In this review paper we survey recent achievements in anomalous heat diffusion, while highlighting open problems and research perspectives. First, we briefly recall the main features of the phenomenon in low-dimensional classical anharmonic chains and outline some recent developments in the study of perturbed integrable systems and the effect of long-range forces and magnetic fields. Selected applications to heat transfer in material science at the nanoscale are described. In the second part, we discuss of the role of anomalous conduction in coupled transport and describe how systems with anomalous (thermal) diffusion allow a much better power-efficiency trade-off for the conversion of thermal to particle current.

Energy transport in one-dimensional chains of particles with three conservation laws is generically anomalous and belongs to the Kardar-Parisi-Zhang dynamical universality class. Surprisingly, some examples where an apparent normal heat diffusion is found over a large range of length scales were reported. We propose a novel physical explanation of these intriguing observations. We develop a scaling analysis that explains how this may happen in the vicinity of an integrable limit, such as, but not only, the famous Toda model. In this limit, heat transport is mostly supplied by quasiparticles with a very large mean free path l. Upon increasing the system size L, three different regimes can be observed: a ballistic one, an intermediate diffusive range, and, eventually, the crossover to the anomalous (hydrodynamic) regime. Our theoretical considerations are supported by numerical simulations of a gas of diatomic hard-point particles for almost equal masses and of a weakly perturbed Toda chain. Finally, we discuss the case of the perturbed harmonic chain, which exhibits a yet different scenario.

We review the state of the art of the problem of heat conduction in one dimensional nonlinear lattices. The peculiar role of finite size and time corrections to the predictions of the hydrodynamic theory is discussed. The emerging scenario indicates that when dealing with systems, whose spatial size is comparable with the mean-free path of peculiar nonlinear excitations, hydrodynamic predictions at leading order are no more predictive. We can conjecture that one should take into account estimates of subleading contributions, which could play a major role in some regions of the parameter space in 'small' systems.

We present a detailed investigation of the denaturation process for intergenic sequences of several bacterial species. The reason for analyzing these specific sequences is that these regions are expected to be denaturated to allow for the intrusion of the transcription factors performing the transcription process of genes. Our study relies upon a well known dynamical model of the DNA double-strand proposed by Dauxois-Peyrard-Bishop [44], applied to the collection of intergenic regions from bacterial species. We have performed extended numerical simulations in the presence of a thermostat (canonical setup) and we have found that all of these indicators essentially identify a typical denaturation temperature. This confirms the reliability and robustness of the denaturation thermodynamics described by this model. We want to remark also that the actual value of the denaturation temperature, as expected, varies from species to species, because of the different structural features of the corresponding intergenic sequences. Another important result reported in this manuscript is that the comparison with simulations in the absence of a thermostat (microcanonical setup) yields sensibly higher, irrealistic denaturation temperatures, thus providing evidence that thermal fluctuations play a crucial role in the cooperative effect yielding the denaturation process.

In this paper, we propose a computational strategy for performing genome-wide analyses of intergenic sequences in bacterial genomes. Following similar directions of a previous paper, where a method for genome-wide analysis of eucaryotic Intergenic sequences was proposed, here we developed a tool for implementing similar concepts in bacteria genomes. This allows us to (i) classify intergenic sequences into clusters, characterized by specific global structural features and (ii) draw possible relations with their functional features.

We present the Multi-Particle-Collision (MPC) dynamics approach to simulate properties of low-dimensional systems. In particular, we illustrate the method for a simple model: A one-dimensional gas of point particles interacting through stochastic collisions and admitting three conservation laws (density, momentum and energy). Motivated from problems in fusion plasma physics, we consider an energy-dependent collision rate that accounts for the lower collisionality of high-energy particles. We study two problems: (i) the collisional relaxation to equilibrium starting from an off-equilibrium state and (ii) the anomalous dynamical scaling of equilibrium time-dependent correlation functions. For problem (i), we demonstrate the existence of long-lived population of suprathermal particles that propagate ballistically over a quasi-thermalized background. For (ii) we compare simulations with the predictions of nonlinear fluctuating hydrodynamics for the structure factors of density fluctuations. Scaling analysis confirms the prediction that such model belong to the Kardar-Parisi-Zhang universality class.

We present a numerical study of dynamical correlations (structure factors) of the long-range generalization of the Fermi-Pasta-Ulam oscillator chain, where the strength of the interaction between two lattice sites decays as a power alpha of the inverse of their distance. The structure factors at finite energy density display distinct peaks, corresponding to long-wavelength propagating modes, whose dispersion relation is compatible with the predictions of the linear theory. We demonstrate that dynamical scaling holds, with a dynamical exponent z that depends weakly on a in the range 1 < alpha < 3. The lineshapes have a non-trivial functional form and appear somehow independent of alpha. Within the accessible time and size ranges, we also find that the short-range limit is hardly attained even for relatively large values of alpha.

The non-local expression proposed by Luciani-Mora-Virmont is implemented in a one dimensional fluid model for the scrape-off layer. Analytical solutions for heat equation are discussed as well as the impact of sheath boundary conditions on the continuity of the temperature profile. The non-local heat flux is compared to the Spitzer-Härm heat flux for different collisionality.

Nonequilibrium and thermal transport properties of the Toda chain, a prototype of classically integrable system, subject to additional (nonintegrable) terms are considered. In particular, we study via equilibrium and nonequilibrium simulations, the Toda lattice with a power-law pinning potential, recently analyzed by Lebowitz and Scaramazza [ArXiv:1801.07153]. We show that, according to general expectations, even the case with quadratic pinning is genuinely non-integrable, as demonstrated by computing the Lyapunov exponents, and displays normal (diffusive) conductivity for very long chains. However, the model has unexpected dynamical features and displays strong finite-size effects and slow decay of correlations to be traced back to the propagation of soliton-like excitations, weakly affected by the harmonic pinning potential. Some novel results on current correlations for the standard integrable Toda model are also reported.

We investigate thermal conduction in arrays of long-range interacting rotors and Fermi-Pasta-Ulam (FPU) oscillators coupled to two reservoirs at different temperatures. The strength of the interaction between two lattice sites decays as a power ? of the inverse of their distance. We point out the necessity of distinguishing between energy flows towards or from the reservoirs and those within the system . We show that energy flow between the reservoirs occurs via adirect transfer induced by long-range couplings and a diffusive process through the chain. To this aim, we introduce a decomposition of the steady-state heat current that explicitly accounts for such direct transfer of energy between the reservoir. For 0 ?<1, the direct transfer term dominates, meaning that the system can be effectively described as a set of oscillators each interacting with the thermal baths. Also, the heat current exchanged with the reservoirs depends on the size of the thermalized regions: In the case in which such size is proportional to the system size N, the stationary current is independent on N.For ?>1, heat transport mostly occurs through diffusion along the chain: For the rotors transport is normal, while for FPU the data are compatible with an anomalous diffusion, possibly with an ?-dependent characteristic exponent.

We numerically investigate out-of-equilibrium stationary processes emerging in a Discrete Nonlinear Schrödinger chain in contact with a heat reservoir (a bath) at temperature TL and a pure dissipator (a sink) acting on opposite edges. Long-time molecular-dynamics simulations are performed by evolving the equations of motion within a symplectic integration scheme. Mass and energy are steadily transported through the chain from the heat bath to the sink. We observe two different regimes. For small heat-bath temperatures TL and chemical-potentials, temperature profiles across the chain display a non-monotonous shape, remain remarkably smooth and even enter the region of negative absolute temperatures. For larger temperatures TL, the transport of energy isstrongly inhibited by the spontaneous emergence of discrete breathers, which act as a thermal wall. A strongly intermittent energy flux is also observed, due to the irregular birth and death of breathers.The corresponding statistics exhibit the typical signature of rare events of processes with largedeviations. In particular, the breather lifetime is found to be ruled by a stretched-exponential law.

By means of hybrid multiparticle collsion-particle-in-cell (MPC-PIC) simulations we study the dynamical scaling of energy and density correlations at equilibrium in moderately coupled two-dimensional (2D) and quasi-one-dimensional (1D) plasmas. We find that the predictions of nonlinear fluctuating hydrodynamics for the structure factors of density and energy fluctuations in 1D systems with three global conservation laws hold true also for 2D systems that are more extended along one of the two spatial dimensions. Moreover, from the analysis of the equilibrium energy correlators and density structure factors of both 1D and 2D neutral plasmas, we find that neglecting the contribution of the fluctuations of the vanishing self-consistent electrostatic fields overestimates the interval of frequencies over which the anomalous transport is observed. Such violations of the expected scaling in the currents correlation are found in different regimes, hindering the observation of the asymptotic scaling predicted by the theory.

Steady nonequilibrium states are investigated in a one-dimensional setup in the presence of two thermodynamic currents. Two paradigmatic nonlinear oscillators models are investigated: an XY chain and the discrete nonlinear Schrödinger equation. Their distinctive feature is that the relevant variable is an angle in both cases. We point out the importance of clearly distinguishing between energy and heat flux. In fact, even in the presence of a vanishing Seebeck coefficient, a coupling between (angular) momentum and energy arises, mediated by the unavoidable presence of a coherent energy flux. Such a contribution is the result of the 'advection' induced by the position-dependent angular velocity. As a result, in the XY model, the knowledge of the two diagonal elements of the Onsager matrix suffices to reconstruct its transport properties. The analysis of the nonequilibrium steady states finally allows to strengthen the connection between the two models.

We study the dynamics of networks with inhibitory and excitatory leak-integrate-and-fire neurons with short-term synaptic plasticity in the presence of depressive and facilitating mechanisms. The dynamics is analyzed by a heterogeneous mean-field approximation, which allows us to keep track of the effects of structural disorder in the network. We describe the complex behavior of different classes of excitatory and inhibitory components, which give rise to a rich dynamical phase diagram as a function of the fraction of inhibitory neurons. Using the same mean-field approach, we study and solve a global inverse problem: reconstructing the degree probability distributions of the inhibitory and excitatory components and the fraction of inhibitory neurons from the knowledge of the average synaptic activity field. This approach unveils new perspectives on the numerical study of neural network dynamics and the possibility of using these models as a test bed for the analysis of experimental data.

We investigate an open XYZ spin-1/2 chain driven out of equilibrium by boundary reservoirs targeting different spin orientations, aligned along the principal axes of anisotropy. We show that by tuning local magnetic fields, applied to spins at sites near the boundaries, one can change any nonequilibrium steady state to a fully uncorrelated Gibbsian state at infinite temperature. This phenomenon occurs for strong boundary coupling and on a critical manifold in the space of the fields amplitudes. The structure of this manifold depends on the anisotropy degree of the model and on the parity of the chain size.

We report about the main dynamical features of a model of leaky integrate-and-fire excitatory neurons with short-term plasticity defined on random massive networks. We investigate the dynamics by use of a heterogeneous mean-field formulation of the model that is able to reproduce dynamical phases characterized by the presence of quasisynchronous events. This formulation allows one to solve also the inverse problem of reconstructing the in-degree distribution for different network topologies from the knowledge of the global activity field. We study the robustness of this inversion procedure by providing numerical evidence that the in-degree distribution can be recovered also in the presence of noise and disorder in the external currents. Finally, we discuss the validity of the heterogeneous mean-field approach for sparse networks with a sufficiently large average in-degree.

The dynamics of neural networks is often characterized by collective behavior and quasi-synchronous events, where a large fraction of neurons fire in short time intervals, separated by uncorrelated firing activity. These global temporal signals are crucial for brain functioning. They strongly depend on the topology of the network and on the fluctuations of the connectivity. We propose a heterogeneous mean-field approach to neural dynamics on random networks, that explicitly preserves the disorder in the topology at growing network sizes, and leads to a set of self-consistent equations. Within this approach, we provide an effective description of microscopic and large scale temporal signals in a leaky integrate-and-fire model with short term plasticity, where quasi-synchronous events arise. Our equations provide a clear analytical picture of the dynamics, evidencing the contributions of both periodic (locked) and aperiodic (unlocked) neurons to the measurable average signal. In particular, we formulate and solve a global inverse problem of reconstructing the in-degree distribution from the knowledge of the average activity field. Our method is very general and applies to a large class of dynamical models on dense random networks.

In this paper we perform a genome-wide analysis of H. sapiens promoters. To this aim, we developed and combined two mathematical methods that allow us to (i) classify promoters into groups characterized by specific global structural features, and (ii) recover, in full generality, any regular sequence in the different classes of promoters. One of the main findings of this analysis is that H. sapiens promoters can be classified into three main groups. Two of them are distinguished by the prevalence of weak or strong nucleotides and are characterized by short compositionally biased sequences, while the most frequent regular sequences in the third group are strongly correlated with transposons. Taking advantage of the generality of these mathematical procedures, we have compared the promoter database of H. sapiens with those of other species. We have found that the above-mentioned features characterize also the evolutionary content appearing in mammalian promoters, at variance with ancestral species in the phylogenetic tree, that exhibit a definitely lower level of differentiation among promoters.

DNA is one of the most investigated systems in science not only for its basic relevance for biology, but also because it exhibits very interesting features associated with its structure and dynamics. These aspects have attracted from the very beginning the interest of physicists, chemists and mathematicians. In particular, much progress has been done during the decades on the understanding of the mechanism of denaturation dynamics in the double-strand of DNA in the presence of structural and topological disorder. We report here the introductory remarks to the seminars, that were presented on this topic in occasion of the conference on "Inhomogeneous Random Systems", held at the Institute Henri Poincare on the 25th of January 2012.

We study the influence of noise on the dynamics of a simple model of excitatory leaky integrate - and - fire neurons in a diluted network. The stochastic process amounts to a random walk with boundaries acting on the external current, whose average value plays the role of a control parameter identifying different dynamical phases. Above a given threshold value one observes a gaussian statistics of synchronous firing events, that changes to an asymmetric long-tail distribution below threshold. For uncorrelated noise the distribution below threshold exhibits an exponential tail for large rare events, while for strongly correlated noise the long-tail turns to a power-law. This interesting dynamical scenario is shown to persist also when short-term plasticity is introduced in the model. Synchronous firing events change to population bursts and the model with plasticity is shown to reproduce quantitatively what observed in in vitro experiments. We also discuss the persistence of this scenario in the thermodynamic limit.