We complement and extend our work on fluctuation relations arising in nonequilibrium systems in steady states driven by Lévy noise [Phys. Rev. E 76, 020101(R) (2006)]. As a concrete example, we consider a particle subjected to a drag force and a Lévy white noise with tail index $μ\in (0,2]$, and calculate the probability distribution of the work done on the particle by the drag force, as well as the probability distribution of the work dissipated by the dragged particle in a nonequilibrium steady state. For $0<μ<2$, both distributions satisfy what we call an anomalous fluctuation relation, characterized by positive and negative fluctuations that asymptotically have the same probability. For $μ=2$, by contrast, the work and dissipated work distributions satisfy the known conventional and extended fluctuation relations, respectively, which are both characterized by positive fluctuations that are exponentially more probable than negative fluctuations. The difference between these different fluctuation relations is discussed in the context of large deviation theory. Experiments that could probe or reveal anomalous fluctuation relations are also discussed.
We show that the condensate occupation of a superfluid Bose liquid quite generally exhibits anomalously large fluctuations at finite temperatures. In three dimensions, the variance scales like T^2 V^{4/3} at low T, generalizing the result obtained by Giorgini, Pitaevskii, and Stringari for a weakly interacting Bose gas. In two dimensions there is only a quasicondensate, and the fluctuations are of the same order as the mean value.
Previously, we have derived a generalization of the canonical fluctuation relation between heat capacity and energy fluctuations $C=β^{2}<δU^{2}>$, which is able to describe the existence of macrostates with negative heat capacities $C<0$. In this work, we extend our previous results for an equilibrium situation with several control parameters to account for the existence of states with anomalous values in other response functions. Our analysis leads to the derivation of three different equilibrium fluctuation theorems: the \textit{fundamental and the complementary fluctuation theorems}, which…
An equilibrium statistical system is known to be stable if the fluctuations of global observables are normal, when their dispersions are proportional to the number of particles, or to the system volume. A general theorem is rigorously proved for the case, when an observable is a sum of linearly independent terms: The dispersion of a global observable is normal if and only if all partial dispersions of its terms are normal, and it is anomalous if and only if at least one of the partial dispersions is anomalous. This theorem, in particular, rules out the possibility that in a stable system with …
We study transient work Fluctuation Relations (FRs) for Gaussian stochastic systems generating anomalous diffusion. For this purpose we use a Langevin approach by employing two different types of additive noise: (i) internal noise where the Fluctuation-Dissipation Relation of the second kind (FDR II) holds, and (ii) external noise without FDR II. For internal noise we demonstrate that the existence of FDR II implies the existence of the Fluctuation-Dissipation Relation of the first kind (FDR I), which in turn leads to conventional (normal) forms of transient work FRs. For systems driven by ext…
We study Fluctuation Relations (FRs) for dynamics that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of standard Brownian motion. We first briefly review the concept of transient work FRs for stochastic dynamics modeled by the ordinary Langevin equation. We then introduce three generic types of dynamics generating anomalous diffusion: Lévy flights, long-time correlated Gaussian stochastic processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the work probability distributions in the simple nonequilibrium sit…
Anomalous dynamics characterized by non-Gaussian probability distributions (PDFs) and/or temporal long-range correlations can cause subtle modifications of conventional fluctuation relations. As prototypes we study three variants of a generic time-fractional Fokker-Planck equation with constant force. Type A generates superdiffusion, type B subdiffusion and type C both super- and subdiffusion depending on parameter variation. Furthermore type C obeys a fluctuation-dissipation relation whereas A and B do not. We calculate analytically the position PDFs for all three cases and explore numericall…
We discuss the conditions under which a population of anomalously diffusing individuals can be characterized by demographic fluctuations that are anomalously scaling themselves. Two examples are provided in the case of individuals migrating by Gaussian diffusion, and by a sequence of Lévy flights.
We study an anomalous behavior of the height fluctuation width in the crossover from random to coherent growths of surface for a stochastic model. In the model, random numbers are assigned on perimeter sites of surface, representing pinning strengths of disordered media. At each time, surface is advanced at the site having minimum pinning strength in a random subset of system rather than having global minimum. The subset is composed of a randomly selected site and its $(\ell-1)$ neighbors. The height fluctuation width $W^2(L;\ell)$ exhibits the non-monotonic behavior with $\ell$ and it has a m…
A Levy walk is a non-Markovian stochastic process in which the elementary steps of the walker consist of motion with constant speed in randomly chosen directions and for a random period of time. The time of flight is chosen from a long-tailed distribution with a finite mean but an infinite variance. Here we consider an open system with boundary injection and removal of particles, at prescribed rates, and study the steady state properties of the system. In particular, we compute density profiles, current and current fluctuations in this system. We also consider the case of a finite density of L…
We study in detail the dynamic scaling of the three-dimensional (3D) Ising model driven through its critical point on finite-size lattices and show that a series of new critical exponents are needed to account for the anomalous scalings originating from breaking of self-similarity of the so-called phases fluctuations. Our results demonstrate that new exponents are generally required for scaling in the whole driven process once the lattice size or an externally applied field are taken into account. These open a new door in critical phenomena and suggest that much is yet to be explored in driven…
A one-dimensional Hamiltonian system with exponential interactions perturbed by a conservative noise is considered. It is proved that energy superdiffuses and upper and lower bounds describing this anomalous diffusion are obtained
The ABC model is a simple diffusive one-dimensional non-equilibrium system which exhibits a phase transition. Here we show that the cumulants of the currents of particles through the system become singular near the phase transition. At the transition, they exhibit an anomalous dependence on the system size (an anomalous Fourier's law). An effective theory for the dynamics of the single mode which becomes unstable at the transition allows one to predict this anomalous scaling.
Recently, Portelli et al (2003) have semi-numerically obtained a functional form of the probability distribution of fluctuations in the total energy flow in a model for fluid turbulence. This follows earlier work suggesting that fluctuations in the total magnetization in the 2D X-Y model for a ferromagnet also follow this distribution. Here, starting from the scaling ansatz that is the basis of the turbulence model we analytically derive the functional form of this distribution and find its single control parameter that depends upon the scaling exponents and system size of the model. Our analy…
For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. Previous works have reveals that this anomalous scaling causes a singularity in the corresponding large deviation function. In order to further understand this problem, we study in this article the scaling of variance in several renewal-reward processes: counting processes with two different power-law decaying waiting time distributions and a Knudsen gas (a heat conduction model). Through analytical and numerical analyses…
We study the effect of fractal initial conditions in closed reactive systems in the cases of both mobile and immobile reactants. For the reaction $A+A\to A$, in the absence of diffusion, the mean number of particles $A$ is shown to decay exponentially to a steady state which depends on the details of the initial conditions. The nature of this dependence is demonstrated both analytically and numerically. In contrast, when diffusion is incorporated, it is shown that the mean number of particles $<N(t)>$ decays asymptotically as $t^{-d_f/2}$, the memory of the initial conditions being now carried…
We consider Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously in time, in order to study fluctuations of time-averaged observables in temporary heterogeneous diffusion process. We find that occupation time statistics is a powerful tool for calculating the time-averaged mean square displacement in the model. We show that the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges. Our model provides anomalous fluctuations of time-averaged diffusivity, which have rele…
The particle-number fluctuations originated from collective excitations are investigated for a three-dimensional, repulsively interacting Bose-Einstein condensate (BEC) confined in a harmonic trap. The contribution due to the quantum depletion of the condensate is calculated and the explicit expression of the coefficient in the formulas denoting the particle-number fluctuations is given. The results show that the particle-number fluctuations of the condensate follow the law $<δN_0^2> \sim N^{22/15}$ and the fluctuations vanish when temperature approaches to the BEC critical temperature.
We study anomalous diffusion for one-dimensional systems described by a generalized Langevin equation. We show that superdiffusion can be classified in slow superdiffusion and fast superdiffusion. For fast superdiffusion we prove that the Fluctuation-Dissipation Theorem does not hold. We show as well that the asymptotic behavior of the response function is a stretched exponential for anomalous diffusion and an exponential only for normal diffusion.
The exponent of anomalous diffusion of virus in cytoplasm of a living cell is experimentally known to fluctuate depending on localized areas of the cytoplasm, indicating heterogeneity of diffusion. In a recent paper (Itto, 2012), a maximum-entropy-principle approach has been developed in order to propose an Ansatz for the statistical distribution of such exponent fluctuations. Based on this approach, here the deviation of the statistical distribution of the fluctuations from the proposed one is studied from the viewpoint of Einstein's theory of fluctuations (of the thermodynamic quantities). T…
Using a simple model, we study the fluctuating dynamics of inextensible, semiflexible polar filaments interacting with active and directed force generating centres such as molecular motors. Taking into account the fact that the activity occurs on time-scales comparable to the filament relaxation time, we obtain some unexpected differences between both the steady-state and dynamical behaviour of active as compared to passive filaments. For the statics, the filaments have a {novel} length-scale dependent rigidity. Dynamically, we find strongly enhanced anomalous diffusion.
We consider work fluctuation relations (FRs) for generic types of dynamics generating anomalous diffusion: Levy flights, long-correlated Gaussian processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the probability distributions of mechanical and thermodynamical work in two paradigmatic nonequilibrium situations, respectively: a particle subject to a constant force and a particle in a harmonic potential dragged by a constant force. We check the transient FR for two models exhibiting superdiffusion, where a fluctuation-dissipation relation does not e…
An autoregressive model with a power-law type memory kernel is studied as a stochastic process that exhibits a self-affine-fractal-like behavior for a small time scale. We find numerically that the root-mean-square displacement for the time interval increases with a power law for small time but saturates at sufficiently large time. The exponent changes with the power exponent of the memory kernel.
The effect of random velocity field on the kinetics of single-species and two-species annihilation reactions is analysed near two dimensions in the framework of the field-theoretic renormalisation group. Fluctuations of particle density are modeled within the approach of Doi. The random incompressible velocity field is generated by stochastically forced Navier-Stokes equation in which thermal fluctuations - relevant below two dimensions - are taken into account.
We investigate the particle-number fluctuations due to the collective excitations created in a two-dimensional (2D) and a quasi-2D Bose-Einstein condensates (BECs) at low temperature. We find that the fluctuations display an anomalous behavior, i. e. for the 2D BEC they are proportional to $N^2$, where $N$ is the total number of particles. For the quasi-2D BEC, the particle-number fluctuations are proportional not only to $N^2$ but also to the square root of the trapping frequency in the strongly-confined direction.
We study the Brownian motion of a classical particle in one-dimensional inhomogeneous environments where the transition probabilities follow quasiperiodic or aperiodic distributions. Exploiting an exact correspondence with the transverse-field Ising model with inhomogeneous couplings we obtain many new analytical results for the random walk problem. In the absence of global bias the qualitative behavior of the diffusive motion of the particle and the corresponding persistence probability strongly depend on the fluctuation properties of the environment. In environments with bounded fluctuations…
It is shown that the Goldstone modes associated with a broken continuous symmetry lead to anomalously large fluctuations of the zero field order parameter at any temperature below T_c. In dimensions 2<d<4, the variance of the extensive spontaneous magnetization scales as L^4 with the system size L, independent of the order parameter dynamics. The anomalous scaling is a consequence of the 1/q^{4-d} divergence of the longitudinal susceptibility. For ground states in two dimensions with Goldstone modes vanishing linearly with momentum, the dynamical susceptibility contains a singular contribution…
Critical Scaling of Finite-Size Fluctuations around Marginal Stability in Long-Range Hamiltonian Systems
Graduate School of Informatics, Kyoto University（京都大学信息学研究科）；Institut Denis Poisson, Université d’Orléans, Université de Tours and CNRS（德尼·普瓦松研究所，奥尔良大学，图尔大学及法国国家科学研究中心）
Recently some experimental evidences have been obtained in favour of the existence of the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state in heavy-fermion superconductor CeCoIn_{5} and organic superconductor -(BETS)_{2}FeCl_{4}. However the unambiguous identification of FFLO state remains very difficult. We present the theoretical studies of the Gaussian fluctuations near the tricritical point (where the FFLO modulation appears) and demonstrate that the behavior of the fluctuational specific heat, paraconductivity and diamagnetism is qualitatively different from the…
Giant number fluctuations (GNF) are an anomaly universally observed in active fluids with polar or nematic order. In this paper, we show that GNF arise in the fluid phase of active Brownian particles (ABP), where the polar order is absent. GNF in ABP extends over a large but finite length which characterizes the growing velocity correlations. To suppress unwanted phase separation and allow ones to explore the disordered fluid phase at large activities, we impart the inertia, or the mass, to the ABP. A linearized hydrodynamic theory captures our findings, but only qualitatively. We find numeric…
Many approaches to modelling reaction-diffusion systems with anomalous transport rely on deterministic equations and ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a generating-functional approach to derive a Gaussian approximation for this intrinsic noise in subdiffusive systems. This results in corrections to the deterministic fractional reaction-diffusion equations. Using this analytical approach, we study the onset of noise-driven quasi-patterns in reaction-subdiffusion systems. We find that subdiffusion can be conducive to the fo…
We analyze the quantum phase transition between a semimetal and a superfluid in a model of attractively interacting fermions with a linear dispersion. The quantum critical properties of this model cannot be treated by the Hertz-Millis approach since integrating out the fermions leads to a singular Landau-Ginzburg order parameter functional. We therefore derive and solve coupled renormalization group equations for the fermionic degrees of freedom and the bosonic order parameter fluctuations. In two spatial dimensions, fermions and bosons acquire anomalous scaling dimensions at the quantum criti…
Momentum-conserving one-dimensional models are known to exhibit anomalous Fourier's law, with a thermal conductivity varying as a power law of the system size. Here we measure, by numerical simulations, several cumulants of the heat flux of a one-dimensional hard particle gas. We find that the cumulants, like the conductivity, vary as power laws of the system size. Our results also indicate that cumulants higher than the second follow different power laws when one compares the ring geometry at equilibrium and the linear case in contact with two heat baths (at equal or unequal temperatures). ke…
We present a unified formulation of a rotationally invariant nonlinear elasticity for a variety of spontaneously anisotropic phases, and use it to study thermal fluctuations in nematic elastomers and spontaneously anisotropic gels. We find that in a thermodynamic limit homogeneous nematic elastomers are universally incompressible, are characterized by a universal ratio of shear moduli, and exhibit an anomalous elasticity controlled by a nontrivial low temperature fixed point perturbative in D=3-epsilon dimensions. In three dimensions, we make predictions that are asymptotically exact.
Anomalous hydrodynamic fluctuations in the quantum XXZ spin chain
King’s College London（伦敦国王学院）；University of Oxford（牛津大学）；New York University（纽约大学）；CNRS, Université Paris-Dauphine, Université PSL（法国国家科学研究中心，巴黎第九大学，巴黎文理研究大学）；University of Ljubljana（卢布尔雅那大学）
Anomalous Diffusion in Driven Electrolytes due to Hydrodynamic Fluctuations
Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·皮尔斯理论物理中心）
Long-wavelength Mermin-Wagner fluctuations prevent the existence of translational long-range order, in two-dimensional systems at finite temperature. Their dynamical signature, which is the divergence of the vibrational amplitude with the system size, also affects disordered solids and washes out the transient solid-like response generally exhibited by liquids cooled below their melting temperature. Through a combined numerical and experimental investigation, here we show that long-wavelength fluctuations are also relevant at high temperature, where the liquid dynamics does not reveal a transi…
The emergence of a finite staggered magnetization in quantum Heisenberg antiferromagnets subject to a uniform magnetic field can be viewed as Bose-Einstein condensation of magnons. Using non-perturbative results for the infrared behavior of the interacting Bose gas, we present exact results for the staggered spin-spin correlation functions of quantum antiferromagnets in a magnetic field at zero temperature. In particular, we show that in dimensions 1 < D <= 3 the longitudinal dynamic structure factor S_parallel(q,omega) describing staggered spin fluctuations in the direction of the staggered m…
By constructing a multicanonical Monte Carlo simulation, we obtain the full probability distribution $ρ_N(r)$ of the degree assortativity coefficient $r$ on configuration networks of size $N$ by using the multiple histogram reweighting method. We suggest that $ρ_N(r)$ obeys a large deviation principle, $ρ_N \left(r-r_N^* \right) \asymp {e^{ - {N^ξ}I\left( {r- r_N^* } \right)}}$, where the rate function $I$ is convex and possesses its unique minimum at $r=r_N^*$, and $ξ$ is an exponent that scales $ρ_N$'s with $N$. We show that $ξ=1$ for Poisson random graphs, and $ξ\geq1$ for scale-free networ…
We study fluctuation properties of a one-dimensional anisotropic Heisenberg model out of equilibrium, focusing in particular on the gapped regime. Within the open-system setting we study large-deviation properties of the spin current. Numerically evaluating the first four current cumulants in a nonequilibrium stationary state at high energies, we find that the first two cumulants scale with the system size in a diffusive way, while the 3rd and the 4th cumulants do not. This means that the model is not an ordinary diffusive spin conductor.
We show a direct connection between Kubo's fluctuation-dissipation relation and Hawking effect that is valid in any dimensions for any stationary or static black hole. The relevant correlators corresponding to the fluctuating part of the force, computed from the known expressions for the anomalous stress tensor related to gravitational anomalies, are shown to satisfy the Kubo relation, from which the temperature of a black hole as seen by an observer at an arbitrary distance is abstracted. This reproduces the Tolman temperature and hence the Hawking temperature as that measured by an observer …
We study the collective dynamics of elongated swimmers in a very thin fluid layer by devising long, filamentous, non-tumbling bacteria. The strong confinement induces weak nematic alignment upon collision, which, for large enough density of cells, gives rise to global nematic order. This homogeneous but fluctuating phase, observed on the largest experimentally-accessible scale of millimeters, exhibits the properties predicted by standard models for flocking such as the Vicsek-style model of polar particles with nematic alignment: true long-range nematic order and non-trivial giant number fluct…
The kinetics of phase and population relaxation in quantum systems induced by noise with anomalously slowly decaying correlation function P (t) ~ (wt)^{- alpha}, where 0 < alpha < 1 is analyzed within continuous time random walk approach. The relaxation kinetics is shown to be anomalously slow. Moreover for alpha < 1 in the limit of short characteristic time of fluctuations w^{-1} the kinetics is independent of w. As alpha \to 1 the relaxation regime changes from the static limit to fluctuation narrowing. Simple analytical expressions are obtained describing the specific features of the kineti…
We study the polar collective dynamics of Janus colloidal particles fueled by an AC electric field. When the density is high enough, the polar interactions between the particles induce a polar orientationally ordered state which exhibits features reminiscent of the Vicsek model such as true long-range order and giant number fluctuations. Independent measurements of the polarity and velocity at the single particle level allowed us to investigate the single particle dynamics within the ordered state. We discovered theoretically-unaddressed statistical properties of the ordered state such as the …
Molecular dynamics simulations and nonequilibrium importance sampling are used to study the heat transport of low dimensional carbon lattices. For both carbon nanotubes and graphene sheets heat transport is found to be anomalous, violating Fourier's law of conduction with a system size dependent thermal conductivity and concomitant nonlinear temperature profiles. For carbon nanotubes, the thermal conductivity is found to increase as the square root of the length of the nanotube, while for graphene sheets the thermal conductivity is found to increase as the logarithm of the length of the sheet.…
The following question is the subject of our work: could a two-dimensional random path pushed by some constraints to an improbable "large deviation regime", possess extreme statistics with one-dimensional Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though non-universal, since the fluctuations depend on the underlying geometry. We consider in details two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a str…
The way tension propagates along a chain is a key to govern many of anomalous dynamics in macromolecular systems. After introducing the weak and the strong force regimes of the tension propagation, we focus on the latter, in which the dynamical fluctuations of a segment in a long polymer during its stretching process is investigated. We show that the response, i.e., average drift, is anomalous, which is characterized by the nonlinear memory kernel, and its relation to the fluctuation is nontrivial. These features are discussed on the basis of the generalized Langevin equation, in which the rol…
Anomalous scaling of heterogeneous elastic lines: a new picture from sample to sample fluctuations
Université Paris Saclay（巴黎萨克雷大学）；Laboratoire de Physique de l’Ecole Normale Supérieure, CNRS, ENS and PSL Université, Sorbonne Université, Université Paris Cité（法国高等师范学院物理实验室，CNRS，ENS，PSL大学，索邦大学，巴黎西岱大学）
We present a comprehensive theory of the dynamics and fluctuations of a two-dimensional suspension of polar active particles in an incompressible fluid confined to a substrate. We show that, depending on the sign of a single parameter, a state with polar orientational order is anomalously stable (or anomalously unstable), with a nonzero relaxation (or growth) rate for angular fluctuations at zero wavenumber. This screening of the broken-symmetry mode in the stable state does lead to conventional rather than giant number fluctuations as argued by Bricard et al., Nature ${\bf 503}$, 95 (2013), b…
We study properties of a random walk in a generalized Sinai model, in which a quenched random potential is a trajectory of a fractional Brownian motion with arbitrary Hurst parameter H, 0< H <1, so that the random force field displays strong spatial correlations. In this case, the disorder-average mean-square displacement grows in proportion to log^{2/H}(n), n being time. We prove that moments of arbitrary order k of the steady-state current J_L through a finite segment of length L of such a chain decay as L^{-(1-H)}, independently of k, which suggests that despite a logarithmic confinement th…
We derive the probability density of a diffusion process generated by nonergodic velocity fluctuations in presence of a weak potential, using the Liouville equation approach. The velocity of the diffusing particle undergoes dichotomic fluctuations with a given distribution $ψ(τ)$ of residence times in each velocity state. We obtain analytical solutions for the diffusion process in a generic external potential and for a generic statistics of residence times, including the non-ergodic regime in which the mean residence time diverges. We show that these analytical solutions are in agreement with …
The single-species annihilation reaction $A + A \rightarrow\varnothing$ is studied in the presence of a random velocity field generated by the stochastic Navier-Stokes equation. The renormalization group is used to analyze the combined influence of the density and velocity fluctuations on the long-time behavior of the system. The direct effect of velocity fluctuations on the reaction constant appears only from the two- loop order, therefore all stable fixed points of the renormalization group and their regions of stability are calculated in the two-loop approximation in the two-parameter $(ε, …
