Fluctuations and Large Deviations
Fluctuations arise in most physical phenomena, and their study has proven once and again to be a fruitful endeavor. The first example is probably Einstein’s determination of molecular scales on the basis of the fluctuating behavior of a mesoscopic particle immersed in a fluid, which opened the door to an experimental verification of the molecular hypothesis. Other examples range from the role of fluctuations to understand critical phenomena beyond mean field theories, to the study of fluctuations of spacetime correlations in glasses and other disordered materials, which has revealed the universal existence of dynamic heterogeneities in these systems. In all cases the statistics of fluctuations encodes essential information to understand the physics of the system of interest. Even further, fluctuations reflect the symmetries of the microscopic world at the macroscale. This is the case, for instance, of the Gallavotti-Cohen fluctuation theorem or the recently introduced isometric fluctuation relation, which express the subtle but enduring consequences of microscopic time reversibility at the macroscopic level. Special attention is due to large fluctuations which, though rare, play a dominant role as they drastically affect the system behavior.
The study of fluctuating behavior provides an alternative way to derive thermodynamic potentials from which to calculate the properties of a system, a path complementary to the usual ensemble approach. This can be extended to systems far from equilibrium, where no bottom-up approach exists yet connecting microscopic dynamics with macroscopic properties. The large-deviation function (LDF) controlling the fluctuations of the relevant macroscopic observables plays in nonequilibrium systems a role akin to the equilibrium free energy, and reflects the phenomenology typical of nonequilibrium physics (e.g., non-local behavior resulting in long-range correlations).
Fourier’s Law and One-Dimensional Fluids
Most one-dimensional (1D) systems -both quantum and classical- display anomalous collective properties. Generally speaking, the dimensional constraint inherent in these systems affects strongly the propagation of fluctuations. These fluctuations, in turn, control the system cooperative behavior, so their abnormal propagation leads to the observed anomalies. This observation underlies most surprising results found in one dimension.
Particularly interesting is the nonequilibrium behavior of 1D classical fluids. An essential feature of these systems is that the particle sequence remains unchanged during their time evolution. This important trait, also known as the single-file constraint, is directly responsible for the anomalous propagation of fluctuations. For instance, in order for a particle in a 1D fluid to move appreciably, a coherent, correlated motion of many particles is needed. This strongly suppresses diffusion: In 1D (stochastic) fluids, the mean square displacement grows now as ~t1/2, much slower than usual Fickian diffusion (~t). This single-file diffusion, which turns out to be relevant in many fields ranging from membrane biophysics to carbon nanotubes, zeolites, DNA, polymers, and nanofluids, etc., has been recently confirmed in experiments with colloids. Common to all these systems is the presence of confining structures (e.g., narrow channels) inducing quasi-1D (single-file) motion.
In the same way anomalous diffusion reflects the strong spatial correlations in 1D fluids, energy transport in one dimension reveals the presence of long-range velocity and current correlations. The simplest nonequilibrium situation where this is put forward is Fourier’s law. When the 1D fluid is subject to a small temperature gradient dT, an energy flux J~-kdT appears from the hot to the cold reservoir. It has been found that the heat conductivity k has unusual large values for 1D fluids. In particular, it is actually believed (though some controversy remains) that any classical 1D fluid with momentum-conserving interactions and nonzero total pressure should exhibit a divergent k in the thermodynamic limit; i.e., k~La as the system size L goes to infinity. Although there exists analytical and numerical evidence supporting this result in both 1D fluids and crystals, there is still no agreement on the theoretical framework and the exponent a>0: Fluctuating hydrodynamics predicts a=1/3, while mode-coupling theories and the Boltzmann equation result in a=2/5, and numerical results are still inconclusive (there is however new strong evidence supporting a=1/3). In this way, the behavior of 1D fluids far from equilibrium and the theoretical approach suitable for their description are still open problems.
Gels and Soft Matter
A gel can be roughly defined as a low-density, disordered, solid material composed by a liquid matrix in which dispersed particles form a very open network. In this way, a gel can be notably elastic and jellylike, as for instance gelatin, or rather solid and rigid, as silica gel. Though gels are materials common to everyday experience, their structural and dynamical properties remain puzzling in many respects. This is mostly due to the wide window of timescales and lengthscales which determine their physical behavior, e.g. from the molecular size of particles in the solvent to supramolecular structures. An important parameter for a gel is the typical lifetime of the interparticle bonds which define the underlying stress-sustaining network. In the limit of very strong, permanent bonds (with typical energies much larger than kT ), diffusion-limited cluster aggregation (DLCA) may lead to the formation of a fractal, system-spanning network usually termed as chemical gel and whose properties follow directly from geometry. In particular, gelation in chemical gels can be unambiguously identified with percolation. On the other hand, weaker bonds (with energies competing directly with kT) result in the formation of physical (or reversible) gels, where the links have a finite lifetime and the transient character of the network results in a complex interplay between structure and dynamics, leading to non-trivial flow properties.
Despite its ubiquity, the nature of physical gelation is still under intense debate and several mechanisms have been proposed to account for this transition, among them geometric percolation, the glass transition, and arrested phase separation. For instance, detailed experiments performed with colloidal particles with tunable interactions revealed that a non-trivial interplay between phase separation and kinetic arrest may produce gel-like structures. Associating polymers constitute another well-studied example of reversible gels. In that case, gels can be obtained far from phase separation, producing viscoelastic materials with highly non-linear rheological properties that are not well understood. In fact, one may speak about different routes to gelation: a nonequilibrium route based on kinetic arrest during spinodal decomposition (an irreversible process), and an equilibrium route in which the gel state is reached from an ergodic phase always allowing for the equilibration of the underlying network structure (as in the case of polymer association). In all cases, the emergence of the gel phase ensues some degree of kinetic arrest (at least for low wave-vector modes) which reflects the appearance of a percolating macroscopic structure capable of slowing down particle motions over long timescales. This kinetic arrest implies in many cases a close similarity between gelation and glass formation. We explain this similarity for the case of equilibrium gels, but discuss also important differences.
Synchronizability is one of the currently leading problems in the fast-growing field of complex networks. A number of studies have been devoted to scrutinize which network topologies are more prone to sustain a stable globally-synchronized state of generic oscillators defined at each of its nodes. This question is of broad interest since many complex systems in fields ranging from physics, biology, computer science, or physiology, can be seen as networks of coupled oscillators, whose functionality depends crucially on the network ability to maintain a synchronous oscillation pattern. In addition, it has been shown that networks with good synchronizability are also ‘good’ for (i) fast random walk spreading and therefore for efficient communication , (ii) searchability in the presence of congestion, (iii) robustness inthe absence of privileged hubs, (iv) performance of neural networks, (v) generating consensus in social networks, etc. Another related and important problem that has received a lot of attention, but that we will not study here, is the dynamics toward synchronized states (see, for example).
In general terms, we can say that the degree of synchronizability is high when all the different nodes in a given network can ‘talk easily’ to each other, or information packets can travel efficiently from any starting node to any target one. It was first observed that adding some extra links to an otherwise regular lattice in such a way that a small-world topology is generated, enhances synchronizability. This was attributed to the fact that the node-to- node average distance diminishes as extra links are added. Afterwards, heterogeneity in the degree distribution was shown to hinder synchronization in networks of symmetrically coupled oscillators, leading to the so-called ‘paradox of heterogeneity’ as heterogeneity is known to reduce in average the node-to-node distance but still it suppresses synchronizability. The effect of other topological features as betweenness centrality, correlation in the degree distribution and clustering has been also analyzed. For example, it has been shown that the presence of weighted links (rather than uniform ones) and asymmetric couplings do enhance further the degree of synchronizability, but here we focus on un-weighted and un-directed links.
Certainly, the main breakthrough was made by Barahona and Pecora who, in a series of papers, established a criterion based on spectral theory to determine the stability of synchronized states under very general conditions. Their main contribution is to link graph spectral properties with network dynamical properties. In particular, they considered the Laplacian matrix, encoding the network topology, and showed that the degree of synchronizability (understood as the range of stability of the synchronous state) is controlled by the ratio between its largest eigenvalue (λN) and the smallest non-trivial one (λ2), i.e. Q = λN/λ2, where N is the total number of nodes. The smaller Q the better the synchronizability.
Note that, as the range of variability of λN is quite limited (it is directly related to the maximum connectivity), minimizing Q is almost equivalent to maximizing the denominator λ2 (i.e the spectral gap) when the degree distribution is kept fixed.
It is worth noticing that, even if the eigenratio Q can be related to (or bounded by) topological properties such as the ones cited above (average path length, betweenness centrality, etc), none of these provides with a full characterization of a given network and therefore they are not useful to determine strict criteria for synchronizability. Nevertheless, they can be very helpful as long as they give easy criteria to determine in a rough way synchronizability properties, without having to resort to lengthly eigenvalue calculations.
Metastability and Nucleation
Many different natural phenomena involve metastable states that, eventually, decay via nucleation. Some familiar examples appear in the flow of electrical current through resistors, relaxation in amorphous materials, glasses and gels, domain wall motion in hysteretic disordered magnets, granular media evolution, earthquake dynamics, protein conformations, or false vacuum states in quantum field theory. There is a great amount of information concerning these situations but a full microscopic theory of metastability and nucleation is elusive. To begin with, there are two main coupled difficulties. One is that metastability concerns dynamics. The systems of interest typically show a complex free-energy landscape with (many) local minima, which are metastable in the sense that they trap the system for a long time. One may imagine that, eventually, relaxation occurs when the system after long wandering finds a proper path between the minima. This results in a complicate coupling of dynamics and thermodynamics. A second difficulty is that many systems of interest cannot reach thermal equilibrium after relaxation. In general, they are open to the environment, which often induces currents of matter or energy, or they are subject to agents which impose opposing tendencies which typically break detailed balance. These perturbations result in a final steady state which cannot be described by a Gibbsian measure, i.e. a nonequilibrium stationary state. Consequently, thermodynamics and ensemble statistical mechanics do not hold in these systems, which is a serious drawback.
These difficulties make the field most suitable for simple-model analysis. Indeed, the (two-dimensional) kinetic Ising model has been the subject of many studies of metastability not only in the case of periodic boundaries but also for finite lattices with free boundaries. The latter try to capture some of the physics of demagnetization in very dense media where magnetic particle sizes typically range from mesoscopic down to atomic levels. On the other hand, if one keeps oneself away from specific models, metastable states are often treated in the literature as rare equilibrium states, at least for times much shorter that the relaxation time. In this way, it has been shown that one may define a metastable state in a properly constrained (equilibrium) ensemble, and that most equilibrium concepts may easily be adapted.
In our work we continue those efforts and investigate metastability (and nucleation) in a nonequilibrium model. In order to deal with a simple microscopic model of metastability, we study a two-dimensional kinetic Ising system, as in previous studies. However, for the system to exhibit nonequilibrium behavior, time evolution is defined here as a superposition of the familiar thermal process at temperature T and a weak completely-random process. This competition is probably one of the simplest, both conceptually and operationally, ways of impeding equilibrium. Furthermore, one may argue that it captures some underlying disorder in nature, induced by random impurities or other causes which are unavoidable in actual samples. The specific origin for such dynamic randomness will vary with the situation considered. A similar mechanism has already been used to model the macroscopic consequences of rapidly-diffusing local defects and quantum tunneling in magnetic materials, for instance.
Coarsening and Phase Separation
There are dynamical phenomena in nonequilibrium systems which involve transformations between different phases. In particular, a very interesting phenomenon is the segregation process that emerges when a nonequilibrium (conserved) system evolves from a disordered phase towards the ordered one. An open question is the effects that nonequilibrium anisotropic conditions induce on this phase separation phenomenon.
Many alloys such as Al-Zn, which are homogeneous at high temperature, undergo phase separation after a sudden quench into the miscibility gap. One first observes nucleation in which small localized regions (grains) form. This is followed by “spinodal decomposition”. That is, some grains grow at the expense of smaller ones, and eventually coarsen, while their composition evolves with time. In addition to theoretically challenging, the details are of great practical importance. For example, hardness and conductivities are determined by the spatial pattern finally resulting in the alloy, and this depends on how phase separation competes with the progress of solidification from the melt.
A complete kinetic description of these highly non-linear processes is lacking. Nevertheless, the essential physics for some special situations is now quite well understood. This is the case when nothing prevents the sys- tem from reaching the equilibrium state, namely, coexistence of two thermodynamic phases. The simplest example of this is the (standard) lattice gas evolving from a fully disordered state to segregation into liquid (particle-rich phase) and gas (particle-poor phase). (Alternatively, using the language of the isomorphic lattice binary alloy, the segregation is into, say Al-rich and Zn-rich phases.) As first demonstrated by means of computer simulations, this segregation, as well as similar processes in actual mixtures exhibit time self-similarity. This property is better defined at sufficiently low temperature, when the thermal correlation length is small. The system then exhibits a single relevant length, the size l(t) of typical grains growing algebraically with time. Consequently, any of the system properties (including the spatial pattern) look alike, except for a change of scale, at different times.
This interesting property is revealed, for example, by the sphericalized structure factor S(k,t) as observed in scattering experiments. After a relatively short transient time, one observes that S(k,t)~J(t) F[k l(t)]. Taking this as a hypothesis, one may interpret J(t) and l(t) as phenomenological parameters to scale along the S and k axes, respectively. The hypothesis is then widely confirmed, and it follows that J(t)~l(t) d where d is the system dimension. It also follows that F(k) = Φ(k)·Ψ(σκ) where Φ and Ψ are universal functions. In fact, Φ describes the diffraction by a single grain, Ψ is a grain interference function, and σ characterizes the point in the (density−temperature) phase diagram where the sample is quenched. It then ensues that Ψ≈1 except at small values of k, so that, for large k, F(k) becomes almost independent of density and temperature, and even the substance investigated.
The grain distribution may also be directly monitored. A detailed study of grains in both microscopy experiments and computer simulations confirms time scale invariance. More specifically, one observes that the relevant length grows according to a simple power law, l(t)∼ta, and one typically measures a=1/3 at late times. This is understood as a consequence of diffusion of monomers that, in order to minimize surface tension, evaporate from small grains of high curvature and condensate onto larger ones (Ostwald ripening). In fact, Lifshitz and Slyozov, and Wagner independently predicted l∼t1/3, which is often observed, even outside the domain of validity of the involved approximations. In some circumstances, one should expect other, non- dominant mechanisms inducing corrections to the Lifshitz-Slyozov-Wagner one. For instance, effective diffusion of grains (Smoluchowski coagula- tion) leads to a=1/6, which may occur at early times; interfacial conduction leads to a=1/4; and, depending on density and viscosity, a fluid capable of hydrodynamic interactions may exhibit crossover with time to viscous (a=1) and then inertial (a=2/3) regimes.
Extending the above interesting picture to more realistic situations is an open question. The assumption that the system asymptotically tends to the coexistence of two thermodynamic (equilibrium) phases is often unjustified in Nature. This is the case, for example, for mixtures under a shear flow, whose study has attracted considerable attention. The problem is that sheared flows asymptotically evolve towards a nonequilibrium steady state and that this is highly anisotropic. Studying the consequences of anisotropy in the behavior of complex systems is in fact an important challenge. Another important example is that of binary granular mixtures under horizontal shaking. The periodic forcing causes in this case phase separation and highly anisotropic clustering.
In this thesis I study the dynamics of some nonequilibrium systems, using both computer simulations and theoretical tools. In particular, the following topics are studied: (i) metastability in a nonequilibrium ferromagnetic system, (ii) the onset of scale-free avalanches during metastable state decay, (iii) phase segregation under anisotropic nonequilibrium conditions, (iv) the microscopic basis of heat conduction and Fourier’s law in low dimensional systems and, finally, (v) a problem related with the physics of absorbing states. The study of these nonequilibrium dynamic phenomena provides an overview of some of the effects that nonequilibrium conditions may induce on the dynamics of complex systems.