Steepest Descent Approximation Research Articles

Steady-state analysis of the stochastic Beverton-Holt growth model driven by correlated colored noises

In this paper, we investigate a single population growth model with Beverton-Holt function, which is driven by cross-correlation between multiplicative and additive colored noises. Firstly, using approximate Fokker-Planck equation, the stationary probability distribution of the model is obtained, and then its the shape structure is discussed in detail. In addition, the influence of noise characteristics on mean, variance and skewness is studied numerically. Finally, an explicit expression of the mean first passage time is given by using the steepest-descent approximation. It is found that: (i) the P-bifurcation occurs when the two noises are positively correlated or zero correlated, but not in the case of negative correlation; (ii) a strong negative correlation degree and correlation time can promote population growth, while the strong positive correlation degree plays an opposite role; (iii) the noise enhanced stability induced by multiplicative noise is different from that induced by the additive one.

A fast algorithm for prestack Gaussian beam migration adopting the steepest descent approximation

In the past, prestack Gaussian beam migration adopted the steepest descent approximation to reduce the dimension of the integrals and speed up the computation. However, the simplified formula by the steepest descent approximation was still in the frequency domain, and it had to be evaluated at each frequency. To solve this problem, we present a fast algorithm by changing the order of the integrals. The innermost integral is regarded as a two-dimensional continuous function with respect to the real part and the imaginary part of the total traveltime. A lookup table corresponding to the value of the innermost integral is constructed at the sampling points. The value of the innermost integral at one imaging point can be obtained through interpolation in the constructed lookup table. The accuracy and efficiency of the fast algorithm are validated with the Marmousi dataset. The application to the Sigsbee2A dataset shows a good result.

Absolute instability: A toy model and an application to the Rayleigh–Bénard problem with horizontal flow in porous media

The concept of absolute instability is surveyed and applied to the study of the Rayleigh–Bénard problem in a horizontal porous layer with longitudinal flow. The survey is aimed to provide a simple introduction to absolute instability by employing a toy model based on a one-dimensional Burgers’ equation. The method of analysis is based on the steepest descent approximation, for large times, of the Fourier integral expressing the wavepacket perturbation of the basic solution. The analysis of Burgers’ equation is a suitable arena for the illustration of the elementary features of absolute instability. Then, the onset of absolute instability in a horizontal porous layer with a prescribed wall temperature difference between the boundaries and subject to a longitudinal flow is analysed. The seepage flow is modelled through Darcy’s law by assuming a finite Darcy–Prandtl number. It is shown that the transition from convective to absolute instability occurs at supercritical conditions, except for the limiting case when the horizontal flow rate is vanishingly small. In this special case, corresponding to the Darcy–Bénard problem, the condition of convective instability yields also absolute instability. The effects of the governing parameters, the Péclet number and the Darcy–Prandtl number, on the onset of absolute instability are studied.

Dynamics of interacting bosons using the Herman–Kluk semiclassical initial value representation

Recent experimental progress in monitoring the dynamics of ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methodology, with propagators expressed as integrals over phase space and using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that application of the Herman–Kluk (HK) propagator conserves unitarity in the semiclassical limit, but a decay of the norm is seen for low particle numbers n. The frozen Gaussian approximation (FGA) (HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the semiclassical limit (). An error is, however, incurred in next-to-next-to-leading order (small parameter ), as seen upon numerical evaluation of the integral and confirmed analytically by considering finite n corrections to the steepest descent calculations. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also using classical information only, reproduces the salient features of the exact solution correctly.

Effects of self-correlation time on stability of metastable state in logistic model

Effects of colored noise on stability of metastable state in logistic model are studied by means of Fox approach and steepest-descent approximation. The expression for mean first-passage time from metastable state to stable state is derived firstly. Based on the expression, we analyze effects of colored noise. Results indicate: that stability of the metastable state always weakens as multiplicative noise intensity and additive noise intensity increase. However, it is always enhanced as self-correlation time of multiplicative noise and cross-correlation degree between multiplicative noise and additive noise increases.

On the derivation of semiclassical expressions for quantum reaction rate constants in multidimensional systems

Expressions for reaction rate constants in multidimensional chemical systems are derived by applying semiclassical approximation to the quantum path integrals of the ImF formulation of reaction rate theory. First, the transverse degrees of freedom orthogonal to the reaction coordinate are treated within the steepest descent approximation, after which the semiclassical approximation is applied to the remaining reaction coordinate. Thus derived, the semiclassical expressions account for the multidimensional nature of quantum effects and accurately incorporate nuclear quantum effects such as multidimensional tunneling and zero point energies. The obtained expressions are applicable in the broad temperature range from the deep tunneling to high-temperature regimes. The present paper provides derivation of the semiclassical instanton expressions proposed by Kryvohuz [J. Chem. Phys. 134, 114103 (2011)].

The mean first-passage time for piecewise nonlinear system driven by colored correlated additive and multiplicative colored noises

In this paper, We have studied the effects of intensity and correlation time of noises on the mean first-passage time in a picecewise nonlinear system driven by multiplicative and additive colored noises with colored cross-correlation. We derived the expression of the mean first-passage time (MFPT) by applying the unified colored approximation method and the steepest-descent approximation. Results show that the MFPT of the system exhibits a mono-peak structure and the “resonance” phenomena enhance with the increase of multiplicative noise intensity. The value of the peak decreases with increasing additive noise intensity and the correlation between the additive and multiplicative noises. However, the MFPT of the system increases with the increase of additive noise intensity. That is, the effects of the additive noise and the multiplicative noise on MFPT are different. Moreover, the negative and passive correlations play different roles in the MFPT.

The effects of correlated time between noises on stability of unstable state in Logistic system

The effects of correlated time between noises on the stability of unstable state in the Logistic system are investigated. Using the steepest-descent approximation, the analytic expression for the mean first-passage time from unstable state to stable state is derived. The numerically calculated results indicate that the additive noise, the multiplicative noise and the correlated time between the additive and the multiplicative noises weaken the stability of unstable state, but the correlation between the additive and the multiplicative noises enhances the stability of unstable state.

Effects of time delay on transition rate of state in an increasing process of Logistic system

The effects of time delay on transition rate from metastable state to stable state in a Logistic system are investigated. On the assumption that there is a delay time in the decay process of cell evolution, the expression of the transition rate is derived in the first order approximation under the condition of the small delay time and the steepest-descent approximation. The numerically calculated results indicate that the additive noise, the multiplicative noise and the correlations between additive and multiplicative noises enhance the system transition from metastable state to stable state. However, such a transition is restrained by the time delay, i.e., the time delay enhances the stability of the system.

Colored Noise Enhanced Stability in a Tumor Cell Growth System Under Immune Response

In this paper, we investigate a mathematical model for describing the growth of tumor cell under immune response, which is driven by cross-correlation between multiplicative and additive colored noises as well as the nonzero cross-correlation in between. The expression of the mean first-passage time (MFPT) is obtained by virtue of the steepest-descent approximation. It is found: (i) When the noises are negatively cross-correlated (λ 0), then the escape is slower than in the case with no correlation. Moreover, in the case of positive cross-correlation, the escape time has a maximum for a certain intensity of one of the noises, i.e., the maximum for MFPT identifies the noise enhanced stability of the cancer state. (ii) The effect of the cross-correlation time τ 3 on the MFPT is completely opposite for λ>0 and λ<0. (iii) The self-correlation times τ 1 and τ 2 of colored noises can enhance stability of the cancer state, while the immune rate β can reduce it.

Time delay to enhance the giant suppression in a bistable system

Effects of time delay on the active rate of a bistable system was investigated. Using the steepest-descent approximation method, the analytical expression of the mean first passage time and the active rate of the bistable system with time delay was derived under the condition of small delay time. The numerical computations show that the time delay enhances the resonance suppression in the system.

Mean first-passage time in a bistable system driven by multiplicative and additive colored noises with colored cross-correlation

The effect of any system parameter (include p, q, λ, τ1, τ2, τ3) on the mean first-passage time is investigated in a symmetric bistable system driven by multiplicative and additive colored noise with colored cross-correlation. Expression of the mean first-passage time (MFPT) has been obtained by applying the steepest-descent approximation. The effects of the additive noise intensity and the multiplicative noise intensity on the MFPT are mostly different only with bigger positive values of the strength of correlations between multiplicative and additive noise; and the MFPT has a maximum with the increase of the multiplicative noise intensity. The effects of the noise self-correlation times on the MFPT are quite different from that of the correlation time between the noises as the strength of correlations between multiplicative and additive noise is positive.

Effects of cross correlated noises on the mean first-passage time of optical bistable system

The effects of multiplicative colored noise and additive white noise on the mean first-passage time （MFPT） T+（xs1→xs2） and T-（xs2→xs1） of an optical bistable system are investigated. The approximate Fokker-Planck equation is obtained based on the Novikov theorem and the Fox approach. The expression of MFPT is obtained in the steepest-descent approximation. Numerical results show that （1） the multiplicative noise intensity Q and additive noise intensity D play the same roles in T+（xs1→xs2） but different roles in T-（xs1→xs2）;（2）T+（xs1→xs2）increases with the increase of the self-correlation time τ but decreases with the increase of the correlation intensity λ；（3）T-（xs2→xs1）decreases with τ increasing， but increases with λ increasing.

Dynamical chiral symmetry breaking, color superconductivity, and Bose–Einstein condensation in an [formula omitted]-invariant supersymmetric Nambu–Jona-Lasinio model at finite temperature and density

We investigate the phenomena of the dynamical chiral symmetry breaking (DCSB), color superconductivity (CSC), and Bose–Einstein condensation (BEC) in a supersymmetric (SUSY) vector-like SU ( N c ) gauge model at finite temperature and density. Both the N = 1 four-dimensional and N = 2 three-dimensional cases are considered. We employ the N = 1 four-dimensional generalized SUSY Nambu–Jona-Lasinio model ( N = 1 generalized SNJL 4 ) with a chemical potential as the model Lagrangian. The N = 2 three-dimensional theory is obtained by a simple dimensional reduction scheme of the four-dimensional counterpart. In order to realize the DCSB and BCS-type CSC in this model, we introduce a SUSY soft mass term. After adopting the method of SUSY auxiliary fields with the Fierz transformation in color and flavor spaces, we discuss several possible breaking schemes of the global symmetries of the model. The integrations of the auxiliary fields of composites in the effective potential are performed by using the steepest descent approximation. Under the finite-temperature Matsubara formalism, the gap equations are derived and solved. The roles of both the boson and fermion sectors in the BEC, DCSB and CSC are examined by the quasiparticle excitation spectra and the gap equations. The physical properties of the DCSB and the CSC are studied in detail. Examination on the physical property of the BEC is beyond scope of this paper. It is found that the BEC, DCSB and CSC can coexist under a condition of model parameters. Several important observations are obtained in the method of the construction of the SUSY BCS-type theory, starting from a SUSY field theoretical framework.

Monte Carlo approach to the decay rate of a metastable system with an arbitrarily shaped barrier

A path integral Monte Carlo method based on the fast-Fourier transform technique combined with the important sampling method is proposed to calculate the decay rate of a metastable quantum system with an arbitrary shape of a potential barrier. The contribution of all fluctuation actions is included which can be used to check the accuracy of the usual steepest-descent approximation, namely, the perturbation expansion of potential. The analytical approximation is found to produce the decay rate of a particle in a cubic potential being about 20% larger than the Monte Carlo data at the crossover temperature. This disagreement increases with increasing complexity of the potential shape. We also demonstrate via Langevin simulation that the postsaddle potential influences strongly upon the classical escape rate.

Mean first-passage time of an asymmetric bistable system driven by colour-correlated noise

In this paper, the effect of every parameter (including p,q,r,λ,τ) on the mean first-passage time (MFPT) is investigated in an asymmetric bistable system driven by colour-correlated noise. The expression of MFPT has been obtained by applying the steepest-descent approximation. Numerical results show that (1) the intensity of multiplicative noise p and the intensity of additive noise q play different roles in the MFPT of the system, (2) suppression appears on the curve of the MFPT with small λ (e.g. λ 0.5), and (3) with different values of r (e.g. r = 0.1, 0.5, 1.5), the effort of τ on the MFPT is diverse.

Mean First-Passage Time in a Bistable Kinetic Model with Stochastic Potentials

The transient properties of a bistable system with the stochastic potentials are investigated. The explicit expressions of the mean first-passage time (MFPT) are obtained by using a steepest-descent approximation. The results show that the MFPT of the system increases with the amplitude Δ of the stochastic potential, decreases with the noise intensity D and the correlation length l. The stochastic potential makes against the particle moving towards the destination.

Transient properties of a bistable kinetic model with quantum corrections

Abstract The transient properties of a Brownian particle moving in a bistable system with quantum corrections are investigated. The Quantum Smoluchowski Equation (QSE) is fully valid for high temperatures; for low temperatures it is valid only in a restricted domain of the state space. The quantum effects in a bistable system stand out for low temperatures. Explicit expressions of the mean first-passage time (MFPT) are obtained by using a steepest-descent approximation. The quantum effects are against the particle moving towards the destination from its original position.

Steepest descent calculation of RNA pseudoknots

We enumerate possible topologies of pseudoknots in single-stranded RNA molecules. We use a steepest-descent approximation in the large N matrix field theory, and a Feynman diagram formalism to describe the resulting pseudoknot structure.

Magnetization of nanomagnet assemblies: Effects of anisotropy and dipolar interactions

We investigate the effect of anisotropy and weak dipolar interactions on the magnetization of an assembly of nanoparticles with distributed magnetic moments, i.e., assembly of magnetic nanoparticles in the one-spin approximation, with textured or random anisotropy. The magnetization of a free particle is obtained either by a numerical calculation of the partition function or analytically in the low and high field regimes, using perturbation theory and the steepest-descent approximation, respectively. The magnetization of an interacting assembly is computed analytically in the range of low and high field, and numerically using the Monte Carlo technique. Approximate analytical expressions for the assembly magnetization are provided which take account of the dipolar interactions, temperature, magnetic field, and anisotropy. The effect of anisotropy and dipolar interactions are discussed and the deviations from the Langevin law they entail are investigated, and illustrated for realistic assemblies with the lognormal moment distribution.