Table of contents

It has been discovered previously that the topological order parameter could be identified from the topological data of the Green's function, namely the (generalized) TKNN invariant in general dimensions, for both non-interacting and interacting systems. In this note, we show that this phenomenon has a clear geometric derivation. This proposal could be regarded as an alternative proof for the identification of the corresponding topological invariant and the topological order parameter.

We study the continuum limit of the entanglement Hamiltonians of a block of consecutive sites in massless harmonic chains. This block is either in the chain on the infinite line or at the beginning of a chain on the semi-infinite line with Dirichlet boundary conditions imposed at its origin. The entanglement Hamiltonians of the interval predicted by conformal field theory (CFT) for the massless scalar field are obtained in the continuum limit. We also study the corresponding entanglement spectra, and the numerical results for the ratios of the gaps are compatible with the operator content of the boundary CFT of a massless scalar field with Neumann boundary conditions imposed along the boundaries introduced around the entangling points by the regularisation procedure.

The thermal properties of antiferromagnetic films—in particular, the square-lattice antiferromagnet—subjected to an external magnetic field pointing into the direction of the staggered magnetization are explored. The effective field theory analysis of the free energy density is carried out to two-loop order. While the emphasis is on finite temperature, we also discuss the behavior of the magnetization and staggered magnetization at zero temperature. Our results imply that the staggered magnetization increases in presence of the magnetic field—reminiscent of magnetic catalysis. Most remarkably, if staggered and magnetic field strength are kept fixed, the magnetization initially grows when temperature increases.

The transfer matrix of the square-lattice eight-vertex model on a strip with vertical lines and open boundary conditions is investigated. It is shown that for vertex weights that obey the relation and appropriately chosen K-matrices this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue . For positive vertex weights, is shown to be the largest transfer-matrix eigenvalue. The corresponding eigenspace is equal to the space of the ground states of the Hamiltonian of a related XYZ spin chain. An essential ingredient in the proofs is the supersymmetry of this Hamiltonian.

We investigate several entanglement-related quantities at finite-temperature criticality in the three-dimensional quantum spherical model, both as a function of temperature T and of the quantum parameter g, which measures the strength of quantum fluctuations. While the von Neumann and the Rényi entropies exhibit a volume-law for any g and T, the mutual information obeys an area law. The prefactors of the volume-law and of the area-law are regular across the transition, reflecting that universal singular terms vanish at the transition. This implies that the mutual information is dominated by nonuniversal contributions. This hampers its use as a witness of criticality, at least in the spherical model. We also study the logarithmic negativity. For any value of g, T, the negativity exhibits an area-law. The negativity vanishes deep in the paramagnetic phase, it is larger at small temperature, and it decreases upon increasing the temperature. For any g, it exhibits the so-called sudden death, i.e. it is exactly zero for large enough T. The vanishing of the negativity defines a 'death line', which we characterise analytically at small g. Importantly, for any finite T the area-law prefactor is regular across the transition, whereas it develops a cusp-like singularity in the limit . Finally, we consider the single-particle entanglement and negativity spectra. The vast majority of the levels are regular across the transition. Only the larger ones exhibit singularities. These are related to the presence of a zero mode, which reflects the symmetry breaking. This implies the presence of sub-leading singular terms in the entanglement entropies. Interestingly, since the larger levels do not contribute to the negativity, sub-leading singular corrections are expected to be suppressed for the negativity.

In a quantum many-body system that possesses an additive conserved quantity, the entanglement entropy of a subsystem can be resolved into a sum of contributions from different sectors of the subsystem's reduced density matrix, each sector corresponding to a possible value of the conserved quantity. Recent studies have discussed the basic properties of these symmetry-resolved contributions, and calculated them using conformal field theory and numerical methods. In this work we employ the generalized Fisher–Hartwig conjecture to obtain exact results for the characteristic function of the symmetry-resolved entanglement ('flux-resolved entanglement') for certain 1D spin chains, or, equivalently, the 1D fermionic tight binding and the Kitaev chain models. These results are true up to corrections of order where L is the subsystem size. We confirm that this calculation is in good agreement with numerical results. For the gapless tight binding chain we report an intriguing periodic structure of the characteristic functions, which nicely extends the structure predicted by conformal field theory. For the Kitaev chain in the topological phase we demonstrate the degeneracy between the even and odd fermion parity sectors of the entanglement spectrum due to virtual Majoranas at the entanglement cut. We also employ the Widom conjecture to obtain the leading behavior of the symmetry-resolved entanglement entropy in higher dimensions for an ungapped free Fermi gas in its ground state.

In single-molecule force spectroscopy experiments, the dependence of the mean unfolding force on the loading rate is used for obtaining information about the energetic and dynamic properties of the system under study. However, it is crucial to understand that different dynamic force spectroscopy (DFS) models are applicable in different regimes, and that different definitions of the unfolding force might be used in those models. Here, for the first time, we discuss three definitions of the unfolding force. We carried out Brownian dynamics simulations in order to demonstrate the difference between these definitions and compare DFS models. Importantly, we derive the dependence of the mean unfolding force for the whole range of the loading rates by unifying three previously reported DFS models. Among the currently available models, this unified model shows the best agreement with the simulated data.

We consider N circles of equal radii, r, having their centers randomly placed within a square domain of size with periodic boundary conditions (). When two or more circles intersect each other, each circle is divided by the intersection points into several arcs. We found the exact length distribution of the arcs. In the limiting case of dense systems and large size of the domain ( in such a way that the number of circle per unit area, n  =  N/L², is constant), the arc distribution approaches the probability density function (PDF) , where is the central angle subtended by the arc. This PDF is then used to estimate the sheet resistance of transparent electrodes based on conductive rings randomly placed onto a transparent insulating film.

We consider the Biswas–Chatterjee–Sen model on Barabasi–Albert networks. This system undergoes a continuous phase transition from a consensus state to a disordered state by increasing a noise parameter q over a critical threshold q_c. The noise parameter is defined as the probability of the affinity between two neighbors being negative, modeling Galam contrarians. We obtained the critical exponent ratios , , and by finite-size scaling data collapses, as well as the critical noises. Our numerical data is consistent with the critical thresholds q_c being a linear function of the inverse of network connectivity z, and with an asymptotic value of q_c  =  0.3418, close to the value of the critical noise for the complete graph.

The mean first passage time, one of the important characteristics for a stochastic process, is often calculated assuming the observation time is infinite. However, in practice, the observation time, T, is always finite and the mean first passage time (MFPT) is dependent on the length of the observation time. In this work, we investigate the observation time dependence of the MFPT of a particle freely moving in the interval [−L,L] for a simple diffusion model and five different models of subdiffusion, the fractional diffusion equation (FDE), scaled Brown motion (SBM), fractional Brownian motion (FBM), stationary Markovian approximation of SBM, and the aging continuous-time random walk model. We find that the MFPT is linearly dependent on T in the small T limit for all the models investigated, while the large-T behavior of the MFPT is sensitive to stochastic properties of the transport model in question. We also discuss the relationship between the observation time, T, dependence and the travel length, L, dependence of the MFPT. Our results suggest the observation time dependency of the MFPT can serve as an experimental measure that is far more sensitive to stochastic properties of transport processes than the mean square displacement.

A number of drawbacks found in the paper 'Bhatia–Thornton fluctuations, transport and ordering in partially ordered Al–Cu alloys' are considered. In particular, it is shown that this paper misleads readers confusing two different approaches—the mean spherical approximation and the random phase approximation.

We consider the effect of uniform driving on the interface between two phases which are described by model C dynamics. The non-driven system has a classical Gaussian interface described by capillary wave theory. The model under driving retains Gaussian statistics but the interface statistics are modified by driving, notably the height fluctuations are suppressed and the correlation length of the fluctuations is increased. The model we introduce can also be used as a model for the effect of activity on interface dynamics.

Bidirectional transport in (quasi) one-dimensional systems generically leads to cluster-formation and small particle currents. This kind of transport can be described by the asymmetric simple exclusion process (ASEP) with two species of particles. In this work, we consider the effect of non-Markovian site exchange times between particles. Different realizations of the exchange process can be considered: The exchange times can be assigned to the lattice bonds or each particle. In the latter case we specify additionally which of the two exchange times is executed, the earlier one (minimum rule) or the later one (maximum rule). In a combined numerical and analytical approach we find evidence that we recover the same asymptotic behavior as for unidirectional transport for most realizations of the exchange process. Differences in the asymptotic behavior of the system have been found for the minimum rule which is more efficient for fast decaying exchange time distributions.

We study the fluctuations of systems modeled by time periodically driven Markov jump processes. We focus on observables defined through time-periodic functions of the system's states or transitions. Using large deviation theory, canonical biasing and Doob transform, we characterize the asymptotic fluctuations of such observables after a large number of periods by obtaining the Markov process that produces them. We show that this process, called driven process, is the optimizer under constraint of the large deviation function for occupation and jumps.

We consider a three-dimensional lattice CP^N−1 model, which corresponds to the lattice Abelian–Higgs model in the infinite gauge-coupling limit. We investigate its phase diagram and critical behavior in the large-N limit. We obtain numerical evidence that the model undergoes a first-order transition for sufficiently large values of N, i.e. for any N  >  2 up to N  =  100. The transition becomes stronger—both the latent heat and the surface tension increase—as N increases. Moreover, on the high-temperature side, gauge fields decorrelate on distances of the order of one lattice spacing for all values of N considered. Our results are consistent with a simple scenario, in which the transition is of first order for any N, including . We critically discuss the analytic large-N calculations that predicted a large-N continuous transition, showing that one crucial assumption made in these computations fails for the model we consider.

We study the simple, linear, Edwards–Wilkinson equation that describes surface growth governed by height diffusion in the presence of spatiotemporally power-law decaying correlated noise. We analytically show that the surface becomes super-rough when the noise correlations spatio/temporal range is long enough. We calculate analytically the associated anomalous exponents as a function of the noise correlation exponents. We also show that super-roughening appears exactly at the threshold point where the local slope surface field becomes rough. In addition, our results indicate that the recent numerical finding of anomalous kinetic roughing of the Kardar–Parisi–Zhang model subject to temporally correlated noise may be inherited from the linear theory.

The percolation properties in anisotropic irreversible deposition of extended objects are studied by Monte Carlo simulations on a triangular lattice. Depositing objects of various shapes and sizes are made by directed self-avoiding walks on the lattice. Anisotropy is introduced by imposing unequal probabilities for placing the objects along different directions of the lattice. The degree of the anisotropy is characterized by the order parameter p  determining the probability for deposition in the chosen (horizontal) direction. For each of the other two directions adsorption occurs with probability . It is found that the percolation threshold increases with the degree of anisotropy, having the maximum values for fully oriented objects. Percolation properties of the elongated shapes, such as k-mers, are more affected by the presence of anisotropy than the compact ones.

Percolation in anisotropic deposition was also studied for a lattice with point-like defects. For elongated shapes a slight decrease of the percolation threshold with the impurity concentration d can be observed. However, for these shapes, significantly increases with the degree of anisotropy. In the case when depositing objects are triangles, results are qualitatively different. The percolation threshold decreases with d, but is not affected by the presence of anisotropy.

We study the random-link matching problem on random regular graphs, along with two relaxed versions of the problem, namely the fractional matching and the so-called 'loopy' fractional matching. We estimate the asymptotic average optimal cost using the cavity method. Moreover, we study the finite-size corrections due to rare topological structures appearing in the graph at large sizes. We estimate these contributions using the cavity approach, and we compare our results with the output of numerical simulations. The analysis also clarifies the meaning of the finite-size contributions appearing in the fully connected version of the problem, which have already been analyzed in the literature.

The jamming transition of non-spherical particles is fundamentally different from the spherical case. Non-spherical particles are hypostatic at their jamming points, while isostaticity is ensured in the case of the jamming of spherical particles. This structural difference implies that the presence of asphericity affects the critical exponents related to the contact number and the vibrational density of states. Moreover, while the force and gap distributions of isostatic jamming present power-law behaviors, even an infinitesimal asphericity is enough to smooth out these singularities. In a recent work (Brito et al 2018 Proc. Natl Acad. Sci. 115 11736–41), we have used a combination of marginal stability arguments and the replica method to explain these observations. We argued that systems with internal degrees of freedom, like the rotations in ellipsoids, or the variation of the radii in the case of the breathing particles fall in the same universality class. In this paper, we review comprehensively the results about the jamming with internal degrees of freedom in addition to the translational degrees of freedom. We use a variational argument to derive the critical exponents of the contact number, shear modulus, and the characteristic frequencies of the density of states. Moreover, we present additional numerical data supporting the theoretical results, which were not shown in the previous work.

With the aging of the world's population and the resulting increase of disabilities, the characteristics of pedestrian flow mixed with wheelchair users has been paid more and more attention. In this study, experiments in funnel-shaped bottlenecks were performed to study the impact of the bottleneck shape and the ratio of wheelchair users on the crowd dynamics. It is found that the increase in wheelchairs in the crowd leads to lower moving efficiency and greater congestion impacting the escape time, time-space relationship and time headway. Under low mixing ratios (<2.35%), less congestion occurred in the 45° bottleneck among the four tested angles (0°, 15°, 30°, 45°). The average speeds of the wheelchair users are the fastest in 45° bottleneck (0.310  ±  0.097 m s⁻¹) until the mixing ratio becomes 7.05%. However, the advantage of the angle disappears when the mixing ratio gets higher. The findings in this study are meaningful for the guidance of pedestrian evacuation through bottlenecks with the presence of wheelchair users.

Axelrod's model for the dissemination of culture combines two key ingredients of social dynamics: social influence, through which people become more similar when they interact, and homophily, which is the tendency of individuals to interact preferentially with similar others. In Axelrod's model, the agents are fixed to the nodes of a network and are allowed to interact with a predetermined set of peers only, resulting in the frustration of a large number of agents that end up culturally isolated. Here we modify this model by allowing the agents to move away from their cultural opposites and stay put when near their cultural likes. The comfort, i.e. the tendency of an agent to stay put in a neighborhood, is determined by the cultural similarity with its neighbors. The less the comfort, the higher the odds that the agents will move apart a fixed step size. We find that the comfort-driven mobility fragments severely the influence network for low initial cultural diversity, resulting in a network composed of only microscopic components in the thermodynamic limit. For high initial cultural diversity and intermediate values of the step size, we find that a macroscopic component coexists with the microscopic ones. The transition between these two fragmentation regimes changes from continuous to discontinuous as the step size increases. In addition, we find that for both very small and very large step sizes the influence network is severely fragmented.

Synchronization among rhythmic elements is modeled by coupled phase-oscillators, each of which has the so-called natural frequency. A symmetric natural frequency distribution induces a continuous or discontinuous synchronization transition, or oscillation, from the nonsynchronized state, for example. It has been numerically reported that asymmetry in the natural frequency distribution introduces new types of bifurcation diagrams with, in the order parameter, oscillation or a discontinuous jump which emerges from a partially synchronized state. We propose a theoretical classification method for five types of bifurcation diagrams including the new ones, paying attention to the generality of the theory. The oscillation and the jump from partially synchronized states are discussed, respectively, by the linear analysis around the nonsynchronized state and by extending the amplitude equation up to the third leading term. The theoretical classification is examined through comparison with the numerically obtained classification.

In this work we study a modified stochastic version of the well known Bass model of innovation diffusion. In the considered model, an innovation spreads through an homogeneously mixing population which is subdivided into four classes of individuals, namely, ignorants, aware, adopters and abandoners. These classes are related to the participation level of each individual in the spreading procedure. An individual in ignorant or aware state becomes an adopter due to the influence of other adopters in the population. On the other hand, any adopter can spontaneously abandon the innovation, thus becoming an abandoner, at constant rate. We measure the impact of the innovation spreading by studying the remaining proportion of population who have never heard about the innovation and those who know about it but they have not adopted it yet. This is accomplished by proving a law of large numbers and a central limit theorem. In addition, we discuss the behavior of the maximum of adopters during the process, as well as the instant of time in which the process reaches this quantity.

We introduce the discrete Green's function to elucidate how resource fluctuations determine flow fluctuations in a network optimizing a global cost function. To enhance the robustness of the network against fluctuations, we develop the schemes of optimal bandwidth allocation in links and optimal resource adjustment in nodes. With the total bandwidth of the network fixed, the approach of optimal bandwidth allocation is to increase the bandwidth in links such that the number of overloaded links or the amount of excess flows in networks under fluctuations can be minimized. Similarly, the approach of optimal resource adjustment is to minimize the number of overloaded links in networks under fluctuations with the total resource change in the network fixed. Compared with the conventional approach of proportionate bandwidth assignment or resource reduction, it is found that the optimized bandwidth allocation or resource adjustment can highly enhance the stability of the networks against fluctuations. The changes of loads and currents prescribed by the optimal bandwidth allocation and resource adjustment schemes are correlated with each other, except for some nodes that exhibit relay effects.

We propose a mean-field vaccination game framework that combines two distinct processes: the simultaneous spreading of two strains of an influenza-like disease, and the adoption of vaccination based on evolutionary game theory presuming an infinite and well-mixed population. The vaccine is presumed to be imperfect such that it shows better efficacy against the original (resident) strain rather than the new one (mutant). The vaccination-decision takes place at the beginning of an epidemic season and depends upon the vaccine-effectiveness along with the cost. Additionally, we explore a situation if the original strain continuously converts to a new strain through the process of mutation. With the aid of numerical experiments, we explore the impact of vaccinating behavior on a specific strain prevalence. Our results suggest that the emergence of vaccinators can create the possibility of infection-prevalence of the new strain if the vaccine cannot bestow a considerable level of efficiency against that strain. On the other hand, the resident strain can continue to dominate under large-scale vaccine avoidance. Moreover, in the case of continuous mutation, the vaccine efficacy against the new strain plays a pivotal role to control the disease prevalence. We successfully obtain phase diagrams, displaying the infected fraction with each strain, final epidemic size, vaccination coverage, and average social payoff considering two-different strategy-update rules and provide a comprehensive discussion to get an encompassing idea, justifying how the vaccinating behavior can affect the spread of a disease having two strains.

Highlights

–We build a mean-field vaccination game scheme to analyze the effect of an imperfect vaccine on a two-strain epidemic spreading taking into account individuals' vaccination behavior.

–En masse vaccine avoidance can enhance the possibility of the original strain prevalence.

–Propensity for vaccination can create the possibility of infection by the new strain if the vaccine is unable to provide a considerable level of efficiency against that strain.

Outbreaks of repeated pandemics and heavy epidemics are daunting threats to human life. This study aims at investigating the dynamics of disease conferring temporary or waning immunity with several forced-control policies aided by vaccination game theory. Considering an infinite and well-mixed homogenous population, our proposed model further illustrates the significance of introducing two well-known forced control techniques, namely, quarantine and isolation, in order to model the dynamics of an infectious disease that spreads within a human population where pre-emptive vaccination has partially been taken before the epidemic season begins. Moreover, we carefully examine the combined effects of these two types (pre-emptive and forced) of protecting measures using the SEIR-type epidemic model. An in-depth investigation based on evolutionary game theory numerically quantifies the weighing impact of individuals' vaccinating decisions to improve the efficacy of forced control policies leading up to the relaxation of the epidemic spreading severity. A deterministic SVEIR model, including vaccinated (V) and exposed (E) states, is proposed having no spatial structure while implementing these intervention techniques. This study uses a mixed control strategy relying on quarantine and isolation policies to quantify the optimum requirement of vaccines for eradicating disease prevalence completely from human societies. Furthermore, our theoretical study justifies the fact that adopting forced control policies significantly reduces the required level of vaccination to suppress emerging disease prevalence, and it also confirms that the joint policy works even better when the epidemic outbreak takes place at a higher transmission rate. Research reveals that the isolation policy is a better disease attenuation tool than the quarantine policy, especially in endemic regions where the disease progression rate is relatively higher. However, a meager progression rate gradually weakens the speed of an epidemic outbreak and, therefore, applying a moderate level of control policies is sufficient to restore the disease-free state. Essentially, positive measures (pre-emptive vaccination) regulate the position of the critical line between two phases, whereas exposed provisions (quarantine or isolation) are rather dedicated to mitigating the disease spreading in endemic regions. Thus, an optimal interplay between these two types of intervention techniques works remarkably well in attenuating the epidemic size. Despite having advanced on the development of new vaccines and control strategies to mitigate epidemics, many diseases like measles, tuberculosis, Ebola, and flu are still persistent. Here, we present a dynamic analysis of the SVEIR model using mean-field theory to develop a simple but efficient strategy for epidemic control based on the simultaneous application of the quarantine and isolation policies.

Highlights

• This model incorporates the elements of mathematical epidemiology and a vaccination game into a single framework.

• A dynamical analysis of the SEIR/V epidemic model equipped with quarantine and isolation policies is introduced.

• The proposed game-theoretic framework rigorously addresses real situations when control policies are adopted simultaneously as well as separately.

• Adopting control policies can significantly reduce the required level of vaccination to suppress disease prevalence.

• A joint policy seems impressive when an epidemic outbreak takes place at a higher transmission rate.

• For a smaller basic reproduction number, the quarantine policy outperforms the isolation policy.

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.

The local balance equations for the density, momentum, and energy of a dilute gas of elastic or inelastic hard spheres, strongly confined between two parallel hard plates are obtained. The starting point is a Boltzmann-like kinetic equation, recently derived for this system. As a consequence of the confinement, the pressure tensor and the heat flux contain, in addition to the terms associated to the motion of the particles, collisional transfer contributions, similar to those that appear beyond the dilute limit. The complexity of these terms, and of the kinetic equation itself, compromise the potential of the equation to describe the rich phenomenology observed in this kind of systems. For this reason, a simpler model equation based on the Boltzmann equation is proposed. The model is formulated to keep the main properties of the underlying equation, and it is expected to provide relevant information in more general states than the original equation. As an illustration, the solution describing a macroscopic state with uniform temperature, but a density gradient perpendicular to the plates is considered. This is the equilibrium state for an elastic system, and the inhomogeneous cooling state for the case of inelastic hard spheres. The results are in good agreement with previous results obtained directly from the Boltzmann equation.

Upon cooling or densification, a supercooled liquid shows drastic slowing down toward its glass-transition point. The physical mechanism behind this slow glassy dynamics has been a matter of discussion for a long time, but there has still been no consensus on its origin. Recently, we have found that for systems mainly interacting with steric repulsions, glassy structural order (or, angular order) generally develops upon cooling and its correlation length, , grows as ( is the bare correlation length, T is the temperature, T₀ is the hypothetical ideal glass transition, and d is the spatial dimensionality). This ordering is difficult to detect by two-body density correlation since it is a consequence of sterically-induced (entropically-driven) many-body correlation that lowers local free energy. Interestingly, the power-law growth of with the exponent of 2/d is reminiscent of the Ising criticality. We also find that the structural relaxation time diverges as (: the microscopic relaxation time, K is a fragility index, is the Boltzmann constant), suggesting that glass transition is a consequence of Ising-like criticality with growing activation energy. Unlike ordinary critical phenomena, the activation energy of particle motion increases in proportion to the root of the correlation volume of , implying that the particle motion is strongly correlated in that volume. This relation indicates that the impact of spatial fluctuations of the order parameter on slow dynamics is not perturbative but intrinsic. Although we need further study to confirm our claim, we hope that the discussion in this article would provide a good starting point for further consideration of the physical nature of glass transition.

We review recent works (Sarao Mannelli et al 2018 arXiv:1812.09066, 2019 Int. Conf. on Machine Learning 4333–42, 2019 Adv. Neural Information Processing Systems 8676–86) on analyzing the dynamics of gradient-based algorithms in a prototypical statistical inference problem. Using methods and insights from the physics of glassy systems, these works showed how to understand quantitatively and qualitatively the performance of gradient-based algorithms. Here we review the key results and their interpretation in non-technical terms accessible to a wide audience of physicists in the context of related works.

Rare events in stochastic processes with heavy-tailed distributions are controlled by the big jump principle, which states that a rare large fluctuation is produced by a single event and not by an accumulation of coherent small deviations. The principle has been rigorously proved for sums of independent and identically distributed random variables and it has recently been extended to more complex stochastic processes involving Lévy distributions, such as Lévy walks and the Lévy–Lorentz gas, using an effective rate approach. We review the general rate formalism and we extend its applicability to continuous time random walks and to the Lorentz gas, both with stretched exponential distributions, further enlarging its applicability. We derive an analytic form for the probability density functions for rare events in the two models, which clarify specific properties of stretched exponentials.

This article presents a review on the extension of the idea of linear response theory (LRT) that is well known in the statistical mechanics of systems near thermal equilibrium state, so as to be applied to the statistics in turbulence. The idea has been applied to the statistics in the inertial subrange of turbulence in an unbounded fluid domain. Recently, the idea was extended to one-point statistics in the inertial sublayer of a wall bounded turbulence. A similarity between the energy flux, from large to small scales, in homogeneous isotropic turbulence and momentum transfer in the wall-normal direction in wall bounded turbulence plays a key role in the extension. This article presents also a discussion on further extension of the scope of LRT to the statistics of single-particle diffusion in turbulence.