Large Deviation Formalism Research Articles

Large deviations in statistics of the local time and occupation time for a run and tumble particle.

We investigate the statistics of the local time T=∫_{0}^{T}δ(x(t))dt that a run and tumble particle (RTP) x(t) in one dimension spends at the origin, with or without an external drift. By relating the local time to the number of times the RTP crosses the origin, we find that the local time distribution P(T) satisfies the large deviation principle P(T)∼e^{-TI(T/T)} in the large observation time limit T→∞. Remarkably, we find that in the presence of drift the rate function I(ρ) is nonanalytic: we interpret its singularity as a dynamical phase transition of first order. We then extend these results by studying the statistics of the amount of time R that the RTP spends inside a finite interval (i.e., the occupation time), with qualitatively similar results. In particular, this yields the long-time decay rate of the probability P(R=T) that the particle does not exit the interval up to time T. We find that the conditional end-point distribution exhibits an interesting change of behavior from unimodal to bimodal as a function of the size of the interval. To study the occupation time statistics, we extend the Donsker-Varadhan large-deviation formalism to the case of RTPs, for general dynamical observables and possibly in the presence of an external potential.

Anomalous Dynamical Phase Transitions of the Ising Model Studied by the Nonequilibrium Ensemble Method: Freezing-by-Heating and Permanent Liquid.

We investigate the nonequilibrium, dynamical phase transition behavior of the one-dimensional Ising model, using the trajectory ensemble approach in the context of the large deviation theory. We introduce a double-biased ensemble, named the s, g-ensemble, based on nonequilibrium steady-state trajectories. The ensemble invokes the time-integrated, trajectory energy as an order parameter, coupled to its conjugate g-field in addition to the dynamical activity and its conjugate field s in the trajectory space. Using the dynamical free energy obtained from the large deviation formalism, we explore the rich behaviors of the dynamical phase transition of the 1D Ising model in the (s, g, T) parameter space, with T being temperature. Among other features, we discover that novel, anomalous dynamical phase transitions are possible due to the decoupling between the dynamical activity and trajectory energy under specific conditions. In particular, we observe that the system exhibits a freezing-by-heating phenomenon as the dynamical activity decreases with temperature under a specific condition. We also find a permanent liquid phase when the equilibrium temperature and the nonequilibrium g-field are exactly balanced against each other. Our results provide a useful tool for exploring the dynamical phase transition phenomena to be investigated in various systems.

Large Deviations at Level 2.5 for Markovian Open Quantum Systems: Quantum Jumps and Quantum State Diffusion

We consider quantum stochastic processes and discuss a level 2.5 large deviation formalism providing an explicit and complete characterisation of fluctuations of time-averaged quantities, in the large-time limit. We analyse two classes of quantum stochastic dynamics, within this framework. The first class consists of the quantum jump trajectories related to photon detection; the second is quantum state diffusion related to homodyne detection. For both processes, we present the level 2.5 functional starting from the corresponding quantum stochastic Schrödinger equation and we discuss connections of these functionals to optimal control theory.

Current statistics and depinning transition for a one-dimensional Langevin process in the weak-noise limit

We consider a particle with a Langevin dynamics driven by a uniform non-conservative force, in a one-dimensional potential with periodic boundary conditions. We are interested in the properties of the system for atypical values of the time-integral of a generalized particle current. To study these, we bias the dynamics, at trajectory level, by a parameter conjugated to the current, within the large-deviation formalism. We investigate, in the weak-noise limit, the phase diagram spanned by the physical driving force and the parameter defining the biased process. We focus in particular on the depinning transition in this two-dimensional phase diagram. In the absence of trajectory bias, the depinning transition as a function of the force is characterized by the standard exponent . We show that for any non-zero bias, the depinning transition is characterized by an inverse logarithmic behavior as a function of either the bias or the force, close to the critical lines. We also report a scaling exponent for the current when considering the depinning transition in terms of the bias, fixing the non-conservative force to its critical value in the absence of bias. Then, focusing on the time-integrated particle current, we study the thermal rounding effects in the zero-current phase when the tilted potential exhibits a local minimum. We derive in this case the Arrhenius scaling, in the small noise limit, of both the particle current and the scaled cumulant generating function. This derivation of the Arrhenius scaling relies on the determination of the left eigenvector of the biased Fokker–Planck operator, to exponential order in the low-noise limit. An effective Poissonian statistics of the integrated current emerges in this limit.

Witnessing nonclassicality through large deviations in quantum optics

This work formulates a nonclassicality criterion for radiation emitted from an open quantum system. The approach is based on the time-integrated photocurrent which is analyzed through the Large Deviation formalism for quantum trajectories.

Giant disparity and a dynamical phase transition in large deviations of the time-averaged size of stochastic populations.

We study large deviations of the time-averaged size of stochastic populations described by a continuous-time Markov jump process. When the expected population size N in the steady state is large, the large deviation function (LDF) of the time-averaged population size can be evaluated by using a Wentzel-Kramers-Brillouin (WKB) method, applied directly to the master equation for the Markov process. For a class of models that we identify, the direct WKB method predicts a giant disparity between the probabilities of observing an unusually small and an unusually large values of the time-averaged population size. The disparity results from a qualitative change in the "optimal" trajectory of the underlying classical mechanics problem. The direct WKB method also predicts, in the limit of N→∞, a singularity of the LDF, which can be interpreted as a second-order dynamical phase transition. The transition is smoothed at finite N, but the giant disparity remains. The smoothing effect is captured by the van-Kampen system size expansion of the exact master equation near the attracting fixed point of the underlying deterministic model. We describe the giant disparity at finite N by developing a different variant of WKB method, which is applied in conjunction with the Donsker-Varadhan large-deviation formalism and involves subleading-order calculations in 1/N.

Occupation-time statistics of a gas of interacting diffusing particles.

The time that a diffusing particle spends in a certain region of space is known as the occupation time, or the residence time. Recently, the joint occupation-time statistics of an ensemble of noninteracting particles was addressed using the single-particle statistics. Here we employ the macroscopic fluctuation theory (MFT) to study the occupation-time statistics of many interacting particles. We find that interactions can significantly change the statistics and, in some models, even cause a singularity of the large-deviation function describing these statistics. This singularity can be interpreted as a dynamical phase transition. We also point out to a close relation between the MFT description of the occupation-time statistics of noninteracting particles and the level 2 large deviation formalism which describes the occupation-time statistics of a single particle.

Accuracy of classical conductivity theory at atomic scales for free fermions in disordered media

The growing need for smaller electronic components has recently sparked the interest in the breakdown of the classical conductivity theory near the atomic scale, at which quantum effects should dominate. In 2012, experimental measurements of electric resistance of nanowires in Si doped with phosphorus atoms demonstrate that quantum effects on charge transport almost disappear for nanowires of lengths larger than a few nanometers, even at very low temperature (4.2K). We mathematically prove, for non-interacting lattice fermions with disorder, that quantum uncertainty of microscopic electric current density around their (classical) macroscopic values is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. This is in accordance with the above experimental observation. Disorder is modeled by a random external potential along with random, complex-valued, hopping amplitudes. The celebrated tight-binding Anderson model is one particular example of the general case considered here. Our mathematical analysis is based on Combes–Thomas estimates, the Akcoglu–Krengel ergodic theorem, and the large deviation formalism, in particular the Gärtner–Ellis theorem.

Efficiency fluctuations in steady-state machines

We study the statistics of the efficiency in a class of isothermal cyclic machines with realistic coupling between the internal degrees of freedom. We derive, under fairly general assumptions, the probability distribution function for the efficiency. We find that the macroscopic efficiency is always equal to the most likely efficiency, and it lies in an interval whose boundaries are universal as they only depend on the input and output thermodynamic forces, and not on the details of the machine. The machine achieves the upper boundary of such an interval only in the limit of tight coupling. Furthermore, we find that the tight coupling limit is a necessary, yet not sufficient, condition for the engine to perform close to the reversible efficiency. The reversible efficiency is the least likely regardless of the coupling strength, in agreement with previous studies. By using a large deviation formalism for the energy currents we derive a fluctuation relation for the efficiency which holds for any number of internal degrees of freedom in the system.

Catching and reversing quantum jumps and thermodynamics of quantum trajectories

A recent experiment by Minev et. al [arXiv:1803.00545] demonstrated that in a dissipative (artificial) 3-level atom with strongly intermittent dynamics it is possible to "catch and reverse" a quantum jump "mid-flight": by the conditional application of a unitary perturbation after a fixed time with no jumps, the system was prevented from getting shelved in the dark state, thus removing the intermittency from the dynamics. Here we offer an interpretation of this phenomenon in terms of the dynamical large deviation formalism for open quantum dynamics. In this approach, intermittency is seen as the first-order coexistence of active and inactive dynamical phases. Dark periods are thus like space-time bubbles of the inactive phase in the active one. Here we consider a controlled dynamics via the (single - as in the experiment - or multiple) application of a unitary control pulse during no-jump periods. By considering the large deviation statistics of the emissions, we show that appropriate choice of the control allows to stabilise a desired dynamical phase and remove the intermittency. In the thermodynamic analogy, the effect of the control is to prick bubbles thus preventing the fluctuations that manifest phase coexistence. We discuss similar controlled dynamics in broader settings.

Large deviations in the presence of cooperativity and slow dynamics.

We study simple models of intermittency, involving switching between two states, within the dynamical large-deviation formalism. Singularities appear in the formalism when switching is cooperative or when its basic time scale diverges. In the first case the unbiased trajectory distribution undergoes a symmetry breaking, leading to a change in shape of the large-deviation rate function for a particular dynamical observable. In the second case the symmetry of the unbiased trajectory distribution remains unbroken. Comparison of these models suggests that singularities of the dynamical large-deviation formalism can signal the dynamical equivalent of an equilibrium phase transition but do not necessarily do so.

Rare behavior of growth processes via umbrella sampling of trajectories.

We compute probability distributions of trajectory observables for reversible and irreversible growth processes. These results reveal a correspondence between reversible and irreversible processes, at particular points in parameter space, in terms of their typical and atypical trajectories. Thus key features of growth processes can be insensitive to the precise form of the rate constants used to generate them, recalling the insensitivity to microscopic details of certain equilibrium behavior. We obtained these results using a sampling method, inspired by the "s-ensemble" large-deviation formalism, that amounts to umbrella sampling in trajectory space. The method is a simple variant of existing approaches, and applies to ensembles of trajectories controlled by the total number of events. It can be used to determine large-deviation rate functions for trajectory observables in or out of equilibrium.

Rare events statistics of random walks on networks: localisation and other dynamical phase transitions

Rare event statistics for random walks on complex networks are investigated using the large deviation formalism. Within this formalism, rare events are realised as typical events in a suitably deformed path-ensemble, and their statistics can be studied in terms of spectral properties of a deformed Markov transition matrix. We observe two different types of phase transition in such systems: (i) rare events which are singled out for sufficiently large values of the deformation parameter may correspond to localised modes of the deformed transition matrix; (ii) ‘mode-switching transitions’ may occur as the deformation parameter is varied. Details depend on the nature of the observable for which the rare event statistics is studied, as well as on the underlying graph ensemble. In the present paper we report results on rare events statistics for path averages of random walks in Erdős–Rényi and scale free networks. Large deviation rate functions and localisation properties are studied numerically. For observables of the type considered here, we also derive an analytical approximation for the Legendre transform of the large deviation rate function, which is valid in the large connectivity limit. It is found to agree well with simulations.

Analysis of the correlation dimension for inertial particles

We obtain an implicit equation for the correlation dimension which describes clustering of inertial particles in a complex flow onto a fractal measure. Our general equation involves a propagator of a nonlinear stochastic process in which the velocity gradient of the fluid appears as additive noise. When the long-time limit of the propagator is considered our equation reduces to an existing large-deviation formalism from which it is difficult to extract concrete results. In the short-time limit, however, our equation reduces to a solvability condition on a partial differential equation. In the case where the inertial particles are much denser than the fluid, we show how this approach leads to a perturbative expansion of the correlation dimension, for which the coefficients can be obtained exactly and in principle to any order. We derive the perturbation series for the correlation dimension of inertial particles suspended in three-dimensional spatially smooth random flows with white-noise time correlations, obtaining the first 33 non-zero coefficients exactly.

Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence

AbstractSmall droplets in turbulent flows can undergo highly variable deformations and orientational dynamics. For neutrally buoyant droplets smaller than the Kolmogorov scale, the dominant effects from the surrounding turbulent flow arise through Lagrangian time histories of the velocity gradient tensor. Here we study the evolution of representative droplets using a model that includes rotation and stretching effects from the surrounding fluid, and restoration effects from surface tension including a constant droplet volume constraint, while assuming that the droplets maintain an ellipsoidal shape. The model is combined with Lagrangian time histories of the velocity gradient tensor extracted from direct numerical simulations (DNS) of turbulence to obtain simulated droplet evolutions. These are used to characterize the size, shape and orientation statistics of small droplets in turbulence. A critical capillary number is identified associated with unbounded growth of one or two of the droplet’s semi-axes. Exploiting analogies with dynamics of polymers in turbulence, the critical capillary number can be predicted based on the large deviation theory for the largest finite-time Lyapunov exponent quantifying the chaotic separation of particle trajectories. Also, for subcritical capillary numbers near the critical value, the theory enables predictions of the slope of the power-law tails of droplet size distributions in turbulence. For cases when the viscosities of droplet and outer fluid differ in a way that enables vorticity to decorrelate the shape from the straining directions, the large deviation formalism based on the stretching properties of the velocity gradient tensor loses validity and its predictions fail. Even considering the limitations of the assumed ellipsoidal droplet shape, the results highlight the complex coupling between droplet deformation, orientation and the local fluid velocity gradient tensor to be expected when small viscous drops interact with turbulent flows. The results also underscore the usefulness of large deviation theory to model these highly complex couplings and fluctuations in turbulence that result from time integrated effects of fluid deformations.

Large-deviation statistics of a diffusive quantum spin chain and the additivity principle.

Using the large-deviation formalism, we study the statistics of current fluctuations in a diffusive nonequilibrium quantum spin chain. The boundary-driven XX chain with dephasing consists of a coherent bulk hopping and a local dissipative dephasing. We analytically calculate the exact expression for the second current moment in a system of any length and then numerically demonstrate that in the thermodynamic limit, higher-order cumulants and the large-deviation function can be calculated using the additivity principle or macroscopic hydrodynamic theory. This shows that the additivity principle can also hold in systems that are not purely stochastic, and can in particular be valid in quantum systems. We also show that in large systems, the current fluctuations are the same as in the classical symmetric simple exclusion process.

On the existence of a glass transition in a random energy model

We consider a generalized version of the random energy model in which the energy of each configuration is given by the sum of N independent contributions (local energies) with finite variances but otherwise arbitrary statistics. Using the large deviation formalism, we find that the glass transition generically exists when local energies have a smooth distribution. In contrast, if the distribution of the local energies has a Dirac mass at the minimal energy (e.g., if local energies take discrete values), the glass transition ceases to exist if the number of energy levels grows sufficiently fast with system size. This shows that the statistical independence of energy levels does not imply the existence of a glass transition.

Baire generic histograms of wavelet coefficients and large deviation formalism in Besov and Sobolev spaces

Histograms of wavelet coefficients are expressed in terms of the wavelet profile and the wavelet density. The large deviation multifractal formalism states that if a function f has a minimal uniform Hölder regularity then its Hölder spectrum is equal to the wavelet density. The purpose of this paper is twofold. Firstly, we compute generically (in the sense of Baire's categories) these histograms in Besov B p s , q ( T ) and L p , s ( T ) spaces, where T is the torus R d / Z d (resp. in the Baire's vector space V = ⋂ ε > 0 , p > 0 B p s ( 1 p ) − ε p , p where s : q ↦ s ( q ) is a C 1 and concave function on R + satisfying 0 ⩽ s ′ ⩽ d and s ( 0 ) > 0 ). Secondly, as an application, we deduce some extra generic properties for the histograms in these spaces, and study the generic validity of the large deviation multifractal formalism in Besov and L p , s spaces for s > d / p (resp. in the above space V).

Universal finite-sample effect on the perturbation growth in chaotic dynamical systems

The finite-sample effect on the growth of moments of the perturbation observed in numerical simulations of chaotic dynamical systems is studied. To numerically estimate the moments, only a limited number of sample trajectories can be utilized, and therefore the moments exhibit pure exponential growth only initially, and give way to relaxed growth thereafter. Such transition is a consequence of the unobservability of rare events in finite sample sets. Using the large-deviation formalism for chaotic time series, we estimate the relaxation time and derive the post-relaxation growth law. We demonstrate that even after the relaxation, each moment still obeys a universal growth law of different type, which reflects physical information on the statistics of chaotic expansion rates.

Phenomenological framework for fluctuations around steady state

A phenomenological framework to describe fluctuations around steady states is formulated. The framework is illustrated for a magnetic system maintained at a nonequilibrium steady state by an oscillating magnetic field, modeled at the mesoscopic level by a Langevin dynamics. The large deviation formalism along the time axis is employed to construct a generalized entropy to describe the fluctuations in the steady state for time averaged observables (state variables). We propose a phenomenological postulate that the fluctuations about the steady state can be obtained from the response of the state variables to 'thermodynamic conjugate forces' (fluctuation-response relation), as in the ordinary thermodynamic fluctuation theory. An experimentally realizable method to study the linear response about the steady state against state variable perturbations is proposed, and illustrated for the driven magnetic system. The notion of a proper state space to describe nonequilibrium steady states is discussed, and to this end, we introduce a dissipation variable to extend the state space for our model system. In the extended state space, we elucidate and study various stability and Maxwell-type relations that follow from our local phenomenological (thermodynamic) framework. Some relevant issues regarding a more general thermodynamic framework are also discussed.