We study the time until first occurrence, the first-passage time, of rare density fluctuations in diffusive systems. We approach the problem using a model consisting of many independent random walkers on a lattice. The existence of spatial correlations makes this problem analytically intractable. However, for a mean-field approximation in which the walkers can jump anywhere in the system, we obtain a simple asymptotic form for the mean first-passage time to have a given number k of particles at a distinguished site. We show numerically, and argue heuristically, that for large enough k, the mean-field results give a good approximation for first-passage times for systems with nearest-neighbour dynamics, especially for two and higher spatial dimensions. Finally, we show how the results change when density fluctuations anywhere in the system, rather than at a specific distinguished site, are considered.
University of California, Berkeley（加州大学伯克利分校）；University of Nottingham（诺丁汉大学）；Kavli Energy NanoScience Institute（卡夫利能源纳米科学研究所）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We present a method to probe rare molecular dynamics trajectories directly using reinforcement learning. We consider trajectories that are conditioned to transition between regions of configuration space in finite time, like those relevant in the study of reactive events, as well as trajectories exhibiting rare fluctuations of time-integrated quantities in the long time limit, like those relevant in the calculation of large deviation functions. In both cases, reinforcement learning techniques are used to optimize an added force that minimizes the Kullback-Leibler divergence between the conditi…
We consider fluctuations of the dissipated energy in nonlinear driven diffusive systems subject to bulk dissipation and boundary driving. With this aim, we extend the recently-introduced macroscopic fluctuation theory to nonlinear driven dissipative media, starting from the fluctuating hydrodynamic equations describing the system mesoscopic evolution. Interestingly, the action associated to a path in mesoscopic phase-space, from which large-deviation functions for macroscopic observables can be derived, has the same simple form as in non-dissipative systems. This is a consequence of the quasi-…
The probability of trajectories of weakly diffusive processes to remain in the tubular neighbourhood of a smooth path is given by the Freidlin-Wentzell-Graham theory of large deviations. The most probable path between two states (the instanton) and the leading term in the logarithm of the process transition density (the quasipotential) are obtained from the minimum of the Freidlin-Wentzell action functional. Here we present a Ritz method that searches for the minimum in a space of paths constructed from a global basis of Chebyshev polynomials. The action is reduced, thereby, to a multivariate …
The idea of rare event sampling is applied to the estimation of the performance of error-correcting codes. The essence of the idea is importance sampling of the pattern of noises in the channel by Multicanonical Monte Carlo, which enables efficient estimation of tails of the distribution of bit error rate. The idea is successfully tested with a convolutional code.
We study the distribution of the minimum free energy (MFE) for the Turner model of pseudoknot free RNA secondary structures over ensembles of random RNA sequences. In particular, we are interested in those rare and intermediate events of unexpected low MFEs. Generalized ensemble Markov-chain Monte Carlo methods allow us to explore the rare-event tail of the MFE distribution down to probabilities like $10^{-70}$ and to study the relationship between the sequence entropy and structural properties for sequence ensembles with fixed MFEs. Entropic and structural properties of those ensembles are co…
We develop a recently proposed importance-sampling Monte Carlo algorithm for sampling rare events and quenched variables in random disordered systems. We apply it to a two dimensional bond-diluted Ising model and study the Griffiths singularity which is considered to be due to the existence of rare large clusters. It is found that the distribution of the inverse susceptibility has an exponential tail down to the origin which is considered the consequence of the Griffiths singularity.
Rare event statistics for random walks on complex networks are investigated using the large deviations formalism. Within this formalism, rare events are realized 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 {\em localized\/} modes of the deformed transition matrix, (ii) "mode-switching transitions…
A method based on multicanonical Monte Carlo is applied to the calculation of large deviations in the largest eigenvalue of random matrices. The method is successfully tested with the Gaussian orthogonal ensemble (GOE), sparse random matrices, and matrices whose components are subject to uniform density. Specifically, the probability that all eigenvalues of a matrix are negative is estimated in these cases down to the values of $\sim 10^{-200}$, a region where naive random sampling is ineffective. The method can be applied to any ensemble of matrices and used for sampling rare events character…
Sequence heterogeneity broadens the binding transition of a polymer onto a linear or planar substrate. This effect is analyzed in a real-space renormalization group scheme designed to capture the statistics of rare events. In the strongly disordered regime, binding initiates at an exponentially rare set of ``good contacts''. Renormalization of the contact potential yields a Kosterlitz-Thouless type transition in any dimension. This and other predictions are confirmed by extensive numerical simulations of a directed polymer interacting with a columnar defect.
Randomized search algorithms for hard combinatorial problems exhibit a large variability of performances. We study the different types of rare events which occur in such out-of-equilibrium stochastic processes and we show how they cooperate in determining the final distribution of running times. As a byproduct of our analysis we show how search algorithms are optimized by random restarts.
A sampling procedure for the transition matrix Monte Carlo method is introduced that generates the density of states function over a wide parameter range with minimal coding effort.
Percolation theory characterizing the robustness of a network has applications ranging from biology, to epidemic spreading, and complex infrastructures. Percolation theory, however, only concern the typical response of a infinite network to random damage of its nodes while in real finite networks, fluctuations are observable. Consequently for finite networks there is an urgent need to evaluate the risk of collapse in response to rare configurations of the initial damage. Here we build a large deviation theory of percolation characterizing the response of a sparse network to rare events. This g…
Many rare event transitions involve multiple collective variables (CVs) and the most appropriate combination of CVs is generally unknown a priori. We thus introduce a new method, contour forward flux sampling (cFFS), to study rare events with multiple CVs simultaneously. cFFS places nonlinear interfaces on-the-fly from the collective progress of the simulations, without any prior knowledge of the energy landscape or appropriate combination of CVs. We demonstrate cFFS on analytical potential energy surfaces and a conformational change in alanine dipeptide.
The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. According to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Lévy walks, a class of stochastic processes with power law distributed step durations, which model complex microscopic dynamics in the single stretch. We derive the…
We develop an efficient method to calculate probabilities of large deviations from the typical behavior (rare events) in reaction--diffusion systems. The method is based on a semiclassical treatment of underlying "quantum" Hamiltonian, encoding the system's evolution. To this end we formulate corresponding canonical dynamical system and investigate its phase portrait. The method is presented for a number of pedagogical examples.
We describe a new approach to the rare-event Monte Carlo sampling problem. This technique utilizes a symmetrization strategy to create probability distributions that are more highly connected and thus more easily sampled than their original, potentially sparse counterparts. After discussing the formal outline of the approach and devising techniques for its practical implementation, we illustrate the utility of the technique with a series of numerical applications to Lennard-Jones clusters of varying complexity and rare-event character.
Particle hopping is a common feature in heterogeneous media. We explore such motion by using the widely applicable formalism of the continuous time random walk and focus on the statistics of rare events. Numerous experiments have shown that the decay of the positional probability density function P (X, t), describing the statistics of rare events, exhibits universal exponential decay. We show that such universality ceases to exist once the threshold of exponential distribution of particle hops is crossed. While the mean hop is not diverging and can attain a finite value; the transition itself …
We study three-dimensional Dirac fermions with weak finite-range scalar potential disorder. In the clean system, the density of states vanishes quadratically at the Dirac point. Disorder is known to be perturbatively irrelevant, and previous theoretical work has assumed that the Dirac semimetal phase, characterized by a vanishing density of states, survives at weak disorder, with a finite disorder phase transition to a diffusive metal with a non-vanishing density of states. In this paper we show that nonperturbative effects from rare regions, which are missed by conventional disorder-averaged …
The graph transformation approach is a recently proposed method for computing mean first passage times, rates, and committor probabilities for kinetic transition networks. Here we compare the performance to existing linear algebra methods, focusing on large, sparse networks. We show that graph transformation provides a much more robust framework, succeeding when numerical precision issues cause the other methods to fail completely. These are precisely the situations that correspond to rare event dynamics for which the graph transformation was introduced.
Rare events are ubiquitous in many different fields, yet they are notoriously difficult to simulate because few, if any, events are observed in a conventiona l simulation run. Over the past several decades, specialised simulation methods have been developed to overcome this problem. We review one recently-developed class of such methods, known as Forward Flux Sampling. Forward Flux Sampling uses a series of interfaces between the initial and final states to calculate rate constants and generate transition paths, for rare events in equilibrium or nonequilibrium systems with stochastic dynamics.…
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 t…
Due to intermittency and conservation, the Abelian sandpile in 2D obeys multifractal, rather than finite size scaling. In the thermodynamic limit, a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution. Thus, rare events shape most of the scaling pattern and preserve a meaning for effective exponents, which can be determined on the basis of numerical and exact results.
We introduce a new Forward-Flux Sampling in Time (FFST) algorithm to efficiently measure transition times in rare-event processes in non-equilibrium systems, and apply it to study the first-order (discontinuous) kinetic transition in the Ziff-Gulari-Barshad model of catalytic surface reaction. The average time for the transition to take place, as well as both the spinodal and transition points, are clearly found by this method.
We discuss the combined effects of overdamped motion in a quenched random potential and diffusion, in one dimension, in the limit where the diffusion coefficient is small. Our analysis considers the statistics of the mean first-passage time $T(x)$ to reach position $x$, arising from different realisations of the random potential: specifically, we contrast the median $\bar T(x)$, which is an informative description of the typical course of the dispersion, with the expectation value $\langle T(x)\rangle$, which is dominated by rare events where there is an exceptionally high barrier to diffusion…
We briefly review simulation schemes for the investigation of rare transitions and we resume the recently introduced Transition Interface Sampling, a method in which the computation of rate constants is recast into the computation of fluxes through interfaces dividing the reactant and product state.
There is great interest in predicting rare and extreme events in complex systems, and in particular, understanding the role of network topology in facilitating such events. In this work, we show that degree dispersion -- the fact that the number of local connections in networks varies broadly -- increases the probability of large, rare fluctuations in population networks generically. We perform explicit calculations for two canonical and distinct classes of rare events: network extinction and switching. When the distance to threshold is held constant, and hence stochastic effects are fairly co…
Rare events are processes that occur upon the emergence of unlikely fluctuations. Unlike what their name suggests, rare events are fairly ubiquitous in nature, as the occurrence of many structural transformations in biology and material sciences is predicated upon crossing large free energy barriers. Probing the kinetics and uncovering the molecular mechanisms of possible barrier crossings in a system is critical to predicting and controlling its structural and functional properties. Due to their activated nature, however, rare events are exceptionally difficult to study using conventional exp…
We calculate the probabilities to find systems of reacting particles in states which largely deviate from typical behavior. The rare event statistics is obtained from the master equation which describes the dynamics of the probability distribution of the particle number. We transform the master equation by means of a generating function into a time-dependent "Schrödinger equation". Its solution is provided by a separation ansatz and an approximation for the stationary part which is of Wentzel-Kramers-Brillouin (WKB) type employing a small parameter. The solutions of the "classical" equations o…
We classify the rare events of structured, memoryful stochastic processes and use this to analyze sequential and parallel generators for these events. Given a stochastic process, we introduce a method to construct a new process whose typical realizations are a given process' rare events. This leads to an expression for the minimum memory required to generate rare events. We then show that the recently discovered classical-quantum ambiguity of simplicity also occurs when comparing the structure of process fluctuations.
We consider super-diffusive Lévy walks in $d \geqslant 2$ dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the $d-$dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of central limit theorem (CLT) for random walks with finite dispersio…
The survival of natural populations may be greatly affected by environmental conditions that vary in space and time. We look at a population residing in two locations (patches) coupled by migration, in which the local conditions fluctuate in time. We report on two findings. First, we find that unlike rare events in many other systems, here the histories leading to a rare extinction event are not dominated by a single path. We develop the appropriate framework, which turns out to be a hybrid of the standard saddle-point method, and the Donsker-Varadhan formalism which treats rare events of atyp…
In the boundary driven symmetric simple exclusion process consider an open set $\ms O$ of density profiles which does not contain the stationary density profile. We prove that the first time the empirical measure visits the set $\ms O$ converges to an exponential distribution.
We consider control of reaction and population systems by deterministically imposed transitions between the states with different numbers of particles or individuals. Even where the imposed transitions are significantly less frequent than spontaneous transitions, they can exponentially strongly modify the rates of rare events, including switching between metastable states or population extinction. We also study optimal control of rare events, and specifically, optimal control of disease extinction for a limited vaccine supply. A comparison is made with control of rare events by modulating the …
In this paper we address the use of rare event computation techniques to estimate small over-threshold probabilities of observables in determin-istic dynamical systems. We demonstrate that the genealogical particle analysis algorithms can be successfully applied to a toy model of atmospheric dynamics, the Lorenz '96 model. We furthermore use the Ornstein-Uhlenbeck system to illustrate a number of implementation issues. We also show how a time-dependent objective function based on the fluctuation path to a high threshold can greatly improve the performance of the estimator compared to a fixed-i…
We develop a spectral formulation and a stationary WKB approximation for calculating the probabilities of rare events (large deviations from the mean) in systems of reacting particles with infinite-range interaction, describable by a master equation. We compare the stationary WKB approximation with a recent time-dependent semiclassical approximation developed, for the same class of problems, by Elgart and Kamenev. As a benchmark we use an exactly solvable problem of the binary annihilation reaction 2A -> 0.
Transition path sampling is a method for estimating the rates of rare events in molecular systems based on the gradual transformation of a path distribution containing a small fraction of reactive trajectories into a biased distribution in which these rare trajectories have become frequent. Then, a multistate reweighting scheme is implemented to postprocess data collected from the staged simulations. Herein, we show how Bayes formula allows to directly construct a biased sample containing an enhanced fraction of reactive trajectories and to concomitantly estimate the transition rate from this …
We introduce a quantum algorithm for efficient biased sampling of the rare events generated by classical memoryful stochastic processes. We show that this quantum algorithm gives an extreme advantage over known classical biased sampling algorithms in terms of the memory resources required. The quantum memory advantage ranges from polynomial to exponential and when sampling the rare equilibrium configurations of spin systems the quantum advantage diverges.
The average time between two occurrences of the same event, referred to as its return time (or return period), is a useful statistical concept for practical applications. For instance insurances or public agency may be interested by the return time of a 10m flood of the Seine river in Paris. However, due to their scarcity, reliably estimating return times for rare events is very difficult using either observational data or direct numerical simulations. For rare events, an estimator for return times can be built from the extrema of the observable on trajectory blocks. Here, we show that this es…
We study the dynamics of perturbations in time delayed dynamical systems. Using a suitable space-time coordinate transformation, we find that the time evolution of the linearized perturbations (Lyapunov vector) can be mapped to the linear Zhang surface growth model [Y.-C. Zhang, J. Phys. France {\bf 51}, 2129 (1990)], which is known to describe surface roughening driven by power-law distributed noise. As a consequence, Lyapunov vector dynamics is dominated by rare random events that lead to non-Gaussian fluctuations and multiscaling properties.
Active matter represents a broad class of systems that evolve far from equilibrium due to the local injection of energy. Like their passive analogues, transformations between distinct metastable states in active matter proceed through rare fluctuations, however their detailed balance violating dynamics renders these events difficult to study. Here, we present a simulation method for evaluating the rate and mechanism of rare events in generic nonequilibrium systems and apply it to study the conformational changes of a passive solute in an active fluid. The method employs a variational optimizat…
Rare transitions in stochastic processes can often be rigorously described via an underlying large deviation principle. Recent breakthroughs in the classification of reversible stochastic processes as gradient flows have led to a connection of large deviation principles to a generalized gradient structure. Here, we show that, as a consequence, metastable transitions in these reversible processes can be interpreted as heteroclinic orbits of the generalized gradient flow. This in turn suggests a numerical algorithm to compute the transition trajectories in configuration space efficiently, based …
Although many computational methods for rare event sampling exist, this type of calculation is not usually practical for general nonequilibrium conditions, with macroscopically irreversible dynamics and away from both stationary and metastable states. A novel method for calculating the time-series of the probability of a rare event is presented which is designed for these conditions. The method is validated for the cases of the Glauber-Ising model under time-varying shear flow, the Kawasaki-Ising model after a quench into the region between nucleation dominated and spinodal decomposition domin…
We introduce a method for elucidating and modifying the functionality of systems dominated by rare events that relies on the automated tuning of their underlying free energy surface. The proposed approach seeks to construct collective variables (CVs) that encode the essential information regarding the rare events of the system of interest. The appropriate CVs are identified using Harmonic Linear Discriminant Analysis (HLDA), a machine-learning based method that is trained solely on data collected from short ordinary simulations in the relevant metastable states of the system. Utilizing the int…
Atypical, rare trajectories of dynamical systems are important: they are often the paths for chemical reactions, the haven of (relative) stability of planetary systems, the rogue waves that are detected in oil platforms, the structures that are responsible for intermittency in a turbulent liquid, the active regions that allow a supercooled liquid to flow... Simulating them in an efficient, accelerated way, is in fact quite simple. In this paper we review a computational technique to study such rare events in both stochastic and Hamiltonian systems. The method is based on the evolution of a fam…
