Langevin-type Equation Research Articles

Testing the energy diffusion approximation for the escape of a Brownian particle from a potential pocket

For the first time, the energy diffusion approximation is confronted at the percent level with the exact numerical modeling of thermal decay of a metastable state. This model is useful in many branches of natural sciences: e.g. in biology, nuclear physics, chemistry, etc. The exact (within the statistical errors about 2%) quasistationary decay rates result from the Langevin equations for the coordinate and conjugated momentum. For the energy (or action) diffusion approach, a Langevin-type equation for the action is constructed, validated, and solved numerically. The comparison of these two approaches is performed for four potentials (two of which are anharmonic) in a wide range of two dimensionless scaling parameters: i) the governing parameter G reflecting how high is the barrier with respect to the temperature and ii) the damping parameter φ expressing the friction strength. It turns out that the energy diffusion approach produces the rate which comes into 50%-agreement with the exact one only at φ < 0.02. Thus, we quantify, for the first time by our knowledge, the condition φ ≪ 1 known in the literature.

Accurate Sampling with Noisy Forces from Approximate Computing

In scientific computing, the acceleration of atomistic computer simulations by means of custom hardware is finding ever-growing application. A major limitation, however, is that the high efficiency in terms of performance and low power consumption entails the massive usage of low precision computing units. Here, based on the approximate computing paradigm, we present an algorithmic method to compensate for numerical inaccuracies due to low accuracy arithmetic operations rigorously, yet still obtaining exact expectation values using a properly modified Langevin-type equation.

Generalized input-output method to quantum transport junctions. I. General formulation

In this work, we put forward a generalized input-output method (GIOM) for studying charge transport in molecular junctions accounting for strong electron-vibration interactions and including electronic and phononic environments. The method radically expands the scope of the input-output theory, which was originally proposed to treat quantum optic problems. Based on the GIOM we derive a Langevin-type equation of motion for molecular operators, which posses a great generality and accuracy, and permits the derivation of a stationary charge current expression involving only two types of transfer rates. Furthermore, we devise the so-called "Polaron Transport in Electronic Resonance" (PoTER) approximation, which allows to feasibly simulate electron dynamics in generic tight-binding models with strong electron-vibration interactions. For short chains, the charge current reduces to known limits and reasonably agrees with exact numerical simulations. For extended junctions the current displays a turnover from phonon-assisted to phonon-suppressed transport. Nevertheless, the onset of ohmic behavior requires extensions beyond the PoTER approximation. As an additional application, we consider a cavity-coupled molecule junction. Here we identify a cavity-induced suppression of charge current in the single-site case, and observe signatures of polariton formation in the current-voltage characteristics in the strong light-matter coupling regime. A critical understanding gained from the GIOM-PoTER scheme is that the single-site vibrationally-coupled model is deceptively simpler, and amenable to approximations than multi-site models. Therefore, benchmarking of methods should not be concluded with the single-site case. The work manifests that the input-output framework, which is normally employed in quantum optics, can serve as a powerful and feasible tool in the realm of electron transport junctions.

Interdependence of Amplitude Roughness Parameters on Rough Gaussian Surfaces

According to international standards, the topography and the quality of machined surfaces can be characterized simultaneously by more than 70 roughness parameters. Despite the increased accuracy of topography measurements by modern instruments, the gained information about the 3D surface is still not well understood. The fact that machined surfaces are in general of Gaussian height distribution motivated the authors to study the interdependence of the standardized amplitude roughness parameters of computer-generated random rough (Gaussian) surfaces. In this contribution, these rough surfaces are created by solving numerically a Langevin-type stochastic differential equation for a defined random process, namely a Gaussian one. This numerical scheme provides rough surfaces of pre-defined statistical features, e.g., given standard deviation and correlation length. The numerical analysis of 17 standardized amplitude roughness parameters collected from 90000 computer-generated rough surfaces revealed so far undetected interdependencies among some of these parameters, namely the results show a strong linear relation between 12 amplitude roughness parameters.

Accuracy of the analytical escape rate for a cusp barrier in the overdamping regime

For the first time, the accuracy of the approximate analytical Kramers formula for the thermal decay rate over a cusp barrier, RK, is checked numerically for the overdamping regime. The numerical quasistationary rate, RD, which is believed to be exact within the statistical errors is evaluated by means of computer modeling of the stochastic Langevin-type dynamical equations. The agreement between RK and RD significantly depends upon the friction strength and the height of the barrier in comparison to the thermal energy. The difference between RK and RD decreases with the dimensionless damping parameter φ, however, does not become smaller than 10-20%. The unexpected growth of the difference between RK and RD with the governing parameter is observed.

Two ways for numerical solution of the Kramers problem for spatial diffusion over an edge-shaped barrier

Thermal decay rate over an edge-shaped barrier at high dissipation is studied numerically through the computer modeling. Two sorts of the stochastic Langevin type equations are applied: (i) the Langevin equations for the coordinate and conjugate momentum (LEqp, the phase space diffusion) and (ii) the reduced Langevin equation (RLE, the spatial diffusion, overdamped motion). The latter method is much faster and self-similar; however, one can doubt about its applicability in the case of an edge-shaped barrier with a discontinuous force. The reason is that a formal condition of the applicability of the RLE is not fulfilled since the curvature of the potential profile at the barrier is equal to infinity. The present numerical study demonstrates that, for large friction, the decay rate calculated using the RLE agrees with the rate resulting from the more exact LEqp. Moreover, it turns out that the influence of the position of the absorbing border is similar to the case of harmonic potential known in the literature.

Long-Range Temporal Correlations in Kinetic Roughening

We introduce a class of generalized Langevin-type growth equations exhibiting long-ranged temporal correlations in kinetic roughening, and investigate memory effects and dynamic scaling behavior based on scaling analysis and numerical simulations. The critical exponents in the weak- and strong-coupling regions are obtained, and the interplay among temporal correlations is discussed. Our results show that these long-range interactions affect nontrivial scaling properties of the temporally correlated growth system.

Thermal escape from a trap over the parabolic barrier: Langevin type approach to energy diffusion regime

Thermally activated escape from a metastable state is a useful tool to account for some features of the micro/nano-motors or superconducting nanowires. In the present work, we consider the case of week friction (action diffusion regime) which corresponds to the latter example. To describe the thermal decay, we apply two approaches both based on the Langevin type equations: the action diffusion (approximate) and the phase space diffusion (exact). For the first time, the quasistationary decay rates obtained numerically for the parabolic barrier from these two approaches are compared quantitatively with each other as well as with the analytical formula.

Two ways for finding the thermal decay rate at weak friction

We model the thermal decay of a metastable state at weak friction (the underdamped regime), the phenomenon which might be met in many branches of natural and technical sciences. The profile of the metastable state is considered as a pure parabola. We quantitatively compare the decay rate obtained from the phase space dynamics with the rate coming from the approximate energy diffusion approach which should be correct in the case of relatively weak friction. Both the phase space and energy diffusion dynamics employ the Langevin-type equations, however for the different stochastic variables. The results of the comparison reveal that the level of agreement between these two approaches depends not only upon the damping parameter (as qualitatively expected) but also upon the thermal energy.

Data-driven inference for stationary jump-diffusion processes with application to membrane voltage fluctuations in pyramidal neurons

The emergent activity of biological systems can often be represented as low-dimensional, Langevin-type stochastic differential equations. In certain systems, however, large and abrupt events occur and violate the assumptions of this approach. We address this situation here by providing a novel method that reconstructs a jump-diffusion stochastic process based solely on the statistics of the original data. Our method assumes that these data are stationary, that diffusive noise is additive, and that jumps are Poisson. We use threshold-crossing of the increments to detect jumps in the time series. This is followed by an iterative scheme that compensates for the presence of diffusive fluctuations that are falsely detected as jumps. Our approach is based on probabilistic calculations associated with these fluctuations and on the use of the Fokker–Planck and the differential Chapman–Kolmogorov equations. After some validation cases, we apply this method to recordings of membrane noise in pyramidal neurons of the electrosensory lateral line lobe of weakly electric fish. These recordings display large, jump-like depolarization events that occur at random times, the biophysics of which is unknown. We find that some pyramidal cells increase their jump rate and noise intensity as the membrane potential approaches spike threshold, while their drift function and jump amplitude distribution remain unchanged. As our method is fully data-driven, it provides a valuable means to further investigate the functional role of these jump-like events without relying on unconstrained biophysical models.

Large-Eddy Simulation of Turbulent Dispersion Effects in Direct Injection Diesel and Gasoline Sprays

&lt;div class="section abstract"&gt;&lt;div class="htmlview paragraph"&gt;In most large-eddy simulation (LES) applications to two-phase engine flows, the liquid-air interactions need to be accounted for as source terms in the respective governing equations. Accurate calculation of these source terms requires the relative velocity “seen” by liquid droplets as they move across the flow, which generally needs to be estimated using a turbulent dispersion model. Turbulent dispersion modeling in LES is very scarce in the literature. In most studies on engine spray flows, sub-grid scale (SGS) models for the turbulent dispersion still follow the same stochastic approach originally proposed for Reynolds-averaged Navier-Stokes (RANS). In this study, an SGS dispersion model is formulated in which the instantaneous gas velocity is decomposed into a deterministic part and a stochastic part. The deterministic part is reconstructed using the approximate deconvolution method (ADM), in which the large-scale flow can be readily calculated. The stochastic part, which represents the impact of the SGS flow field, is assumed to be locally homogeneous and isotropic and, therefore, governed by a Langevin-type equation. The model is applied to the spray G and spray H conditions defined by the engine combustion network (ECN) group. Simulation results are compared with the available experimental data for spray characteristics such as penetration rates, mixture fraction profile, and droplet velocity and Sauter mean diameter (SMD) distributions. Simulations with no dispersion and the commonly used RANS-type stochastic model are also performed for comparison purposes. Results show that the turbulent dispersion has a considerable impact on quantitative spray characteristics such as projected liquid volume (PLV) fraction, droplet SMD and velocity, and fuel vapor mixture fractions. On the other hand, the macroscopic spray characteristics such as liquid- and vapor-phase penetrations are not significantly affected by the dispersion modeling. The proposed SGS model also improves the prediction of spray and ignition characteristics at the spray conditions studied in this work.&lt;/div&gt;&lt;/div&gt;

Langevin equations from experimental data: The case of rotational diffusion in granular media.

A model has two main aims: predicting the behavior of a physical system and understanding its nature, that is how it works, at some desired level of abstraction. A promising recent approach to model building consists in deriving a Langevin-type stochastic equation from a time series of empirical data. Even if the protocol is based upon the introduction of drift and diffusion terms in stochastic differential equations, its implementation involves subtle conceptual problems and, most importantly, requires some prior theoretical knowledge about the system. Here we apply this approach to the data obtained in a rotational granular diffusion experiment, showing the power of this method and the theoretical issues behind its limits. A crucial point emerged in the dense liquid regime, where the data reveal a complex multiscale scenario with at least one fast and one slow variable. Identifying the latter is a major problem within the Langevin derivation procedure and led us to introduce innovative ideas for its solution.

Universal property of the housekeeping entropy production.

The entropy production of a nonequilibrium system with broken detailed balance is a random variable whose mean value is nonnegative. The housekeeping entropy production, which is a part of total entropy production, is associated with the heat dissipation in maintaining a nonequilibrium steady state. We derive a Langevin-type stochastic differential equation for the housekeeping entropy production. The equation allows us to define a housekeeping entropic time τ. Remarkably it turns out that the probability distribution of the housekeeping entropy production at a fixed value of τ is given by the Gaussian distribution regardless of system details. The Gaussian distribution is universal for any systems, whether in the steady state or in the transient state and whether they are driven by time-independent or time-dependent driving forces. We demonstrate the universal distribution numerically for model systems.

Influence of the transport regime on the energetic particle density profiles upstream and downstream of interplanetary shocks

The spatial distribution of energetic particles accelerated at shocks depends on their transport properties. Analyses of energetic particle fluxes measured by spacecraft upstream of interplanetary shock waves have pointed out the presence of spatial profiles different from those expected for normal diffusion. We propose that anomalous, superdiffusive transport can help to interpret the observed energetic particle profiles both upstream and downstream of shock waves. We set up a numerical model to compute the energetic particle profiles on both sides of the shock: particles are injected at the shock and then propagate according to a Gaussian random walk in the case of normal diffusion and according to a Lévy random walk in the case of superdiffusion. The latter is characterized by a nonlinear growth of the mean square displacement of particles and by a power law distribution of free path lengths. A Langevin type equation is solved numerically, and energetic particle spatial profiles are obtained for a steady state configuration. A number of numerical solutions are obtained: in the case of normal diffusion, the well known exponential profile upstream of the shock is recovered. In the case of superdiffusion, varying the power exponent and the scale time characterizing the power law distribution of free path times, it is found that power law upstream profiles and spatially nonconstant downstream profiles are obtained. A preliminary comparison between the obtained numerical results and interplanetary shock observations by the ACE spacecraft is carried out, and good agreement between the energetic particle profiles is found. This shows that the present superdiffusive model can be helpful for interpreting the overall time/space trends of particles accelerated at interplanetary shock waves.

Dynamics of a stochastic system driven by cross-correlated sine-Wiener bounded noises

The sine-Wiener noise, as one new type of bounded noise and a natural tool to model fluctuations in dynamical systems, has been applied to problems in a variety of areas, especially in biomolecular networks and neural models. In this paper, by virtue of the Novikov theorem, Fox’s approach, and the ansatz of Hanggi, an approximate Fokker–Planck equation is derived for an one-dimensional Langevin-type equation with cross-correlated sine-Wiener noise. Meanwhile, the dynamical characters of a bistable system driven by cross-correlated sine-Wiener noise are investigated by applying the approximate theoretical method. For the bistable system, the cross-correlation intensity $$\lambda $$ can induce the reentrance-like phase transition, while the other noise intensities and the self-correlation time, except for the self-correlation time of additive bounded noise, can induce the first-order-like phase transition. The transition from the stable state to another one can be accelerated by $$\alpha $$ (additive bounded noise intensity), $$\tau _1$$ (the self-correlation time of the multiplicative bounded noise), and $$\tau _2$$ (the self-correlation time of the additive bounded noise) and can be restrained with $$\lambda $$ and $$\tau _3$$ (self-correlation time of the cross-correlation bounded noise). It is interesting that the noise-enhanced stability phenomenon is observed with D (multiplicative bounded noise intensity) varying for the positive correlation ($$\lambda >0$$) and is enhanced as $$\lambda $$ increases. The numerical results are in basic agreement with the theoretical predictions.

Stochastic motion of vortex filaments in He II under random force

Langevin dynamics are applied to describe the stochastic motion of vortex filaments in He II under random force. The article describes a functional formalism, which is a modification of the method developed earlier by Migdal to deal with the stochastic dynamics of classical vortex filaments. In particular, starting with the Langevin-type equation, the functional Fokker–Planck equation for the characteristic functional was obtained. Based on this equation, and under the assumption that the random force correlator satisfies the fluctuation-dissipation theorem, thermodynamic equilibrium in a system of chaotic quantized vortices was investigated. Additionally, the case of stationary helium and the case of counterflow with the constant relative velocity of the normal and superfluid components were considered. Some physical consequences of the results obtained are also discussed.

Boundedness and power-type decay of solutions for a class of generalized fractional Langevin equations

In this paper we study the long-time behavior of solutions for a general class of Langevin-type fractional integro-differential equations. The involved fractional derivatives are either of Riemann–Liouville or Caputo type. Reasonable sufficient conditions under which the solutions are bounded or decay like power functions are established. For this purpose, we combine and generalize some well-known integral inequalities with some crucial estimates. Our findings are supported by examples and special cases.

Uncertainty dynamics in a model of economic inequality

In this article, we consider a stylized dynamic model to describe the economics of a population, expressed by a Langevin-type kinetic equation. The dynamics is defined by a combination of terms, one of which represents monetary exchanges between individuals mutually engaged in trade, while the uncertainty in barter (trade exchange) is modelled through additive and multiplicative stochastic terms which necessarily abide dynamical constraints. The model is studied to estimate three meaningful quantities, the inequality Gini index, the social mobility and the total income of the population. In particular, we investigate the time evolving binary correlations between any two of these quantities.

Pre-equilibrium Effects of the Hot Nuclei De-excitation via GDR Emission --- Theoretical Approach

The hot rotating nuclei could be formed in the complete and incomplete fusion reaction of two heavy ions. At low bombarding energies the reaction goes via compound nucleus formation and subsequent evaporation of charged particles, neutrons and $\gamma$-rays. However, with increasing the energy of the projectile, the emission of particles during the equilibration process becomes more and more probable. This effect can be estimated by the Heavy-Ion Phase-Space Exploration (HIPSE) code which describes the production of clusters of various size from nucleons initially in the target or projectile. This dynamic evolution finalizes with the compound nuclei, quasi-fission or multi-fragmentation products. The hot rotating nuclei produced in fusion reaction can de-excitate by evaporation of particles and emission of $\gamma$-rays from the Giant Dipole Resonance, or by fission into two fragments. These processes, evaporation and fission, are described within statistical codes such as GEMINI++ or in dynamical approaches by solving the transport equations of Langevin type. In the present article we will concentrate on the possible effect of the pre-equilibrium emission on the strength function of the effective Giant Dipole Resonance, which can be described within Thermal Shape Fluctuation Model (TSFM) approach.

Geometric integrator for Langevin systems with quaternion-based rotational degrees of freedom and hydrodynamic interactions.

We introduce new Langevin-type equations describing the rotational and translational motion of rigid bodies interacting through conservative and non-conservative forces and hydrodynamic coupling. In the absence of non-conservative forces, the Langevin-type equations sample from the canonical ensemble. The rotational degrees of freedom are described using quaternions, the lengths of which are exactly preserved by the stochastic dynamics. For the proposed Langevin-type equations, we construct a weak 2nd order geometric integrator that preserves the main geometric features of the continuous dynamics. The integrator uses Verlet-type splitting for the deterministic part of Langevin equations appropriately combined with an exactly integrated Ornstein-Uhlenbeck process. Numerical experiments are presented to illustrate both the new Langevin model and the numerical method for it, as well as to demonstrate how inertia and the coupling of rotational and translational motion can introduce qualitatively distinct behaviours.