Portal:Complex Systems Digital Campus/E-Department on Complex matter

Portal:Complex_Systems_Digital_Campus/E-Department_on_Complex matter The field of complex and non-equilibrium systems is currently driven by a large body of new experiments and theoretical ideas in various branches of physics, from condensed matter physics up to ultra-cold atomic physics and biology. Beyond their apparent diversity, these systems share a common characteristic: the emergence of complex collective behaviors from the interaction of elementary components. Glassy dynamics, out-of-equilibrium systems, the emergence of self-organized or self-assembled structures, criticality, percolating systems, domain wall propagation and pinning of elastic walls, nonlinear systems, turbulence and fracture propagation are some subjects of complex matter that can be addressed only with the tools developed for the study of systems of interacting entities. Understanding these phenomena also requires the development of new theoretical methods in statistical physics and the design of new types of experiments.

Main challenges 1.	Non-equilibrium statistical physics 2.	Damage and fracture of heterogeneous materials 3.	Glassy dynamics: glasses, spin glasses and granular media 4.	Bifurcations in turbulence: from dynamo action to slow dynamics

1. Non-equilibrium statistical physics

The long lasting interest for non-equilibrium phenomena has recently experienced a noticeable revival, because of the concomitant development of novel theoretical ideas (especially on the symmetries of non-equilibrium fluctuations) and of new areas of applications, ranging from many examples in condensed matter physics to other branches of physics (heavy ion collisions, the early universe) and to other sciences, including biology (manipulations of single molecules). Non-equilibrium phenomena also play an important part in many of the interdisciplinary applications of statistical physics (modeling the collective behavior of animals, or social and economic agents). A physical system may be out of equilibrium for either of the following two reasons: Slow dynamics. The microscopic dynamics of the system is reversible, so that the system possesses a bona fide equilibrium state. The dynamics of some of the degrees of freedom is however too slow for these variables to equilibrate within the duration of the experiment. The system is therefore in a slowly evolving non-equilibrium state for a very long time (forever in some model systems). The characteristic features of this regime of non-equilibrium relaxation, including the violation of the fluctuation dissipation theorem, have been the subject of intense activity over the last decade. This kind of situation is commonly referred to as aging (see the part on glassy dynamics). Driven dynamics. The dynamics of the system is not reversible, usually because of some macroscopic driving caused by external forces. For instance, an electric field induces a non-zero current across the system. This driving violates the reversibility of the underlying stochastic dynamics. The system reaches a non-equilibrium stationary state, where it stays forever. There are also systems (at least model systems) where the lack of reversibility lies entirely at the microscopic level, and does not rely on any macroscopic external driving. The paradigm of such a situation is the celebrated voter model. One of the most salient advances of the last decade has been the discovery of a whole series of general results concerning the symmetries of spontaneous fluctuations in non-equilibrium states. These theorems, associated with names such as Gallavotti, Cohen, Evans and Jarzynski, have been applied and/or tested in many circumstances, both by theory and experiment. Most recent efforts in this area have been devoted to interacting particle systems. This broad class of stochastic systems is commonly used to model a wide range of non-equilibrium phenomena (chemical reactions, ionic conduction, transport in biological systems, traffic and granular flows). Many interacting particle systems can be investigated by analytical methods, whereas some of them have even been solved exactly. 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 can be used out-of-equilibrium. This is in particular the case for the framework of critical behavior, scale invariance, finite-size scaling, which have provided (largely numerical) evidence for universality in non-equilibrium systems. It is possible to investigate systems in which the non-equilibrium character stems not from the presence of gradients imposed, for instance, by boundary reservoirs, but because of the breaking of microreversibility - 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. It has been studied experimentally thanks to a model experiment consisting in a sheared layer of metallic disks. 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.

2. Damage and fracture of heterogeneous materials

To understand the interrelation between microstructure and mechanical properties has been one of the major goals of materials science over the past few decades. Quantitative predictive models are even more necessary when extreme conditions – in terms of temperature, environment or irradiation, for example – or long-time behavior are considered. While some properties, such as elastic moduli, are well approximated by the average of the properties of the various microstructural components, none of the properties related to fracture – elongation, stress to failure, fracture toughness –follow such an easy rule, mostly: (i) because of the high stress gradient in the vicinity of a crack tip, and (ii) because, as the more brittle elements of microstructure break first, one is dealing with extreme statistics. As a result, there is no way that a material can be replaced by an “effective equivalent” medium in the vicinity of a crack tip. This has several major consequences: Size effects in material failure

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 nonlinear and statistical physics (percolation theory, random fuse models, etc.). Crack growth in heterogeneous materials

Crack propagation is the fundamental mechanism leading to material failure. While continuum elastic theory allows the precise description of crack propagation in homogeneous brittle materials, we are still far from understanding the case of heterogeneous media. In such 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. Let us finally add that the mechanisms become significantly more complex when the crack growth velocity increases and becomes comparable to the sound velocity, as in impact or fragmentation problems, for instance. Plastic deformation in glassy materials

Because of high stress enhancement at crack tips, fracture is generally accompanied by irreversible deformations, even in the most brittle amorphous materials. While the physical origin of these irreversible deformations is now well understood in metallic materials, it remains mysterious in amorphous materials like oxide glasses, ceramics or polymers, where dislocations cannot be defined.

3. Glassy dynamics

Glasses

The physics of glasses concerns not only the glasses used in everyday life (silicates), but a whole set of physical systems such as molecular glasses, polymers, colloids, emulsions, foams, Coulomb glasses, dense assemblies of grains, etc. Understanding the formation of these amorphous systems, the so-called glass transition, and their out-of-equilibrium behavior is a challenge which has resisted a substantial research effort in condensed matter physics over the last decades. This problem is of interest to several fields from statistical mechanics and soft matter to material sciences and biophysics. Several fundamental open questions emerge: is the freezing due to a true underlying phase transition, or is it a mere crossover with little universality in the driving mechanism? What is the physical mechanism responsible for the slowing down of the dynamics and glassiness? What is the origin of the aging, rejuvenation and memory effects? What are the common concepts that emerge to describe the various systems evoked above, and what remains specific to each of them? Interestingly, however, evidence has mounted recently that the viscous slowing down of super-cooled liquids and other amorphous systems might be related to the existence of genuine phase transitions of a very singular nature. Contrary to usual phase transitions, the dynamics of glass-formers dramatically slows down with nearly no changes in structural properties. We are only just beginning to understand the nature of the amorphous long-range order that sets in at the glass transition, the analogies with spin-glasses and their physically observable consequences. One of the most interesting consequences of these ideas is the existence of dynamical heterogeneities (DH), which have been discovered to be (in the space-time domain) the counterpart of critical fluctuations in standard phase transitions. Intuitively, as the glass transition is approached, increasingly larger regions of the material have to move simultaneously to allow flow, leading to intermittent dynamics, both in space and in time. The existence of an underlying phase transition and of dynamical heterogeneities should significantly influence the rheological and aging behaviors of these materials, which are indeed quite different from those of simple liquids and solids. As a consequence, progress in the understanding of glassy dynamics should trigger several technological advances. An important example where the peculiar properties of glasses are used in technology is the stocking of nuclear waste. From an experimental point of view, the major challenges for the future have been transformed not only because progress in the domain has led to radically new questions, but also because new experimental techniques now allow to investigate physical systems at a microscopic scale. In previous decades the, focus was mainly on the behavior of timescales and of global properties. New challenges for the years to come are: i) To study the local dynamical properties in order to unveil what changes in the way molecules evolve and interact makes the dynamics glassy, in particular why the relaxation time of supercooled liquids increases by more than 14 orders of magnitude in a small temperature window; ii) To provide direct and quantitative evidence that glassy dynamics is (or is not) related to an underlying phase transition; iii) To study the nature of the dynamical heterogeneities (correlation between their size and their time evolution, fractal dimensions, etc.); iv) To investigate the nature of the out-of-equilibrium properties of glasses, such as violation of the fluctuation-dissipation theorem, intermittence, etc. From a theoretical point of view, the major challenge is to construct and develop the correct microscopic theory of glassy dynamics. This will consist both in unveiling the underlying physical mechanisms that give rise to slow and glassy dynamics and in obtaining a quantitative theory that can be compared to experiments. The main focus will again be on local dynamic properties, their associated length-scale and their relation to the growing timescales and the global properties of glassy dynamics. Spin glasses

The expression “spin glasses" was invented to describe certain metallic alloys of a non-magnetic metal with few, randomly substituted, magnetic impurities. Experimental evidences were obtained for a low temperature phase characterized by a non-periodic freezing of the magnetic moments with a very slow and strongly history-dependent response to external perturbations. Basic fundamental ingredients of spin glasses are disorder and frustration. The frustration consists in the fact that the energy of all the pairs of spins cannot be minimized simultaneously. The theoretical analysis of spin glasses lead to the celebrated Edwards-Anderson model: classical spins on the sites of a regular lattice with random interactions between nearest-neighbor spins. This has led to many developments over the years, and the concepts developed for this problem have found applications in many other fields, from structural glasses and granular media to problems in computer science (error correction codes, stochastic optimization, neural networks, etc.). The program of developing a field theory of spin glasses is extremely hard, with steady, slow progress. The theory is not yet able to make precise predictions in three dimensions. Numerical simulations face several difficulties: we cannot equilibrate samples of more than a few thousand spins, the simulation must be repeated for a large number of disorder samples (due to non-self-averaging), and the finite size corrections decay very slowly. Spin glasses also constitute an exceptionally convenient laboratory frame for experimental investigations of glassy dynamics. The dependence of their dynamical response on the waiting time (aging effect) is a widespread phenomenon observed in very different physical systems such as polymers and structural glasses, disordered dielectrics, colloids and gels, foams, friction contacts, etc.

Granular Media close to the Jamming transition

Common experience indicates that as the volume fraction of hard grains is increased beyond a certain point, the system jams, stops flowing and is able to support mechanical stresses. The dynamical behavior of granular media close to the ’jamming transition’ is very similar to that of liquids close to the glass transition. Indeed, granular media close to jamming display a similar dramatic slowing-down of the dynamics as well as other glassy features like aging and memory effect. One of the main features of the dynamics in glass-forming systems is what is usually called the cage effect, which accounts for the different relaxation mechanisms: at short times, any given particle is trapped in a confined area by its neighbors, which form the so-called effective cage, leading to a slow dynamics; at sufficiently long times, the particle manages to leave its cage, so that it is able to diffuse through the sample by successive cage changes, resulting in a faster relaxation. Contrary to standard critical slowing down, this slow glassy dynamics does not seem related to a growing static local order. For glass-formers it has been proposed instead that the relaxation becomes strongly heterogeneous and dynamic correlations build up when approaching the glass transition. The existence of such a growing dynamic correlation length is very important in revealing some kind of criticality associated with the glass transition. One can, for example, study the dynamics of a bi-disperse monolayer of disks under two different mechanical forcings, i.e. cyclic shear and horizontal vibrations. In the first case, a “microscopic” confirmation of the above similarity has been obtained and the second can provide the experimental evidence of a simultaneous divergence of length and time scales precisely at the volume fraction for which the system loses rigidity (jamming transition).

4. Bifurcations in turbulence: from dynamo action to slow dynamics

Dynamo action Dynamo action consists in the emergence of a magnetic field through the motion of an electrically conducting fluid. It is believed to be at the origin of the magnetic fields of planets and most astrophysical objects. One of the most striking features of the Earth's dynamo, revealed by paleomagnetic studies, is the observation of irregular reversals of the polarity of its dipole field. A lot of work has been devoted to this problem, both theoretically and numerically, but the range of parameters relevant for natural objects are out of reach of numerical simulations for a long time to come, in particular because of turbulence. In industrial dynamos, the path of the electrical currents and the geometry of the (solid) rotors are completely prescribed. As this cannot be the case for planets and stars, experiments aimed at studying dynamos in the laboratory have evolved towards relaxing these constraints. The experiments in Riga and Karlsruhe showed in 2000 that fluid dynamos could be generated by organizing favourable sodium flows, but the dynamo fields had simple time dynamics. The search for more complex dynamics, such as exhibited by natural objects, has motivated most teams working on the dynamo problem to design experiments with less constrained flows and a higher level of turbulence. In 2006, the von Karman sodium experiment (VKS) was the first to show regimes where a statistically stationary dynamo self-generates in a fully turbulent flow. It then evidenced other dynamical regimes for the first time, including irregular reversals - as in the Earth - and periodic oscillations -as in the Sun. These complex regimes, involving a strong coupling between hydrodynamic and MHD, need to be studied in detail. In particular, they reveal that although the dynamo magnetic field is generated by the turbulent fluctuations, it behaves as a dynamical system with a few degrees of freedom. Theoretical predictions regarding the influence of turbulence on the mean-flow dynamo threshold are scarce. Small velocity fluctuations produce little impact on the dynamo threshold. Predictions for arbitrary fluctuation amplitudes can be reached by considering the turbulent dynamo as an instability (driven by the mean flow) in the presence of a multiplicative noise (turbulent fluctuations). In this context, fluctuations can favor or impede the magnetic field growth, depending on their intensity or correlation time. We can use direct and stochastic numerical simulations of the MHD equations to explore the influence of turbulence on the dynamo threshold. Bifurcations in turbulence At high Reynolds numbers, some systems undergo a turbulent bifurcation between different mean topologies. Moreover, this turbulent bifurcation can conserve memory of the system history. These aspects of the turbulent bifurcation recall classical properties of bifurcation of low-dimensionality systems, but the bifurcation dynamics is really different, probably because of the presence of very large turbulent fluctuations. Future studies will be concerned with the universal relevance of the concept of multistability in average along time of states of highly fluctuating systems and by the transitions between these states (e.g. magnetic inversions of the Earth, climate changes between glacial and interglacial cycles). The slow dynamics of turbulent systems, in the case where exchanges of stability can be observed for some global quantities or some averaged properties of the flow, should also be studied, and an attempt made to construct nonlinear or stochastic models of those transitions. In the case of turbulent flows with symmetry, it is also possible to construct a statistical mechanics, and develop a thermodynamical approach to the equilibrium states of axisymmetric flows at some fixed coarse-grained scale. This allows the definition of a mixing entropy and derivation of Gibbs states of the problem by a procedure of maximization of the mixing entropy under constraints of conservation of the global quantities. From the Gibbs state, one can define general identities defining the equilibrium states, as well as relations between the equilibrium states and their fluctuations. This thermodynamics should be tested in turbulent flows, e.g. von Karman flow. Effective temperatures can be measured and preliminary results show that they depend on the considered variable, as in other out-of-equilibrium systems (glass). Finally, we can derive a parameterization of inviscid mixing to describe the dynamics of the system at the coarse-grained scale. The corresponding equations have been numerically implemented and can be used as a new subgrid scale model of turbulence.

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