Weak Noise Limit Research Articles

Evaluation of transition rates from nonequilibrium instantons

Equilibrium rate theories play a crucial role in understanding rare, reactive events. However, they are inapplicable to a range of irreversible processes in systems driven far from thermodynamic equilibrium like active and biological matter. Here we develop an efficient numerical method to compute the rate constant of rare nonequilibrium events in the weak-noise limit based on an instanton approximation to the stochastic path integral and illustrate its wide range of application. We demonstrate excellent agreement of the instanton rates with numerically exact results for a particle under a nonconservative force. We also study phase transitions in an active field theory. We elucidate how activity alters the stability of the two phases and their rates of interconversion in a manner that can be well described by modifying classical nucleation theory. Published by the American Physical Society 2024

The fluctuational transition mechanism of non-hyperbolic chaotic invariant sets

In order to reveal the general escape mechanism of non-hyperbolic chaotic invariant sets under both weak noise limit and finite noise intensity, the simplest example of Hénon map, which represents the stretching and folding of the phase space, is taken to study the escape mechanism under two kinds of global bifurcation: the fractal boundary crisis and the attractor contact crisis. In this paper, we revealed the general exit mechanism by analyzing the escape paths derived from the shooting method and Monte Carlo simulation. Finally, to further demonstrate universality, an example of high-dimensional differential dynamical system, namely the transient chaos Duffing oscillator, is examined, which underscores the main idea of this paper analyzing specific deterministic structures not only enhances the comprehensibility of the escape process, but also allows for predictions of general escape behavior under weak noise intensity.

Stochastic path power and the Laplace transform

Transition probabilities for stochastic systems can be expressed in terms of a functional integral over paths taken by the system. Approximately evaluating this integral by the saddle point method in the weak-noise limit leads to a remarkable mapping between dominant stochastic paths through the potential V and conservative, Hamiltonian mechanics with an effective potential −|∇V|2 . The conserved ‘energy’ in this effective system has dimensions of power. We show that this power, H, can be identified with the Laplace parameter of the time-transformed dynamics. As H → 0, corresponding to the long-time limit, the equilibrium Boltzmann density is recovered. However, keeping H finite leads to insights into the non-equilibrium behaviour of the system. Moreover, it facilitates the explicit summation over families of trajectories, which is far harder in the time domain, and turns out to be essential for making contact with the long-time limit in some cases. We illustrate the validity of these results using simple examples that can be explicitly solved by other means.

Geometric Stochastic Resonance in an Asymmetric T-Shaped Chamber

The investigation of a Brownian particle subjected to an AC force that diffuses within a T-shaped chamber was conducted. This T-shaped chamber is composed of a strip cavity and a trapezoidal cavity positioned below it. The interplay between the AC force and asymmetric geometry creates a spatially bistable potential perpendicular to the AC force. With the assistance of noise, the particles can transition between two stable states and oscillate along the AC force at corresponding amplitudes at every spatially stable state. The asymmetric geometry facilitates the trapezoid cavity’s ability to more easily trap the Brownian particle than the upper strip cavity in the weak noise limit. Our observations reveal that proper noise can ensure the particle’s efficient trapping within the upper strip cavity and synchronization with the AC force, indicating the occurrence of geometric stochastic resonance. The T-shaped chamber serves as a simplified model, aiding in the further understanding of geometric stochastic resonance induced by irregular geometries and enabling the manipulation of microscopic particles in various small-scale systems.

Computing large deviation prefactors of stochastic dynamical systems based on machine learning

We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise. We aim to consider a next-to-leading-order approximation for more accurate calculation of the mean exit time by computing large deviation prefactors with the aid of machine learning. More specifically, we design a neural network framework to compute quasipotential, most probable paths and prefactors based on the orthogonal decomposition of a vector field. We corroborate the higher effectiveness and accuracy of our algorithm with two toy models. Numerical experiments demonstrate its powerful functionality in exploring the internal mechanism of rare events triggered by weak random fluctuations.

Brownian dynamics of Dirac fermions in twisted bilayer graphene

Brownian dynamics of Dirac fermions in twisted bilayer graphene is investigated within the framework of semiclassical relativistic Langevin equations. We find that under the influence of orthogonal, commensurate ac drives in the periodic ratchet potential of a substrate, the charge carriers in the system exhibit pronounced random dynamics, tuned by the twist angle, making twisted bilayer graphene distinct from monolayer graphene. It is shown that as threshold twist angle matches the optimal angle, deterministic running states appear in the limit of weak thermal noise where the diffusion rate is enhanced significantly compared to bare thermal diffusion. Analysis of the real space trajectories and diffusion coefficient illustrates the significant role of thermal noise in the random motion of Dirac fermions. In addition, we find that the Brownian particle shows remarkable ratchet effect as a net current.

Stochastic analysis of a bistable piezoelectric energy harvester with a matched electrical load

We present the analysis of a bistable piezoelectric energy harvester with matched electrical load, subject to random mechanical vibrations. The matching network optimizes the average energy transfer to the electrical load. The system is described by a set of nonlinear stochastic differential equations. A perturbation method is used to find an approximate solution of the stochastic system in the weak noise limit, and this solution is used to optimize the circuit parameters of the matching network. In the strong noise limit, the state equations are integrated numerically to determine the average power absorbed by the load and the power efficiency. Our analysis shows that the application of a properly designed matching network improves the performances by a significant amount, as the power delivered to the load improves of a factor about 17 with respect to a direct connection.

Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network

We discuss the analysis of a piezoelectric energy harvester for random mechanical vibrations, and we assess the performance improvement guaranteed by interposing a matching network between the transducer and the electrical load, in terms of average output power and power efficiency. The mathematical model describing the harvester is a system of stochastic differential equations, where both cases of linear and nonlinear devices are considered. In the linear case, the power delivered to the load is increased by a factor of about 20 with respect to the direct connection, with a similar increase in the conversion efficiency. In the nonlinear case, we use a moment closure technique to calculate the first- and second-order moments of the electro-mechanical variables in the weak noise limit. Moment calculation is used to determine the optimal values of the matching network components that maximize the performance. In the strong noise limit, the state equations are integrated numerically to determine the same performance metrics. Our analysis shows that a properly designed matching network improves the performance by a significant amount, especially at low noise intensity.

Giant Diffusion of Nanomechanical Rotors in a Tilted Washboard Potential

We present an experimental realization of a biased optical periodic potential in the low friction limit. The noise-induced bistability between locked (torsional) and running (spinning) states in the rotational motion of a nanodumbbell is driven by an elliptically polarized light beam tilting the angular potential. By varying the gas pressure around the point of maximum intermittency, the rotational effective diffusion coefficient increases by more than 3 orders of magnitude over free-space diffusion. These experimental results are in agreement with a simple two-state model that is derived from the Langevin equation through using timescale separation. Our work provides a new experimental platform to study the weak thermal noise limit for diffusion in this system.

Coherence of oscillations in the weak-noise limit.

In a noisy environment, oscillations lose their coherence, which can be characterized by a quality factor. We determine this quality factor for oscillations arising from a driven Fokker-Planck dynamics along a periodic one-dimensional potential analytically in the weak-noise limit. With this expression, we can prove for this continuum model the analog of an upper bound that has been conjectured for the coherence of oscillations in discrete Markov network models. We show that our approach can also be adapted to motion along a noisy two-dimensional limit cycle. Specifically, we apply our scheme to the noisy Stuart-Landau oscillator and the thermodynamically consistent Brusselator as a simple model for a chemical clock. Our approach thus complements the fairly sophisticated extant general framework based on techniques from Hamilton-Jacobi theory with which we compare our results numerically.

Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model

Abstract. We study the impact of applying stochastic forcing to the Ghil–Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and α-stable Lévy – α∈(0,2) – noise laws, examine the statistics of transition times, and estimate the most probable transition paths. While the Gaussian noise case – used here as a reference – has been carefully studied in a plethora of investigations on metastable systems, much less is known about the Lévy case, both in terms of mathematical theory and heuristics, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. Lévy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to −α, in apparent agreement with rigorous results obtained for additive noise in a related – yet different – reaction–diffusion equation and in simpler models. This can be better understood by treating the Lévy noise as a compound Poisson process. The transition paths are studied in a projection of the state space, and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition paths cross at the single unstable edge state on the basin boundary. In the case of Lévy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance.

Swimmer dynamics in externally driven fluid flows: The role of noise

We theoretically investigate the effect of random fluctuations on the motion of elongated microswimmers near hydrodynamic transport barriers in externally-driven fluid flows. Focusing on the two-dimensional hyperbolic flow, we consider the effects of translational and rotational diffusion as well as tumbling, i.e. sudden jumps in the swimmer orientation. Regardless of whether diffusion or tumbling are the primary source of fluctuations, we find that noise significantly increases the probability that a swimmer crosses one-way barriers in the flow, which block the swimmer from returning to its initial position. We employ an asymptotic method for calculating the probability density of noisy swimmer trajectories in a given fluid flow, which produces solutions to the time-dependent Fokker-Planck equation in the weak-noise limit. This procedure mirrors the semiclassical approximation in quantum mechanics and similarly involves calculating the least-action paths of a Hamiltonian system derived from the swimmer's Fokker-Planck equation. Using the semiclassical technique, we compute (i) the steady-state orientation distribution of swimmers with rotational diffusion and tumbling and (ii) the probability that a diffusive swimmer crosses a one-way barrier. The semiclassical results compare favorably with Monte Carlo calculations.

Noise-induced transition in the Zeldovich–Semenov model with local and global bifurcations

The noise-induced transition of the Zeldovich–Semenov model in a continuous stirred tank reactor is investigated under small random perturbations. The deterministic model will exhibit mono- and bistable characteristics via local and global bifurcations. In the bistable zone, based on the Freidlin–Wentzell large deviation theory, the stochastic preference is explained by analyzing the required action of the fluctuational path. For the case of monostability, in the weak noise limit, the emergence of the switching line gives rise to the sudden switch of the optimal path and the sliding cycle will appear via the sliding bifurcation, which is verified by numerical methods. In addition, when there is no saddle in phase space, stochastic excitation with large-amplitude spikes is studied. On the quasi-threshold manifold, the point with the minimum quasi-potential plays the same role as the saddle, which means that the optimal path will undergo a large excursion by crossing this special point. These phenomena are verified by employing stochastic simulations.

Reduction of a stochastic model of gene expression: Lagrangian dynamics gives access to basins of attraction as cell types and metastabilty.

Differentiation is the process whereby a cell acquires a specific phenotype, by differential gene expression as a function of time. This is thought to result from the dynamical functioning of an underlying Gene Regulatory Network (GRN). The precise path from the stochastic GRN behavior to the resulting cell state is still an open question. In this work we propose to reduce a stochastic model of gene expression, where a cell is represented by a vector in a continuous space of gene expression, to a discrete coarse-grained model on a limited number of cell types. We develop analytical results and numerical tools to perform this reduction for a specific model characterizing the evolution of a cell by a system of piecewise deterministic Markov processes (PDMP). Solving a spectral problem, we find the explicit variational form of the rate function associated to a large deviations principle, for any number of genes. The resulting Lagrangian dynamics allows us to define a deterministic limit of which the basins of attraction can be identified to cellular types. In this context the quasipotential, describing the transitions between these basins in the weak noise limit, can be defined as the unique solution of an Hamilton-Jacobi equation under a particular constraint. We develop a numerical method for approximating the coarse-grained model parameters, and show its accuracy for a symmetric toggle-switch network. We deduce from the reduced model an approximation of the stationary distribution of the PDMP system, which appears as a Beta mixture. Altogether those results establish a rigorous frame for connecting GRN behavior to the resulting cellular behavior, including the calculation of the probability of jumps between cell types.

Conundrum of weak-noise limit for diffusion in a tilted periodic potential.

The weak-noise limit of dissipative dynamical systems is often the most fascinating one. In such a case fluctuations can interact with a rich complexity, frequently hidden in deterministic systems, to give rise to phenomena that are absent for both noiseless and strong fluctuations regimes. Unfortunately, this limit is also notoriously hard to approach analytically or numerically. We reinvestigate in this context the paradigmatic model of nonequilibrium statistical physics consisting of inertial Brownian particles diffusing in a tilted periodic potential by exploiting state-of-the-art computer simulations of an extremely long timescale. In contrast to previous results on this longstanding problem, we draw an inference that in the parameter regime for which the particle velocity is bistable the lifetime of ballistic diffusion diverges to infinity when the thermal noise intensity tends to zero, i.e., an everlasting ballistic diffusion emerges. As a consequence, the diffusion coefficient does not reach its stationary constant value.

Bifurcation of singularities of fluctuational paths for a noise-driven overdamped two-well system.

Noise-induced escape in a 2D generalized Maier-Stein model with two parameters μ and α is investigated in the weak noise limit. With the WKB approximation, the patterns of extreme paths and singularities are displayed. By employing the Freidlin-Wentzell action functional and the asymptotic series, critical parameters α inducing singularity bifurcation are determined analytically for μ=1. The switching line will appear with singularities and is equivalent to the sliding set in the Filippov system. The pseudo-saddle-node bifurcation on the switching line is found. Then, when -1<μ<1, it is found that all bifurcation values α will decrease as μ decreases and the second-order bifurcation values are bigger than all first-order ones. In addition, the variation of the switching line is also analyzed and a new switching line will emerge when the location of the minimum quasi-potential on the boundary changes. At last, when the noise is anisotropic, only the noise intensity ratio will affect the bifurcation value α.

Roughness in the periodic potential enhances transport in a driven inertial ratchet.

We study the effects of roughness in the asymmetric periodic potential on the transport and diffusion of an inertial Brownian particle driven by a time-periodic force in a Gaussian environment. We find that moderate roughness leads to the loss of transient anomalous diffusion, and it helps to establish normal diffusion in the weak noise limit. We uncover a contrasting effect of roughness on the transport of particles in the weak and moderate to large noise limit. In the weak noise limit, small amplitude roughness results in the increase of directed transport, whereas in the moderate to large noise limit, roughness hinders transport. The deterministic dynamics of the system reveals that the purely periodic system under smooth potential transits into a chaotic system due to the moderate roughness in the potential. Therefore our calculations demonstrate the constructive role of roughness in the transport of particles in the inertial regime.

Construction of stochastic hybrid path integrals using operator methods

Stochastic hybrid systems involve the coupling between discrete and continuous stochastic processes. They are finding increasing applications in cell biology, ranging from modeling promoter noise in gene networks to analyzing the effects of stochastically-gated ion channels on voltage fluctuations in single neurons and neural networks. We have previously derived a path integral representation of solutions to the associated differential Chapman–Kolmogorov equation, based on integral representations of the Dirac delta function, and used this to determine ‘least action’ paths in the noise-induced escape from a metastable state. In this paper we present an alternative derivation of the path integral based on operator methods, and show how this provides a more efficient and flexible framework for constructing hybrid path integrals in the weak noise limit. We also highlight the important role of principal eigenvalues, spectral gaps and the Perron–Frobenius theorem. Finally, we carry out a loop expansion of the associated moment generating functional in the weak noise limit, analogous to the semi-classical limit for quantum path integrals.

Exact classical noise master equations: Applications and connections

All quantum systems are subject to noise and imperfections due to stray fields, inhomogeneities or drifting experimental controls. An understanding of the effects of noise and decoherence is critical to the progress towards fully functional quantum devices. In this perspective, we focus on noise in quantum systems which are modelled by a dynamic stochastic parameter in the Hamiltonian. We will outline exact evolution equations describing the ensemble average dynamics for a variety of common noise types and their connections. We will also highlight an approximate evolution equation valid in the weak noise limit for an arbitrary classical stochastic field. This framework should serve as a starting point for identifying signatures of differing noise types and optimisation of robust control protocols.

Statistics of the first passage area functional for an Ornstein–Uhlenbeck process

We consider the area functional defined by the integral of an Ornstein–Uhlenbeck process which starts from a given value and ends at the time it first reaches zero (its equilibrium level). Exact results are presented for the mean, variance, skewness and kurtosis of the underlying area probability distribution, together with the covariance and correlation between the area and the first passage time. Among other things, the analysis demonstrates that the area distribution is asymptotically normal in the weak noise limit, which stands in contrast to the first passage time distribution. Various applications are indicated.