Physicalia Mag. 27 (2005) 3 pp 281-288

RESEARCH ACTIVITIES IN STATISTICAL PHYSICS IN BELGIUM P. Gaspard Université Libre de Bruxelles, Physics Department and Centre for Nonlinear Phenomena and Complex Systems, Campus Plaine CP 231, B-1050 Brussels

Foreword The goal of statistical physics is to deduce quantitative predictions on the global and collective properties of a system from the laws ruling the entities composing the system. Since this goal concerns many different types of systems and that a variety of methods exist to carry out this programme, statistical physics has evolved into a vast field of research extending from mathematics and natural sciences such as chemistry, biology, geophysics, up to economy and the social sciences, including the engineering sciences. The new developments have led to the emergence of closely related fields such as nonlinear physics and the physics of complex systems, which will also been reviewed in this report. At the European level, these fields are represented by the Division of Statistical and Nonlinear Physics of the European Physical Society and their research activities are supported by Institutions such as the European Commission and the European Science Foundation. Generally speaking, the community is developing new conceptual and mathematical methods for the understanding of broad classes of natural systems sharing similar properties and structures: the so-called universality classes. In this regard, the community greatly participates to the mathematization of the natural sciences which is a major trend today after the historical advent of fast computers. Therefore, statistical physics plays a very unique role in the cross-fertilization between the sciences. In particular, the growing interest for biological physics is essentially driven by the statistical physics community. Moreover, it is notorious that econophysics is a spin-off of statistical physics [11]. Belgium has always occupied a central position on the international stage of statistical physics with many original contributions to both equilibrium and nonequilibrium statistical mechanics. Today, Belgium is very present in the current renewal of the field, especially, concerning the fundamental problems of statistical mechanics which leads to unexpected interactions with dynamical systems theory and simulation methods. The current landscape of statistical physics in Belgium can be subdivided into the following topics: - foundations of thermodynamics and statistical mechanics; - stochastic processes, chaos, and fractals; - phase transitions and critical phenomena; - interfaces, wetting, and soft matter; - quantum statistical mechanics and applications to condensed-matter physics; - computational statistical physics and nanosystems; - nonequilibrium statistical mechanics and irreversibility; - nonlinear physics and nonequilibrium instabilities; - the physics of complex systems; 281

which are reviewed in detail below. In Belgium, the main activity concerns theoretical and computational aspects of statistical physics [1-20]. This activity is carried out jointly with related experimental work in some cases [6,10,11,16,19,20]. Foundations of thermodynamics and statistical mechanics Belgium has a historical place in the development of thermodynamics and modern statistical mechanics. Concerning thermodynamics, great advances have been carried out in irreversible thermodynamics with the prediction of dissipative structures far from the thermodynamic equilibrium by the work of the late Nobel Prize Ilya Prigogine. The extension of thermodynamics to systems with a free-energy functional depending on gradients has also been investigated [12]. On the other hand, modern statistical mechanics owes much to the pioneering contributions of the late Professor Léon Van Hove in the mid fifties. The central problem of statistical physics is to derive and characterize the macroscopic and mesoscopic properties of a system from the microscopic dynamics of the entities composing this system. In this regard, statistical physics is a unique combination of microscopic ingredients (the entities and their dynamics) with statistical considerations (law of large numbers, central-limit theorem, and large deviations as the size of the system or the duration of the process increases). This leads to questions which are answered within probability theory, the theory of statistical ensembles, ergodic theory, as well as modern dynamical systems theory. This latter is an extension of traditional ergodic theory enriched by the study of chaotic dynamical systems. A central preoccupation is the existence of an invariant probability measure describing the statistical properties of the system. This concerns not only the systems ruled by classical and quantum mechanics but also the dissipative dynamical systems such as coupled map lattices. These fundamental questions are the subject of intense research [3,7,9,13]. Recent trends also concern systems with non-extensive properties [3,11,18]. Furthermore, important work has been devoted to the understanding of the second law of thermodynamics, especially, during the last few years under the impetus of contributions from dynamical systems theory and chaos theory which have considerably extended the scope of traditional ergodic theory. These new developments have led to the discovery of large-deviation dynamical relationships between irreversible properties such as transport coefficients or entropy production, and the characteristic quantities of the microscopic dynamics such as its dynamical randomness, forward or backward in time. Different versions of large-deviation dynamical relationships have been investigated, from the escape-rate formalism for transient processes to the fluctuation theorem for nonequilibrium steady states. The hydrodynamic modes of relaxation as well as the entropy production have been related to the breaking of the time-reversal symmetry in the statistical description of the underlying dynamics [5,7,13]. Stochastic processes, chaos, and fractals Both stochastic processes and dynamical chaos share the feature of presenting dynamical randomness. Their difference holds in the absence or presence of determinism in the rules of their time evolution. Stochastic processes are studied since the beginning of the XXth century and the early work on Brownian motion. They have been discovered in many nonequilibrium processes in physics, chemistry, and biology. In Belgium, nonequilibrium chemical reactions have been studied by the theory of stochastic processes, in particular, to understand how the molecular fluctuations can be amplified during the formation of 282

dissipative structures at nonequilibrium transitions [5,13]. More recently, the study of nonequilibrium steady states described by an invariant probability measure has been developed in probabilistic cellular automata and nonequilibrium processes [7,13]. Noise-induced phenomena have also been much studied [5,13]. Dynamical chaos and fractals have come to the forefront of research at the end of the seventies and beginning of the eighties, in particular, thanks to the advent of fast and affordable computers which allowed the systematic investigation of nonlinear behavior. The study of nonlinear deterministic dynamical systems has revealed the existence of abrupt bifurcations as well as random time evolutions known as dynamical chaos. In this regard, a connection with statistical mechanics and ergodic theory was established: On the one hand, methods of statistical physics such as the Liouville equation have been applied to the chaotic dynamical systems and, on the other hand, the new concepts of Lyapunov exponents, Kolmogorov-Sinai entropy, and fractal dimensions have been studied in dissipative dynamical systems as well as Hamiltonian systems such as the Lorentz gases which are revelant for the study of transport properties as diffusion [11,13]. An invariant probability measure can be associated with a chaotic dynamical system in very much the same way as in equilibrium statistical mechanics. However, new types of invariant probability measures with unusual fractal properties were discovered, which has significantly enriched our understanding of both dissipative dynamical systems and nonequilibrium statistical mechanics [13,19]. Fractals which do not originate from low-dimensional deterministic dynamical systems have also been studied in such processes as dendritic formations, growth processes, random surfaces, and material science [6,10,11,19]. Phase transitions and critical phenomena The phase transitions are among the most important phenomena which are specific to statistical physics. They arise from the interaction between many particles or entities composing a system. Equilibrium phase transitions are the paradigmatic examples of such phenomena, but transitions have also been much studied in nonequilibrium systems where they are also known under the name of bifurcations. The critical behavior of the system close to a transition is a fascinating topic to which very important efforts are devoted. Both macroscopic and fluctuating properties are investigated at equilibrium and nonequilibrium transitions. Critical phenomena are studied by a variety of methods including free-energy functional theory, renormalization group theory, conformal field theory, stochastic-geometric methods of percolation theory, as well as numerical simulations [6,7,8,9,10,11,16]. At equilibrium, phase transitions and critical phenomena are studied in bulk phases, at interfaces, in soft materials, and in quantum phases such as superconductivity and Bose-Einstein condensation [1,4,6,7,11,16]. In nonequilibrium systems, transitions and the associated critical properties are investigated in systems undergoing stationary or oscillating instabilities, in sandpile models, in coupled map lattices, and in disordered and complex systems [5,7,8,9,11,13,19]. In this context, new phenomena have been studied such as noise-induced phase transitions [5] and self-organized criticality [5,7,11]. Interfaces, wetting, and soft matter Interface phenomena are much studied today since the bulk properties of many materials are well known. Furthermore, many fundamental processes occur at the interface between two or more thermodynamic phases, or at the surface of liquids or solids. The interfaces can be characterized by different thermodynamic properties such

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as surface tension, adsorption properties, the angle of contact between three phases, and the transitions in these thermodynamic properties. Wetting is one of such interfacial phenomena which are currently under study. Partial wetting is characterized by a nonvanishing angle of contact of a liquid droplet in contact with a solid for instance. Complete wetting occurs when this angle vanishes. The angle of contact as well as other related properties are controlled by equilibrium thermodynamics and, in particular, undergo thermodynamic transitions. These properties are theoretically studied by free-energy functional theory or density functional theory as well as numerical simulation methods in different systems including hydrocarbons and other organic materials, polydispersed fluids, or BoseEinstein condensates [4,6,16,20]. The aforementioned methods also apply to soft matter which shares many common aspects with interfaces. In particular, the weak van der Waals forces play a central role in both soft matter and interfaces. For these reasons, the free-energy functional theory or density functional theory are also used for the study of soft matter such as polymers or liquid crystals [16]. These methods often combined with molecular-dynamics simulations can predict their phase transitions and other equilibrium properties [16]. Furthermore, soft materials are also studied for the variety of structures they can generate: this is the case of droplets and foams [10]. Quantum statistical mechanics and applications to condensed-matter physics The quantum properties of matter manifest themselves at low temperatures as it is the case for superconductivity or superfluidities. In electronic systems, the quantum effects are already dominant for ordinary conductors at room temperature. The study of the properties of these condensed phases is today a major activity closely related to statistical physics. The spectrum of activities ranges from theoretical and mathematical work in quantum statistical mechanics to technology-oriented studies of material properties. Theoretical topics of interest are studied such as the quantum fluctuations, the characterization of the quantum phases in fermionic and bosonic systems, or the pathintegral formalism in condensed phases [1,7]. The study of electronic properties is one of the main activities. The electronic density functional theory, the Thomas-Fermi theory, as well as Ginzburg-Landau equations are developed and used for the understanding of different materials such as carbon nanotubes and nano-superconductors [2,4]. The following specific systems are currently under investigation: spin-polarized fermions in magnetic fields, superfluid Fermi gases, high-Tc superconductors, colossal magnetoresistive materials, stacked quantum dots, electron-electron correlations in quantum dots, polarons and bipolarons [1,2,4,7,11]. Furthermore, an active program is carried out on vortex matter in nanostructured superconductors and hybrid magnetic-superconducting systems [2]. This program investigates in particular how vortices are influenced by the finite size and shape of the superconductor [2]. An important effort has been recently devoted to the Bose-Einstein condensation and boson-fermion mixtures [1,4,7]. Besides, transport by solitons is also studied [4]. Computational statistical physics and nanosystems Computational methods play a growing role in statistical physics thanks to the availability of cheap computers. Several groups are working with dedicated clusters of computers which have recently appeared beside the mainframes of each university 284

computing centers. Contrary to neighboring countries, no supercomputing centers have been developed at the federal or regional levels in the fields of statistical or condensedmatter physics. A variety of computational methods are currently used in statistical physics for the study of both equilibrium and nonequilibrium properties. First of all, the quantum ab initio methods such as the electronic density functional method are used for such properties as the atomic configuration, the electronic ground-state energy, and the electronic density of states [4]. For the equilibrium properties of atomic or molecular systems, Monte-Carlo and molecular-dynamics simulations are used in particular, for the study of polymers, liquid crystals, interfacial systems, nanostructured materials, biological systems, as well as classical low-dimensional systems such as dusty plasmas which undergo Wigner crystallization, melting, and structural transitions [2,4,16,17,20]. Nonequilibrium properties are studied by a variety of methods depending on the time scale of interest. For fast phenomena on the time scale of the Newtonian motion, molecular-dynamics simulations are used in studies of polymers, nanoparticles impacting a surface, solids damaged by radiations, friction in carbon nanotubes, nonequilibrium instabilities such as the Rayleigh-Benard convective instability, shock waves, pistons or ratchets between particles reservoirs [5,13,16,17,19]. These latter are much studied as examples of nonequilibrium systems displaying unexpected irreversible behavior of fundamental importance for our understanding of the second law of thermodynamics [5,13,19]. On longer time scales, kinetic Monte-Carlo methods are developed for the study of mesoscopic properties in nonequilibrium processes such as surface diffusion or spreading, growth processes, heterogeneous catalysis and diffusion-reaction processes on lattices where the molecular fluctuations are important [11,13,19,20]. Furthermore, multiscale modeling and hybrid methods are also used for nonequilibrium and complex systems [16,17,19] Other powerful methods such as lattice gas automata have been developed for the purpose of simulating turbulent fluids and other macroscopic nonequilibrium phenomena [18,19]. Nonequilibrium statistical mechanics and irreversibility Nonequilibrium statistical mechanics has the purpose of predicting the properties of systems undergoing irreversible changes as it is the case during timedependent relaxations toward the thermodynamic equilibrium or in nonequilibrium steady states if the system sustains fluxes of matter or energy. A major preoccupation is here to understand the transport properties such as the electric and heat conductivities, the viscosities, diffusion, and also the chemical reaction rates. All these properties are irreversible in the sense that they contribute to the entropy production. The study of the transport properties and the reaction rates is thus closely related to the understanding of the second law of thermodynamics. One of the main approaches developed in nonequilibrium statistical mechanics is the derivation of master or kinetic equations. These equations rule quantities providing a reduced description of the system such as the distribution function of the degrees of freedom of interest for the problem under study. Master or kinetic equations have been derived for plasmas, for dense fluids, and for nonequilibrium chemical reactions [5,13,14]. In each case, a H-theorem can be deduced which provides a justification of the law of increase of the entropy under nonequilibrium conditions. Recent interest has focused on the nonequilibrium fluctuation properties which can be deduced from the master equation [5,7,13]. 285

An important effort has recently been devoted to anomalous transport properties which are of great importance in many systems and especially in plasmas [14]. The study of anomalous transport in plasmas introduces connections with many different fields, in particular: kinetic theory, the theory of turbulence, Langevin equations, and purely stochastic processes. Among the latter, the theories of continuous time random walks, Lévy flights, and the methods of fractional differential equations have been under intense study in recent years [14]. More direct connections have been established between the microscopic dynamics and the transport properties without using of the intermediate kinetic level of description. Such connections are possible by deriving directly the macroscopic hydrodynamic equations from Newton's equations. Such derivations are performed for diffusion and reaction in Lorentz gases and related models such as the multibaker map or for heat conductivity in chains of coupled anharmonic oscillators [9,13]. Recent developments of nonequilibrium statistical mechanics have been possible thanks to the progress in dynamical systems theory and its new concepts of entropy per unit time, Lyapunov exponents, and other large-deviation dynamical properties. These properties characterize the disorder which is generated in time by the trajectories of the system. The formalism which is used is similar to equilibrium statistical mechanics but with space replaced by time. In this way, relationships have been established between the transport coefficients and the large-deviation quantities characterizing dynamical randomness and sensitivity to initial conditions in the underlying microscopic dynamics [13]. Several approaches exist and, in particular, the escape-rate formalism and the thermostated-system approach. The so-called fluctuation theorem is another one of those large-deviation dynamical relationships, here, connecting the entropy production to the breaking of time-reversal symmetry of the invariant probability measure describing of each nonequilibrium steady state [7,13]. Further aspects of the second law of thermodynamics are studied in Brownian motors and pistons [5,13]. These devices can be described by kinetic theory and simulated by molecular dynamics. The effects of molecular fluctuations and the presence of two different time scales in their dynamics modify the standard thermodynamic description which is challenged by these studies [5,13]. Nonlinear physics and nonequilibrium instabilities As previously mentioned, nonlinear physics is closely related to statistical physics since the raise of interest for dynamical chaos and nonequilibrium phenomena in the late seventies and early eighties. The methods of statistical physics and ergodic theory are of great importance for the understanding of the random time evolution of chaotic dissipative dynamical systems. Because of the sensitivity to initial conditions characterized by the Lyapunov exponents, the typical trajectories of these systems become irregular on long times, which requires the use of a statistical description based on a probability measure invariant under the dynamics. The situation is very similar to the one encountered in equilibrium statistical physics and similar methods have thus been developed. These methods have been recently extended to the study of coupled map lattices and granular materials [9,10,11,13,14,19]. Statistical aspects are also the feature of turbulence in hydrodynamics and magnetohydrodynamics. Turbulence may be understood as a high-dimensional chaos but the methods used in this field are specific to the existence of a Kolmogorov-type cascade and are based on statistical moments, scaling arguments, and numerical simulations [7,14,19]. An important activity is devoted to nonequilibrium instabilities in fluids, reacting systems, and nonlinear optics. The nonequilibrium instabilities appear beyond 286

certain thresholds in the nonequilibrium boundary conditions imposed on the system. Because of the coupling between different possible transport and reaction properties, a large variety of instabilities are possible and currently studied [12,13,15,19]. These instabilities are traditionally studied for their macroscopic properties although the study of their fluctuation properties have been undertaken during the last decades as aforementioned. Furthermore, several instabilities which are specific to nanosystems are currently studied at interfaces and heterogeneous catalysis [6,13,19]. The physics of complex systems The methods of statistical physics can be applied to any systems composed of many interacting entities as long as the rules of the interactions and motions of the entities are known. Very often, the collective behavior and self-organization of the total system is captured by assuming simple rules eventhough the system can be complex in essence. This concerns systems in biology, ecoethology, medicine, epidemiology, economy, computer science, linguistics, and social sciences. Specific examples are traffic, economic systems and games, the modeling of social insects and arthropods in ecoethology, population dynamics, genomics, error correcting codes, neural networks, the statistical physics of learning, and more recently the topological and scaling properties of natural and artificial networks [5,6,7,8,10,11,13,19]. In this context, statistical methods of time series analysis have been developed for the study of meteorology and climatic time series, the stock markets and currency exchange rates, DNA or RNA nucleotide series, texts in linguistics, among others [8,10,11,13,19]. These new applications of statistical physics are revealed at different levels of conception from physical considerations based on energetics up to the more abstract mathematical analogies. We should add that the study of biological systems and other complex systems from the viewpoint of statistical physics is an expanding activity which can be expected to develop further in the coming years. Conclusion Statistical physics is a vast and ramified field of research which is intensively active in Belgium. Its foundations hold in thermodynamics and the concept of invariant probability measure which provides the basis of the statistical description of a given system. Its methods have proved to be extremely powerful since they are applied to a broad and diversified range of problems. Referring to the story that the work of Quetelet in statistics inspired Maxwell when he postulated his velocity distribution we can conclude saying that fruitful relationships between statistical physics and the other natural sciences have great perspectives. References to research groups: [1] Theoretische Fysica van de Vaste Stoffen, Departement Fysica, Universiteit Antwerpen, http://www.tvs.ua.ac.be/ [2] Theorie van de Gecondenseerde Materie, Departement Fysica, Universiteit Antwerpen,

http://www.cmt.ua.ac.be/ [3] Wiskunde Natuurkunde, Departement Fysica, Universiteit Antwerpen, http://www.physics.ua.ac.be/DptUA/Nederlands/Onderzoek/wisnat.html [4] Theoretische Studie der Materie, Departement Fysica, Universiteit Antwerpen,

http://www.physics.ua.ac.be/DptUA/Nederlands/Onderzoek/tsm.html [5] Theoretical Physics Group, Hasselt University, Diepenbeek,

http://www.uhasselt.be/theophys/ [6] Interfaces and Wetting Group,Laboratory for Solid State Physics and Magnetism,

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Department of Physics and Astronomy, K U Leuven,

http://fys.kuleuven.be/vsm/joi/joi.html [7] Mathematical Physics Group, Institute for Theoretical Physics, Department of Physics and Astronomy, K U Leuven,

http://itf.fys.kuleuven.be/itf/html/res_math_main.php?lang=en [8] Disordered Systems Group, Institute for Theoretical Physics, Department of Physics and Astronomy, K U Leuven,

http://itf.fys.kuleuven.be/nn/main.html [9] FYMA-Theoretical Physics, Université Catholique de Louvain, Louvain-la-Neuve,

http://www.fyma.ucl.ac.be/ [10] GRASP-Group for Research and Applications in Statistical Physics, Institut de Physique, Université de Liège,

http://www.grasp.ulg.ac.be/ [11] Centre Interfacultaire S.U.P.R.A.T.E.C.S., Institut de Physique, Université de Liège,

http://www.ulg.ac.be/supras/ [12] Groupe de Thermomécanique des Phénomènes Irréversibles, Institut de Physique, Université de Liège,

http://www.ulg.ac.be/thermoir/ [13] Laboratory for Nonlinear Physics and Statistical Mechanics, Department of Physics, Université Libre de Bruxelles,

http://www.ulb.ac.be/cenoliw3 [14] Laboratory for Statistical Physics and Plasmas, Department of Physics, Université Libre de Bruxelles,

http://www.ulb.ac.be/sciences/spp/ [15] Laboratory for Theoretical Nonlinear Optics, Department of Physics, Université Libre de Bruxelles,h

http://www.ulb.ac.be/sciences/ont/ [16] Laboratory of Soft-Matter Physics, Department of Physics, Université Libre de Bruxelles,

http://www.ulb.ac.be/sciences/polphy/ [17] Laboratory for the Physics of Irradiated Solids and Nanostructures, Department of Physics, Université Libre de Bruxelles,

http://www.ulb.ac.be/rech/inventaire/unites/ULB149.html [18] Lattice Gas Automata Group, Department of Physics, Université Libre de Bruxelles,

http://poseidon.ulb.ac.be/lga_en.html [19] Interdisciplinary Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles,

http://www.ulb.ac.be/cenoliw3 [20] Centre de Recherche en Modélisation Moléculaire, Faculté des Sciences, Université de Mons-Hainaut,

http://crmm.umh.ac.be/

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