# Thématique : Complex Matter

**2.2.3. Complex matter
**

**Contributors:**Francesco Ginelli (ISC-PIF), Ivan Junier (ISC-PIF)

The roadmap item about complex matter (considered as “Object” of research) can be found here. This document provides the overall structure of the corresponding digital research group and the digital university. It is articulated around the scientific "grands objets" (research fields) and around the teaching implementation (basic and advanced courses).

**Introduction**

The field of complex matter is currently driven by a large body of new experiments and theoretical ideas in various branches of physics and biology. The former concerns systems that are composed of many interacting entities, such as those found e.g.in condensed matter, statistical physics or ultra- cold atomic physics. The latter aims at understanding the functioning of living systems from the single molecule level up to collective behaviors of cells and animals. Beyond their apparent diversity, these systems share a common characteristic: the emergence of complex collective behaviors from the interaction of elementary components. As a consequence, experts in these fields have been developing tools that are particularly well adapted for the multi-scale modelling and the multi-scale analysis of complex systems. Applications to social sciences issues (econophysics, social physics, etc.) are constantly on the rise.

The emergence of self-organized or self-assembled structures, criticality, percolating systems, out- of-equilibrium systems, disordered systems, nonlinear systems, active matter, fluid dynamics and turbulence, are some of the subjects of complex matter that are useful to tackle complex systems encountered in biology and more generally in multi-agent systems. Tools coming from signal analysis and statistical inference are powerful tools to extract the fundamental interactions that are responsible for an emerging behavior at a particular scale.

The goal of a digital research group, combined with a digital university, is to develop a platform that will be useful both to experts in complex matter and scientists/engineers involved in other fields of complexity sciences (from biology to multi-agent systems such as in social systems). The aim is therefore twofold. First, techniques issued from the field of complex matter must be visible to the rest of the complex system community. Due to its high level of formalization, this body of expertise is going to play a crucial role in the development of a quantitative approach to complexity. On the other way around, complex matter experts need to be informed of the challenges coming from the different fields of complex systems so they can be aware of the potentiality of their methods for investigating other fields of knowledge.

**2.2.3.1. Research fields (Grands objets)**

The following is a list of relevant research fields which contribute to the complex matter paradigma. They have been selected according to their transdisciplinary span and the potential interest of their ideas and tools for the complex system community at large.

**Soft condensed matter**

The field of soft condensed matter deals with the physics of easily deformable matter such as liquids, colloids, polymers, gels, foams (some part of granular matter may be included) — see

(Chaikin and Lubensky, 1995) for a general presentation of the field. The predominant feature of these systems is that the energy scale of their microscopic dynamics is the thermal energy, which is a pretty low energy scale with respect to the chemical bond energy encountered in solid states. This softness of the interaction and the diversity of the molecules that are at play results on a rich zoology of emerging collective behavior. Development of analytical tools, such as filed theory and the renormalization group (Lesnes, 1998) has allowed a powerful treatment of these systems. Interestingly, collective behaviors encountered in various biological systems (e.g. tissue formation) can be described using such tools.

Equilibrium approaches give hint on the natural trend of a system and may explain many of the self- organizing processes that can be encountered in biological systems, e.g. the formation of bilipidic membrane, the folding of amino acids and nucleic acids, the osmotic stability of a cell. Neverthess, a distinctive feature of biological systems comes from the presence of active processes. Non- equilibrium approaches based on (discrete and continuous) dynamical equation of conservation and out-of-equilibrium fluctuation theories have been developed in order to deal with phenomena such as morphogenesis, active molecular transport (e.g. kinesins, dinesins along microtubules), mitotic separation of chromosomes, cellular division, cellular structure and cellular motility (active polymerization).

Non-equilibrium approaches are also used to investigate the properties of biomolecules at the single molecule level, giving new insights into how proteins and nucleic acids do interact and dynamically process (e.g. the study of the replisome using single molecule techniques) (Ritort, 2006).

**Non-equilibrium phase transitions and critical phenomena**

The long lasting interest for non-equilibrium statistical physics has recently experienced a noticeable revival through the development of new methods and new areas of applications, especially including complex biological systems which, generically, persist in an out of equilibrium state.

Most recent efforts in this area have been devoted to interacting particle systems (Marro and Dickman, 1999). This broad class of stochastic systems is commonly used to model a wide range of non-equilibrium phenomena (chemical reactions, epidemiology, transport in biological systems, traffic and granular flows, social and economic systems, growth phenomena, etc…). Interacting particle systems can be investigated by a combination of numerical and analytical methods; some of them have even been solved exactly (see e.g. (Derrida, 1998)).

Biophysics has been very useful to understand living matter from the single molecule level up to the collective behavior of interacting cells (e.g. a tissue). At the single molecule level, powerful single-molecule techniques (e.g.fuorescence techniques or micro-twizzers allowing to apply pN forces) together with novel non-equilbirum theories have provided a better understanding of both the dynamics and the thermodynamics properties of complex biomolecules (nucleic acids and amino acids) in vivo. At the multi-cellular level, the advance of powerful visualizing tools (e.g. confocal microscopes) and the use of powerful theoretical tools for describing active matter and collective behavior have provided an unprecedented view of the genesis of living systems, and how they evolve and interact in space and time.

On the other way around, biology has been proved to be extremely useful to physics. Indeed, living systems have evolved for millions of years under very different non-equilibrium conditions. In this regard, the research of the thermal behavior of biological matter have shown an unprecedented richness of details, which are likely to have important consequences in the complex organization of living matter.

From the theoretical side, although the usual formalism of equilibrium statistical physics does not apply to out-of-equilibrium systems, it is now well-known that many of the tools developed in equilibrium settings can also be used out-of-equilibrium. This is in particular the case for the framework of critical behaviour, where concepts such as scale invariance and finite-size scaling have provided (largely numerical) evidence for universality in non-equilibrium systems (Hinrichsen, 2000). It is possible to investigate systems in which the non-equilibrium character stems not from the presence of gradients imposed, for instance, by boundary conditions, but because of the breaking of micro-reversibility – that is to say, time-reversal invariance – at the level of the microscopic dynamics in the bulk.

A large part of the research activity on non-equilibrium statistical physics is also centred on the various phase transitions observed in many contexts. Indeed, many non-equilibrium situations can be mapped onto each other, revealing a degree of universality going well beyond the boundaries of any particular field: for example, self-organized criticality in stochastic (toy) sand piles has been shown to be equivalent to linear interface depinning on random media, as well as to a particular class of absorbing phase transitions in reaction-diffusion models. Another prominent example is the jamming transition which bridges the fields of granular media and glassy materials (Mari et al., 2009). Synchronization and dynamical scaling are, likewise, very general phenomena which can be related to each other and to the general problem of understanding universality out of equilibrium. Non-equilibrium phase transitions and critical phenomena play a key role in many different fields of science, ranging from public health issues (infection spreading) to engineering problems (material failures, crack growth in hetereogeneous materials).

In recent years, the study of infection spreading in human societies has largely relied on concepts developed in mathematics and tout of equilibrium statistical physics (absorbing phase transitions, directed percolation, Levy flights, etc..).

Concerning applications to material sciences, statistical physics offer an approach to material failures and crack propagations in hetereogeneous media which goes beyond the standard continuum elastic theory. In brittle materials, for example, cracks initiate on the weakest elements of the micro- structures. As a result, toughness and life-time display extreme statistics (Weibull law, Gumbel law), the understanding of which requires approaches based on probabilistic theories. Moreover, in hetereogeneous materials crack growth often displays a jerky dynamics, with sudden jumps spanning over a broad range of length-scales. This is also suggested from the acoustic emission accompanying the failure of various materials and – at much larger scale – the seismic activity associated with earthquakes. This intermittent “crackling” dynamics cannot be captured by standard continuum theory. Furthermore, growing cracks create a structure of their own. Such roughness generation has been shown to exhibit universal morphological features, independent of both the material and the loading conditions, reminiscent of interface growth problems. This suggests that some approaches issued from statistical physics may succeed in describing the failure of heterogeneous materials.

**Disordered systems**

The statistical description of disordered systems is a very active field at the interface between mathematical physics and probability theory. It concerns systems that are either composed of heterogeneous particles, which can be a molecule or an economic agent, or composed of identical interacting particles that see different environments. Spin glasses models are the classical paradigma for these systems (Mézard et al., 1987), and the last 30 years have witnessed a spectacular development of their theoretical understanding. Various powerful methods have been developed to understand the correlations arising in these disordered systems, the associated phase transitions (replica theory, cavity method) and, from a dynamical point of view, how they explore their rugged energy landscape in search of local minima (mode coupling theory, dynamical

heterogeneities approach, fluctuation dissipation out of equilibrium).

These models describe the fundamental properties of a surprising large number of different systems ranging from diluted magnetic materials and glassy systems to neural networks. In recent years it has also been realized that many NP complex combinatorial optimization problems are intimately related to the spin glasses formalism. Spin glasses formalism has been applied to the study of folding processes in biomolecules (proteins, RNA, DNA) and to the formalization of social and economic systems (Challet et al., 2005).

Tools from the field of disordered systems are particularly promising for working out microscopic laws underlying the phenomena we are investigating or experiencing. More precisley, most experiments consist in measuring averaged quantities, correlation functions or consist in observing specific emerging patterns. The question is to find out which entities interact together and to determine the nature of the interaction. A paradigmatic example concerns the functioning of neural networks. In this case, the experiences consist in measuring the (complex) correlation between the spiking activity of neurones. The problem then is to determine the interacting neurons and the nature of their interaction (e.g. repression or activation) (Mézard, 2008).

**Collective behavior in active matter systems**

Active particles are able to self propel themselves by absorbing energy from their environment and transforming it into movement along a preferred direction. The general understanding of the collective properties of these self-propelled particles is the focus of a fast- increasing body of research in statistical physics and beyond (Ramaswamy, 2010). Under the vocable of active matter, one has observed the emergence of a research community involving physicists, biologists, engineers from IT (ad-hoc networks, Hua, 2009), and from swarm robotics (Sahin, 2005).

The collective coherent motion of a large number of these self propelled particles (generally known as flocking) is indeed an ubiquitous phenomena in nature. Examples of large scale structures emerging in such systems range from bird flocks and fish schools to cells growth, bacteria aggregates and segregation phenomena in a driven monolayer of elongated granular matter.

A cornerstone in the theoretical approach to active matter is the concept of universality, that is the idea that, despite the many individual differences existing between these systems, it could be possible to classify them according to their symmetries and conservation laws, thus delimiting broad universality classes. Minimal models – such as the celebrated Vicsek model (Vicsek et al., 1995) – are representative of these classes, and current research involving physicists and mathematicians is focused on the formulation of the proper continuum description (via Langevin equations) of active matter (Toner, 2005).

In recent years this approach has indeed revealed that active matter yields novel fascinating collective properties. Among these, let us mention the existence of true long-range order in 2D systems with XY symmetry, the ubiquity of giant number fluctuations, long-range correlations, and density segregation in the orientationally-ordered phases of collections of self-propelled particles (Chate 2006, 2008).

A closely related fundamental problem in social animal behavior is understanding how the behavior at the group scale (such as the complex aerial displays of starling flocks) results from the behavior and interactions among individuals (Parrish 1997). Modeling efforts aimed at specific animal groups thus represent the tool to link ibehavioral patterns at individual level to the behavior at group level, and should be performed in close conjunction with experimental or field data (to which models could be confronted). Obviously, this activity is closely related to and benefits from the theoretical knowledge accumulated in studying minimal models.

Furthermore, it is only recently that modern imaging and tracking techniques have started to allow for the study of large groups. In this context, a remarkable set of data is obtained in the context of

the EU FP7 StarFlag project which succeeded in fully reconstructing the 3D positions and velocities of individual birds in large groups of starlings (Ballerini 2008). Further experimental efforts on animal groups include the study of fish schools, sheep herds, insect swarms, etc. Collective phenomena in human crowds and more generally displacements patterns in human are an object of many theoretical and experimental studies too (Helbing, 2000).

Due to the ubiquity of mobile phones, smartphones and other Wi-Fi devices, as much as the ever increasing importance of cloud and diffused computing, the study of networks of mobile and intercommunicating agents is also a fundamental topic in current IT research. The collective motion paradigma is also at the heart of control problems for swarm robotics. In this context, proper behavioural rules at the individual level are studied and devised so that large group of simple robots self organize to perform the needed task at the collective level.

Multi agent models are also at the heart of many studies performed on social insects (ants, termites, bees, etc…). The objective here is the understanding the individual level processes which allow these groups to take decisions at the collective level and coordinate their activities in order to construct complex architectures(Theraulaz 2002). The concept of swarming intelligence is often evoked for these systems.

Of particular importance, finally, are the studies of the collective dynamics in bacterial populations and multicellular organisms, as much as the characterization of the dynamics of cytoskeletal filaments and molecular motors ("active gels") inside a single cell (Voituriez, 2006).

**Granular Matter**

Systems that are driven by molecular agitation can be modeled using tools that have been developed and tested for more than 100 years. In contrast, granular systems are composed of macroscopic particles that loose energy during interactions via inelastic collisions. Although the apparent simplicity of these systems, the characterization and understanding of their properties is only recent, raising new challenges for characterizing the collective behavior of systems which needs continuous injection of energy in order to evolve (de Gennes, 1999, Mehta, 1994).

The field of static granular matter has generated much activity for what concerns the geometrical characterization of force networks. From the dynamical point of view, driven granular matter tackles different fields such as the problem of avalanches, the characterization of traffic jams, the dynamic of sand piles, the genesis of pattern formation in natural phenomena (e.g. the formation of sand dunes), the mixing and segregation mechanisms in vibrated granular matter (e.g. mixing of aliments in the agro-industry). Theoretical development has revealed links with statistical physics of disordered systems (e.g. the concept of jamming), but also with fluid mechanics and thermodynamics. New challenges regards the extension of fluctuation dissipation relations, one of the tenets of equilibrium statistical mechanics, to granular systems. These relations allow one to relate the response of a system to external perturbations with its unperturbed fluctuations, and bears connections with a wide range of questions, from transport properties to resilience and robustness issues.

**Turbulence and fluidodynamics**

Turbulence describes the complex and unpredictable motions of a fluid (Monin, 1971, 1975). Examples are ubiquitous, from the volutes of a rising smoke to the dynamical patterns formed by cream mixing in a coffee cup and of far reaching importance; understanding the nature of turbulent flows is a central issue in a wide range of disciplines, from engineering (aeronautics, combustion, etc.) to earth sciences (climate theory, weather forecast, etc.).

While the dynamics of an incompressible Newtonian fluid is described by the Navier-Stokes equations, very little is known analytically about their solutions. Indeed, when the inertial

contribution to the Navier-Stokes equation outbalance the the viscous one, the fluid exhibit a transition between laminar (regular) and turbulent (chaotic) flow.

Approaches to turbulence have so far relied on scaling approaches, as in the celebrated K41 theory, which describe the energy cascade from larger to smaller structures by an inertial mechanism, or on reduced models (shell models, Burger equation, etc.). Direct numerical simulations of the Navier-Stokes equations, on the other hand, are prohibitively difficult, since the number of degrees of freedom which is needed to describe the flow grows with a power of the ratio between inertia and vicosity (the so called reynolds number). Still nowadays, the most refined pseudo-spectral numerical techniques are not able to describe turbulence at the Reynolds numbers achievable by best experimental facilities. A complementary approach developed in the last decades, finally, regards turbulence as a manifestation of deterministic chaos, and uses the typical quantifiers of nonlinear dynamics (Lyapunov exponents, fractal dimensions, etc., Eckmann 1985) to describe its properties (Bohr, 1998).

Nowadays, turbulence is still a substantially open problem (Procaccia, 2008), and there are many fascinating problems in many disciplines closely related to turbulence and fluidodynamics. At the more theoretical level, many efforts are being devoted to the problem of intermittency in turbulence and bifurcations in turbulent flows, once again a problem closely related with chaos theory. Magnetohydrodynamic turbulence (the emergence of a magnetic field through the turbulent motion of an electrically conducting fluid, (Biskamp 2003) plays an important role in many astrophysical and geophysical problems, such as the generation of the earth magnetic fields. Finally, fluidodynamics problems plays a central role in atmospheric sciences, oceanography and climate sciences. In particular, recent developments in chaos theory show interesting links with the goal of enhancing the predictability time in weather and climate forecasts (Kalnay 1997).

**2.2.3.2. Tools and methods (Implémentation)**

Systems in the field of complex matter share common characteristics and hence raises transversal questions. Generic tools must be available for the community. Mathematics, statistichal mechanics and nonlinear dynamics offer a formal language by which a quantitative investigation of complex systems may be carried on. We roughly distinguish between basic tools (roughly teached at the high end of undergraduate sudies) and more advanced ones, suitable for master level, PhD and permanent formation of active researcher.

**Towards a complex systems curriculum studiorum**

In the following we propose a list of basic formal methods commonly employed by theoretical physicist in the study of complex systems. They can be teached at an undergraduated level and represent a common mathematical language for a quantitative analysis of complex systems. We believe their teaching should lie at the heart of a complex system curriculum studiorum.

**Elements of linear algebra**

- Linear vector spaces, matrices, eigensystems, linear algebric equations__ Calculus
- Real and complex numbers, differentiation, integration, function series summation, Fourier series, ordinary differential equations, partial differential equations, distribution theory.
- Information theory, Bayesan theory, Markovian processes, Langevin equations and Fokker Planck approach, stochastic calculus (Ito and Stratonovich approaches)

- Numerical methods
- Introduction to general purpose programming language(s) (C++ ?) and basic numerical tools: linear algebraic equations, least squares method, numerical integration, fast fourier transforms, eigenvalues problems, maximization and minimization, roots finding, random numbers.
Elements of mechanics

- Dynamics, Hamiltonian formulation, basics of fluidodynamics, basics of electromagnetism
- Elements of statistical mechanics
- Emerging behaviors from local interactions (Ising systems): application to non-physical systems. Statistical ensembles — partition functions. Intro to out-of-equilibrium methods (Response theory).
- Dynamical systems theory
- Periodic and chaotic motion, elements of ergodic theory, time series analysis
**Advanced topics****Working out microscopic laws from macroscopic observables**This is a central issues in the study of emergent systems. Let us mention here an important problem that most scientists/engineers have faced or will be facing. In order to investigate a system composed of heterogeneous agents that interact with each other, we first start by measuring average quantities such as correlation functions. We then want to work out the microscopic laws that give rise to these macroscopic correlations. From the point of view of statistical physics techniques, this can be seen as an Inverse (Ising) problem and very promising tools have been developed in order to work out the microscopic details. This has been used for instance for working out neural couplings between retinal ganglion cells (Cocco, 2009), identification of direct residue contacts in protein- protein interaction (Weigt et al., 2009).

**Numerical methods**Discretizing the continuous description: for either solving partial derivative équations or simulating large systems (Finite Element Methods, Lattice Boltzmann methods, (Biased) Monte Carlo methods, Genetic algorithm, Stochastic integration, Molecular Dynamics, Agent-based approaches, …).

Numerical simulations of complex systems: heterogeneous agents, heterogeneous interactions, integration of multiple spatial and temporal scales (coarse-graingin methods, parallelisation methods).

**Response theories**By measuring the way of a thermal system responds to an external small perturbation, one can have access to the way it relaxes when it is not perturbed (Einsitein-Smoluchowski relation). More generally, Onsager relations describe the interplay between flows and forces through the phenomenological concept of transport coefficients. These relations have been formalized using fluctuation theory, showing that linear transport coefficients are related to the time dependence of equilibrium fluctuations in the conjugate flux (linear response theory). Onsager reciprocity relations further provide symmetry relations between transport coefficients mixing different types of perturbation allowing to understand the mutual influence of different parameters and perturbations.

Linear (equivalently equilibrium) response theory is nowadays well established. New challenges come from the description of active matter, and more generally from out-of equilibrium systems and non-linear effects, i.e. how a system responds to strong fluctuations. Recent progress, which must taught as n advanced topics, has been made in both fields. For instance, general linear fluctuation dissipation relations have been derived for stationary states (Marini Bettolo Marconi, 2008, Baiesi et al., 2009). Fluctuations theorems have revealed general symmetry properties for the fluctuations of systems that are driven arbitrarily far away from equilibrium, highlighting deep connections between statistical non-equilibrium behaviors and equilibrium properties (Ritort, 2008). These approaches must be used as an approach to tackle the case of athermal systems (energy needs to be continuously injected) and more generally to systems where there is no separation scale between microscopic elements and macroscopic description (granular matter, embryogenesis, general collective behaviors auch as in social phenomena). This must be a way to understand how such systems tend to evolve with respect to their interaction with the environment.

**An overview of the complex matter French community**

There is a huge community, most of it belonging to statistical physics, working in the field of complex matter. In the following, rather then centrating on individual researcher names, we choose to list some of the labs that have been very active in these fields

Granular Matter: ESPCI, LPS (ENS Paris), ENS Lyon, SPEC (CEA/Saclay)

Soft condensed matter: SPEC (CEA/Saclay), IPHT (CEA/Scalay), LPTMC Jussieu, ESPCI, Laboratoire Matière et Systèmes complexes (MSC), ISC

Disordered systems: LPTMS (Orsay), LPS ENS (Paris), IPHT and SPEC (CEA/Saclay), Laboratoire de physique ENS (Lyon)

Out of equilibrium critical phenomena: LPT (ENS Paris), ESPCI, IPHT and SPEC (CEA/Saclay) Active Matter: SPEC (CEA/Saclay), Institut Curie, ESPCI, ISC

Animal behavior: CRCA Toulouse

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