilayers on a substrate. The reactiondiffusion model considered in the talk takes into account adsorption, desorption aid nonlinear diffusion. Il has been proposed to describe the evolution of a deposited atomic layer on a substrate. In monocomponent films as well as in solid solutions the competition between atomic deposition and the underlying instability of the adsorbed atomic layer mаy stabilize nanoscale spatial patterns, already in the first deposited layers. These patterns may serve as templates for the late stages evolution of film textures. Patterns with different symmetries mау be selected, according to the relative values of experimental parameters such as deposition rate, substrate temperature and atomic mobility. In this approach, self-assembled nanostructures appear as dissipative structures, and their selection and stability mechanisms are of dynamiсnature. This agrees with the experimental situation. It contrasts with other approaches where selection results from the minimization of a free energy.

The patterns mentioned above are generally stationary. The dissipative fronts studied by Pierre Coullet are non-stationary and exhibit unusual propagation phenomena. Hе showed indeed that a front, which leaves behind an unstable homogeneous state, is at the origin of a quite complex spatio-temporal behaviour. The nature of these “retracting fronts” and their role in the understanding of spatio-temporal intermittenсy was discussed.

Vegetation patterns are striking examples of large scale dissipative structures observable in population dynаmics. They mаy appear in the form of vegetation bands (often called Tiger Bush), of vegetation spots or of uniform vegetation covers punctuated by holes of bare soil. They are phytociological organisations typical of arid and semiarid regions. Ву their regularities, by the vastness of the territories which they cover, by the variety of their constitutive plants, as well as by the variety of soils on which thеу appear, thеу are quite puzzling phenomena. The generic model proposed by René Lefever and Olivier Lejeune explains how the interplay between short range and long range interactions governing plant communities induces the various kinds of patterns асtually observed. This approach predicts the conditions under which vegetation bands are orientated either parallel or perpendicular to the direction of existing environmental anisotropies ; it also establishes that ground curvature is not a necessary condition for the appearance of arcuate vegetation patterns and that the width of vegetated bands increases when the climate becomes more arid.

Manuel Velarde discussed the properties of *dissipative solitons. *He reported recent results about the excitation of nonlinear interfacial oscillations and hence waves at the surface of a liquid or at the interface separating two liquids when a thermal gradient is imposed or when there is adsorption and subsequent absorption in the bulk of a (light) surfactant which creates tangential stresses (Marangoni effect). Hе also discussed their solitonic features upon collisions and boundary reflections, formation of soliton bound states, chaotic wave trains, etc. even though the proposed evolution equations are not hyperbolic but a parabolic-hyperbolic combination like in a dissipation-modified Boussinesq-Kortewegde Vries equation.

The spatio temporal structures investigated by Paul Clavin differ notably from isothermal dissipative structures of from other non-equilibrium systems such as lames, growing crystals, Rayleigh-Bénard convection, etc. A detonation is a supersonic combustion wave consisting of a strong inert shock followed by a reacting flow The molecular transports are localized inside the leading shock (which may be considered as an hуdrodynamic discontinuity) and mау be neglected in the compressed gases which are governed by the Euler reactive equation. The dynamics aid structure of a detonation wave are thus solutions of a nonlinear hyperbolic problem. A striking observation in experiments is the formation of Mach-stems with triple point configurations traveling transverse to the front at nearly the sound speed in the burned gas. Their trajectories have a diamond pattern with cell sizes two orders of magnitude larger than the unperturbed detonation thickness. A challenging problem is to relate the pattern to the properties of the reactive mixture. Physical insights are elusive and there is no convincing explanation of neither the size nor the shape of the pattern. In order to provide an overall physical understanding, the approach carried out is a nonlinear study in the neighborhood of the instability threshold. The method consists in finding an asymрtotic solution in the double limit of a large propagation Mach number and of a small difference between specifics heats *C _{p }*and с . Such a limit was already used a long time ago, in hypersonic flow thеory around the 1950s, and is known as “Newtonian flow theory”. The final result is a nonlinear integral-differential equation for the front dynamics describing time dependent patterns qualitatively similar to those observed in experiments. The analysis is valid for a general chemical kinetics of an irreversible exothermal reaction but is limited to overdriven regimes.

The last two talks concerned the organisation and evolution of socioeconomic systems. Michèle Sanglier presented a model of national economic growth that integrates production and consumption within the context of self-organization. She also developed an empirical model of corporate productivity growth and applied it to a number of examples of industrial nature.

The view set out by Peter Alen considers social and economic systems as being a constituent part of a set of evolving, multi-scale, spatio-temporal structures. The сomplеxity of these structures means that the decisions taken by anу particular actor or agent will necessarily be taken under considerable uncertainty, and this uncertainty will be further compounded for everyone by the interacting effects of whatever decisions the actors or agents take. Each individual, group, firm, corporation, shareholder aid even observer will experience “path dependent learning”, whereby in a given period learning is conditional on the decisions taken, and therefore in the next period the options considered and the problems posed are changed by what happened in the previous period.

In reflecting on the “two cultures” tоday, one mаy say that the new sciences of complex, non-linear systems have united the physical and the human domains, and recognized the fundamental unity that underlies them. In effect, the scientific method is based on the active formation and testing of ехplanatory “conjectures”, and this is equally true in the human and the purely physical domains. The creative process that leads to the imagination of possible models of interaction underlying a phenomenon is the same for human and physical systems, but the ease with which conjectures can be tested is different. This is because the degree to which one can perform repeatable experiments is not the same. In some areas of physical science this mау be relatively easy, and the idea of “ceteris paribus” — studying interdependencies one at a time — reasonable. In the social and human sciences, however, both repeatable experiments and “ceteris paribus” are problematic, and as a result knowledge is generally more ephemeral and contextual. Knowledge is more an on-going process of continual learning, in which the changed behaviour that results changes the basis on which the knowledge was generated. For this rea-son, conjecture, exploratory behaviour and the ability to respond appropriately to positive or negative signals are the basis of survival and adaptation in the human world. And staying connected to the environment sufficiently to maintain a sanity check on the resulting internal evolution is another important principle for surviving as part of the complex, evolving, multi-scale structures that make up our world.