Search

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 of these models, we find that the variances show an anomalous scaling when the exponent of the power law is -3. For a counting process with the power-law exponent smaller than -3, this anomalous scaling does not take place: this indicates that the processes only fluctuate around the expectation with an error that is compatible with a standard large deviation scaling. In this case, we argue that anomalous scaling appears in higher order cumulants. Finally, many-body particles interacting through soft-core interactions with the boundary conditions employed in the Knudsen gas are studied using numerical simulations. We observe that the variance scaling becomes normal even though the power-law exponent in the boundary conditions is -3., Comment: discussion of the anomalous fluctuations with memory effects

We study the large-time asymptotic of renewal-reward processes with a heavy-tailed waiting time distribution. It is known that the heavy tail of the distribution produces an extremely slow dynamics, resulting in a singular large deviation function. When the singularity takes place, the bottom of the large deviation function is flattened, manifesting anomalous fluctuations of the renewal-reward processes. In this article, we aim to study how these singularities emerge as the time increases. Using a classical result on the sum of random variables with regularly varying tail, we develop an expansion approach to prove an upper bound of the finite-time moment generating function for the Pareto waiting time distribution (power law) with an integer exponent. We perform numerical simulations using Pareto (with a real value exponent), inverse Rayleigh and log-normal waiting time distributions, and demonstrate similar results are anticipated in these waiting time distributions., Comment: discussion of the finite-time asymptotic of the rate function

Turbulence locally created in laminar pipe flows shows sudden decay or splitting after a stochastic waiting time. In laboratory experiments, the mean waiting time was observed to increase double-exponentially as the Reynolds number (Re) approaches its critical value. To understand the origin of this double-exponential increase, we perform many pipe flow direct numerical simulations of the Navier-Stokes equations, and measure the cumulative histogram of the maximum axial vorticity field over the pipe (turbulence intensity). In the domain where the turbulence intensity is not small, we observe that the histogram is well-approximated by the Gumbel extreme-value distribution. The smallest turbulence intensity in this domain roughly corresponds to the transition value between the locally stable turbulence and a meta-stable (edge) state. Studying the Re dependence of the fitting parameters in this distribution, we derive that the time scale of the transition between these two states increases double-exponentially as Re approaches its critical value. On the contrary, in smaller turbulence intensities below this domain, we observe that the distribution is not sensible to the change of Re. This means that the decay time of the meta-stable state (to the laminar state) is stochastic but Re-independent in average. Our observation suggests that the conjecture made by Goldenfeld et al. to derive the double-exponential increase of turbulence decay time is approximately satisfied in the range of Re we studied. We also discuss using another extreme-value distribution, Fr\'echet distribution, instead of the Gumbel distribution to approximate the histogram of the turbulence intensify, which reveals interesting perspectives., Comment: 18 pages, 10 figures

Colloids coupled to a bath of swimming cells generically display enhanced diffusion. This transport dynamics stems from a subtle interplay between the active and passive particles that still resists our understanding despite decades of intense research. Here, we tackle the root of the problem by providing a quantitative characterisation of the single scattering events between a colloid and a bacterium. Based on our experiments, we build a minimal model that quantitatively predicts the geometry of the scattering trajectories, and enhanced colloidal diffusion at long times. This quantitative confrontation between theory and experiments elucidates the microscopic origin of enhanced transport. Collisions are solely ruled by stochastic contact interactions responsible both for genuine anomalous diffusion at short times and enhanced diffusion at long times with no ballistic regime at any scale.

We analyse a first-order dynamical phase transition that takes place in the Fredrickson--Andersen (FA) model. We construct a two-dimensional spin system whose thermodynamic properties reproduce the dynamical large deviations of the FA model and we analyse this system numerically, comparing our results with finite-size scaling theory. This allows us to rationalise recent results for the FA model, including the exponential divergence of its susceptibility at phase coexistence. We also discuss a simple interfacial model that reproduces quantitatively the behaviour of the FA model at coexistence., Comment: 32 pages

Active fluids operate by constantly dissipating energy at the particle level to perform a directed motion, yielding dynamics and phases without any equilibrium equivalent. The emerging behaviors have been studied extensively, yet deciphering how local energy fluxes control the collective phenomena is still largely an open challenge. We provide generic relations between the activity-induced dissipation and the transport properties of an internal tracer. By exploiting a mapping between active fluctuations and disordered driving, our results reveal how the local dissipation, at the basis of self-propulsion, constrains internal transport by reducing the mobility and the diffusion of particles. Then, we employ techniques of large deviations to investigate how interactions are affected when varying dissipation. This leads us to shed light on a microscopic mechanism to promote clustering at low dissipation, and we also show the existence of collective motion at high dissipation. Overall, these results illustrate how tuning dissipation provides an alternative route to phase transitions in active fluids., Comment: 26 pages, 7 figures

An equivalence between non equilibrium steady states (NESS) driven by a time-independent force and stochastic pumps (SP) stirred by a time-varying conservative force is studied for general many-body diffusive systems. When the particle density and current of NESS are imitated by SP time-averaged counterparts, we prove that the entropy production rate in NESS is always greater than that of SP, provided that the conductivity of the particle current is concave as a function of the particle density. Searching for a SP protocol that saturates the entropy production bound reveals an unexpected connection with traffic waves, where a high density region propagates against the direction of the particle current., Comment: 9 pages, 3 figures

Turbulent flows in a thin layer can develop an inverse energy cascade leading to spectral condensation of energy when the layer height is smaller than a certain threshold. These spectral condensates take the form of large-scale vortices in physical space. Recently, evidence for bistability was found in this system close to the critical height: depending on the initial conditions, the flow is either in a condensate state with most of the energy in the two-dimensional (2-D) large-scale modes, or in a three-dimensional (3-D) flow state with most of the energy in the small-scale modes. This bistable regime is characterised by the statistical properties of random and rare transitions between these two locally stable states. Here, we examine these statistical properties in thin-layer turbulent flows, where the energy is injected by either stochastic or deterministic forcing. To this end, by using a large number of direct numerical simulations (DNS), we measure the decay time $\tau_d$ of the 2-D condensate to 3-D flow state and the build-up time $\tau_b$ of the 2-D condensate. We show that both of these times $\tau_d,\tau_b$ follow an exponential distribution with mean values increasing faster than exponentially as the layer height approaches the threshold. We further show that the dynamics of large-scale kinetic energy may be modeled by a stochastic Langevin equation. From time-series analysis of DNS data, we determine the effective potential that shows two minima corresponding to the 2-D and 3-D states when the layer height is close to the threshold.

The dynamics and structure of nonequilibrium liquids, driven by non-conservative forces which can be either external or internal, generically hold the signature of the net dissipation of energy in the thermostat. Yet, disentangling precisely how dissipation changes collective effects remains challenging in many-body systems due to the complex interplay between driving and particle interactions. First, we combine explicit coarse-graining and stochastic calculus to obtain simple relations between diffusion, density correlations and dissipation in nonequilibrium liquids. Based on these results, we consider large-deviation biased ensembles where trajectories mimic the effect of an external drive. The choice of the biasing function is informed by the connection between dissipation and structure derived in the first part. Using analytical and computational techniques, we show that biasing trajectories effectively renormalizes interactions in a controlled manner, thus providing intuition on how driving forces can lead to spatial organization and collective dynamics. Altogether, our results show how tuning dissipation provides a route to alter the structure and dynamics of liquids and soft materials., Comment: 21 pages, 7 figures

Universality, where microscopic details become irrelevant, takes place in thermodynamic phase transitions. The universality is captured by a singular scaling function of the thermodynamic variables, where the scaling exponents are determined by symmetries and dimensionality only. Universality can persist even for non-equilibrium phase transitions. It implies that a hydrodynamic approach can capture the singular universal scaling function, even far from equilibrium. In particular, we show these results for phase transitions in the large deviation function of the current in diffusive systems with particle-hole symmetry. For such systems, we find the scaling exponents of the universal function and show they are independent of microscopic details as well as boundary conditions., Comment: 11 pages, 3 figures

Active work measures how far the local self-forcing of active particles translates into real motion. Using Population Monte Carlo methods, we investigate large deviations in the active work for repulsive active Brownian disks. Minimizing the active work generically results in dynamical arrest; in contrast, despite the lack of aligning interactions, trajectories of high active work correspond to a collectively moving, aligned state. We use heuristic and analytic arguments to explain the origin of dynamical phase transitions separating the arrested, typical, and aligned regimes., Comment: 14 pages, 9 figures

The fluctuations of turbulence intensity in a pipe flow around the critical Reynolds number is difficult to study but important because they are related to turbulent-laminar transitions. We here propose a rare-event sampling method to study such fluctuations in order to measure the time-scale of the transition efficiently. The method is composed of two parts: (i) the measurement of typical fluctuations (the bulk part of an accumulative probability function) and (ii) the measurement of rare fluctuations (the tail part of the probability function) by employing dynamics where a feedback control of the Reynolds number is implemented. We apply this method to a chaotic model of turbulent puffs proposed by Barkley and confirm that the time-scale of turbulence decay increases super-exponentially even for high Reynolds numbers up to Re = 2500, where getting enough statistics by brute-force calculations is difficult. The method uses a simple procedure of changing Reynolds number that can be applied even to experiments., Comment: 12 pages, 9 figures

We analyze large deviations of the time-averaged activity in the one dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multi-canonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts., Comment: 20 pages, 10 figures

Rare trajectories of stochastic systems are important to understand -- because of their potential impact. However, their properties are by definition difficult to sample directly. Population dynamics provides a numerical tool allowing their study, by means of simulating a large number of copies of the system, which are subjected to selection rules that favor the rare trajectories of interest. Such algorithms are plagued by finite simulation time- and finite population size- effects that can render their use delicate. In this paper, we present a numerical approach which uses the finite-time and finite-size scalings of estimators of the large deviation functions associated to the distribution of rare trajectories. The method we propose allows one to extract the infinite-time and infinite-size limit of these estimators which -- as shown on the contact process -- provides a significant improvement of the large deviation functions estimators compared to the the standard one., Comment: 12 pages, 10 figures. Second part of pair of companion papers, following Part I arXiv:1607.04752

The Giardin\`a-Kurchan-Peliti algorithm is a numerical procedure that uses population dynamics in order to calculate large deviation functions associated to the distribution of time-averaged observables. To study the numerical errors of this algorithm, we explicitly devise a stochastic birth-death process that describes the time evolution of the population probability. From this formulation, we derive that systematic errors of the algorithm decrease proportionally to the inverse of the population size. Based on this observation, we propose a simple interpolation technique for the better estimation of large deviation functions. The approach we present is detailed explicitly in a two-state model., Comment: 13 pages, 1 figure. First part of pair of companion papers, Part II being arXiv:1607.08804

We discuss the Giardin\`a-Kurchan-Peliti population dynamics method for evaluating large deviations of time averaged quantities in Markov processes [Phys. Rev. Lett. \textbf{96}, 120603 (2006)]. This method exhibits systematic errors which can be large in some circumstances, particularly for systems with weak noise, with many degrees of freedom, or close to dynamical phase transitions. We show how these errors can be mitigated by introducing control forces within the algorithm. These forces are determined by an iteration-and-feedback scheme, inspired by multicanonical methods in equilibrium sampling. We demonstrate substantially improved results in a simple model and we discuss potential applications to more complex systems., Comment: 18 pages, 9 figures

On the example of a mean-field Fredrickson-Andersen kinetically constrained model, we focus on the known property that equilibrium dynamics take place at a first-order dynamical phase transition point in the space of time-realizations. We investigate the finite-size properties of this first order transition. By discussing and exploiting a mapping of the classical dynamical transition -an argued glassiness signature- to a first-order quantum transition, we show that the quantum analogy can be exploited to extract finite-size properties, which in many respects are similar to those in genuine mean-field quantum systems with a first-order transition. We fully characterize the finite-size properties of the order parameter across the first order transition.

We propose a computational method for large deviation statistics of time-averaged quantities in general Markov processes. In our proposed method, we repeat a response measurement against external forces, where the forces are determined by the previous measurement as feedback. Consequently, we obtain a set of stationary states corresponding to an exponential family of distributions, each of which shows rare events in the original system as the typical behavior. As a demonstration of our method, we study large deviation statistics of one-dimensional lattice gas models., Comment: 8 pages, 5 figures

We study a Brownian particle on a moving periodic potential. We focus on the statistical properties of the work done by the potential and the heat dissipated by the particle. When the period and the depth of the potential are both large, by using a boundary layer analysis, we calculate a cumulant generating function and a biased distribution function. The result allows us to understand a Zon-Cohen singularity for an extended fluctuation theorem from a view point of rare trajectories characterized by a negative inverse temperature of the biased distribution function., Comment: 11 pages, 3 figures

The cumulant generating function of time-averaged current is studied from an operational viewpoint. Specifically, for interacting Brownian particles under non-equilibrium conditions, we show that the first derivative of the cumulant generating function is equal to the expectation value of the current in a modified system with an extra force added, where the modified system is characterized by a variational principle. The formula reminds us of Einstein's fluctuation theory in equilibrium statistical mechanics. Furthermore, since the formula leads to the fluctuation-dissipation relation when the linear response regime is focused on, it is regarded as an extension of the linear response theory to that valid beyond the linear response regime. The formula is also related to previously known theories such as the Donsker-Varadhan theory, the additivity principle, and the least dissipation principle, but it is not derived from them. Examples of its application are presented for a driven Brownian particle on a ring subject to a periodic potential., Comment: 21 pages, 2 figures; In ver. 2, a new appendix has been added

For Brownian motion of a single particle subject to a tilted periodic potential on a ring, we propose a formula for experimentally determining the cumulant generating function of time-averaged current without measurements of current fluctuations. We first derive this formula phenomenologically on the basis of two key relations: a fluctuation relation associated with Onsager's principle of the least energy dissipation in a sufficiently local region and an additivity relation by which spatially inhomogeneous fluctuations can be properly considered. We then derive the formula without any phenomenological assumptions. We also demonstrate its practical advantage by numerical experiments., Comment: 4 pages, 1 figure; In ver. 2, the organization of the paper has been revised. In ver. 3, substantial revisions have been done