Some applications of stochastic averaging method for quasi Hamiltonian systems in physics

Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response statistics. In the present paper, the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced. The applications of the stochastic averaging method in studying the dynamics of active Brownian particles, the reaction rate theory, the dynamics of breathing and denaturation of DNA, and the Fermi resonance and its effect on the mean transition time are reviewed. Supported by the National Natural Science Foundation of China (Grant Nos. 10772159 and 10802074), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060335125), and the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y7080070)     

Microcanonical variational transition-state theory for reaction rates in dissipative systems

Upper bounds for the classical escape rate of a particle trapped in a metastable well and interacting with a dissipative medium are derived based on the periodic orbits of a reduced two-degree-of-freedom Hamiltonian involving the unstable normal mode and a collective bath mode. It is shown that even in what is usually thought of as the spatial diffusion limit the reactive flux can involve an energy diffusion term due to energy transfer from the dissipative media, in addition to the standard spatial diffusion term.     

Energy diffusion controlled reaction rate of reacting particle driven by broad-band noise

The energy diffusion controlled reaction rate of a reacting particle with linear weak damping and broad-band noise excitation is studied by using the stochastic averaging method. First, the stochastic averaging method for strongly nonlinear oscillators under broad-band noise excitation using generalized harmonic functions is briefly introduced. Then, the reaction rate of the classical Kramers' reacting model with linear weak damping and broad-band noise excitation is investigated by using the stochastic averaging method. The averaged It? stochastic differential equation describing the energy diffusion and the Pontryagin equation governing the mean first-passage time (MFPT) are established. The energy diffusion controlled reaction rate is obtained as the inverse of the MFPT by solving the Pontryagin equation. The results of two special cases of broad-band noises, i.e. the harmonic noise and the exponentially corrected noise, are discussed in details. It is demonstrated that the general expression of reaction rate derived by the authors can be reduced to the classical ones via linear approximation and high potential barrier approximation. The good agreement with the results of the Monte Carlo simulation verifies that the reaction rate can be well predicted using the stochastic averaging method.     

Modified Kramers formulas for the decay rate in agreement with dynamical modeling

Accuracy of the Kramers approximate formulas for the thermal decay rate of the metastable state is studied for the anharmonic shapes of the potential pocket and the barrier. This is done by the comparison with the quasistationary rate resulting from the dynamical modeling. Disagreement between the Kramers rate and the dynamical one is shown to reach 15% in the cases when much better agreement is expected. Corrections to the Kramers formulas accounting for the higher derivatives of the potential are obtained. The small parameters are the ratios of the thermal energy to the stiffnesses at the extremes of the potential. The distance between the potential barrier and the absorptive border is accounted for as well. This corrected Kramers rate is demonstrated to agree with the dynamical rate typically within 2%. Probably the most interesting result is that despite the corrections are derived in the case of the overdamped Brownian motion, the above 2% agreement holds even in the case of medium friction.     

Stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems

The stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems is investigated. First, the stochastic optimal control problem of a partially observable nonlinear quasi-integrable Hamiltonian system is converted into that of a completely observable linear system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle....     

非高斯噪声驱动下一维双稳系统的逻辑操作

本文以成功率作为逻辑随机共振的测度, 主要研究了非高斯噪声激励下一维双稳系统的逻辑随机共振现象, 并用平均首次通过时间的方法对此现象的机理进行了解释. 研究结果表明: 只有在适当的噪声强度带或关联时间带上, 成功率才会达到共振峰值. 通过对系统参数进行优化, 提高了系统实现逻辑操作的可靠性. 关键词： 逻辑随机共振 一维双稳系统 非高斯噪声 平均首次通过时间     

非高斯噪声驱动下非对称双稳系统的平均首次穿越时间与随机共振研究

研究了乘性非高斯噪声和加性高斯白噪声共同激励下非对称双稳系统的平均首次穿越时间和随机共振问题. 利用路径积分法和两态模型理论,推导出平均首次穿越时间和信噪比的表达式. 研究结果表明:势阱非对称性对两个不同方向的平均首次穿越时间的影响是不同的. 信噪比是加性噪声强度和势阱非对称性的非单调函数,系统出现了随机共振现象;信噪比是乘性噪声强度的单调函数,没有共振峰出现. 这说明该系统中乘性噪声强度和加性噪声强度对信噪比的影响是不同的. 关键词： 非高斯噪声 非对称双稳系统 平均首次穿越时间 随机共振     

Stability of motions near resonances in quasi-integrable Hamiltonian systems

Nekhoroshev's theorem on the stability of motions in quasi-integrable Hamiltonian systems is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators; furthermore we prove the confinement of orbits in resonant regions, in the general case of nonisochronous systems, by using the elementary idea of energy conservation instead of more complicated mechanisms. An application of Nekhoroshev's theorem to the study of perturbed motions inside resonances is also provided.Partially supported by Ministere della Pubblica Istruzione.Partially supported by Grant N.S.F. DMS 85-03333 and by Ministero della Pubblica Istruzione.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Hamilton系统Noether理论的新型逆问题

研究Hamilton系统Noether理论新型的逆问题,得到利用Noether理论从已知的第一积分构建Hamilton函数和对称性的一般解法和若干特殊解法,提出由Hamilton函数直接导出守恒量的两条推论.举例说明所得结果的应用.     

基于Kramers逃逸速率的调参随机共振机理

根据Kramers逃逸速率的特性,阐明了随机共振信号的频率被限制在Kramers逃逸速率极限值一半的范围内,这种限制是制约大频率信号产生随机共振的原因.在进一步揭示二次采样随机共振频率尺度变换机理的基础上,证明了二次采样频率尺度可以把任意信号频率映射变换到随机共振频率尺度上的结论.相对于二次采样变换方法,由于双稳系统参数的调节很难使Kramers逃逸速率的一半达到实际信号的大频率,因此系统参数只能在随机共振的小参数频率范围内调节来实现随机共振. 关键词： 双稳随机共振 二次采样频率变换 系统参数调节 Kramers逃逸速率     

Simple generalization of Kramers theory to finite barrier height in spatial diffusion regime

Motivated by the escape process at low reduced barrier heights (measured in units of $k_{B}T$) is still a stationary one, the Kramers theoretical method in spatial diffusion regime should be applicable to this process. The Kramers theory is generalized to finite barrier height in a simple manner. The integration constant is redetermined by introducing metastable equilibrium state concept and continuous condition of the probability at the joint point of the potential barrier and potential well. The parabolic barrier with local frequency is replaced by a parabolic barrier with nonlocal frequency. The modified Kramers theory is confirmed by a cubic potential case. The maximal relative error in the spatial diffusion regime is less than 3% for the applied parameters.     

Kramers time in weakly nonpotential systems

We consider a bistable Fokker-Planck system with a known stationary distribution and a small nonpotential part in the drift force. We perform a perturbation calculation of its Kramers time, _K, and compare it with the corresponding time, _K ⁽⁰⁾ , for the potential system which has the same stationary distribution. We show that _K/ _K ⁽⁰⁾ depends only on the properties of the drift force close to the saddle-point.The authors would like to dedicate this work to their colleagues Y. Orlov, R. Nazarian, and V. Brailovski.     

Uniform expansion of the transition rate in Kramers' problem

Kramers' model of diffusion over potential barriers, e.g., chemical reactions, based on the noise activated escape of a particle from a potential well, is considered. Kramers derived escape rates valid for intermediate and large damping, and in a separate analysis, for small damping. In the small damping limit, Kramers' intermediate result reduces to the transition state rate which does not agree with the small damping result. A new escape rate is derived that is uniformly valid for all values of the damping coefficient. The new rate reduces to Kramers' results in the appropriate limits and, in particular, connects Kramers' intermediate and small damping results.This work was partially supported by the Air Force Office of Scientific Research under Grant No. AFOSR-83-0086, U.S. Department of Energy under Grant No. DE-AC02-78ERO-4650, and the National Science Foundation under Grant No. MCS-83-00562. One of us (BJM) gratefully acknowledges the support of a John Simon Guggenheim Memorial Foundation Fellowship.     

Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems

We develop canonical perturbation theory for a physically interesting class of infinite-dimensional systems. We prove stability up to exponentially large times for dynamical situations characterized by a finite number of frequencies. An application to two model problems is also made. For an arbitrarily large FPU-like system with alternate light and heavy masses we prove that the exchange of energy between the optical and the acoustical modes is frozen up to exponentially large times, provided the total energy is small enough. For an infinite chain of weakly coupled rotators we prove exponential stability for two kinds of initial data: (a) states with a finite number of excited rotators, and (b) states with the left part of the chain uniformly excited and the right part at rest.     

A statistical model for predicting thermal chemical reaction rate

A simple model based on the statistics of individual atoms [Europhys. Lett. 94 40002 （2011）] or molecules [Chin. Phys. Lett. 29 080504 （2012）] was used to predict chemical reaction rates without empirical parameters, and its physical basis was further investigated both theoretically and via MD simulations. The model was successfully applied to some reactions of extensive experimental data, showing that the model is significantly better than the conventional transition state theory. It is worth noting that the prediction of the model on ab initio level is much easier than the transition state theory or unimolecular RRKM theory.     

双层耦合非对称反应扩散系统中的超点阵斑图

通过线性耦合Brusselator模型和Lengyel-Epstein模型,数值研究了双层耦合非对称反应扩散系统中图灵模之间的相互作用以及斑图的形成机理.模拟结果表明,合适的波数比以及相同的对称性是两个图灵模之间达到空间共振的必要条件,而耦合强度则直接影响了图灵斑图的振幅大小.为了保证对称性相同,两个图灵模的本征值高度要位于一定的范围内.只有失稳模为长波模时,才能对另一个图灵模产生调制作用,并形成多尺度时空斑图.随着波数比的增加,短波模子系统依次经历黑眼斑图、白眼斑图以及时序振荡六边形斑图的转变.研究表明失稳图灵模与处于短波不稳定区域的高阶谐波模之间的共振是产生时序振荡六边形的主要原因.     

Application of non-Hermitian Hamiltonian model in open quantum optical systems

Non-Hermitian systems have observed numerous novel phenomena and might lead to various applications. Unlike standard quantum physics, the conservation of energy guaranteed by the closed system is broken in the non-Hermitian system, and the energy can be exchanged between the system and the environment. Here we present a scheme for simulating the dissipative phase transition with an open quantum optical system. The competition between the coherent interaction and dissipation leads to the second-order phase transition. Furthermore, the quantum correlation in terms of squeezing is studied around the critical point. Our work may provide a new route to explore the non-Hermitian quantum physics with feasible techniques in experiments.     

约束哈密顿系统在相空间中的精确不变量与绝热不变量

研究小干扰力作用下约束哈密顿系统对称性的摄动问题.建立了非保守约束哈密顿系统的正则方程,在增广相空间中研究了系统的对称性与精确不变量.基于力学系统的高阶绝热不变量的概念,给出了系统的各阶绝热不变量的形式及存在条件,并建立了绝热不变量与对称变换之间的对应关系 关键词： 约束哈密顿系统 对称性 摄动 不变量     

Mean First-Passage Time in the Stochastic Theory of Biochemical Processes. Application to Actomyosin Molecular Motor

Many studies performed in recent years indicate a rich stochastic dynamics of transitions between a multitude of conformational substates in native proteins. A slow character of this dynamics is the reason why the steady-state kinetics of biochemical processes involving protein enzymes cannot be described in terms of conventional chemical kinetics, i.e., reaction rate constants. A more sophisticated language of mean first-passage times has to be used. A technique of summing up the stochastic dynamics diagrams is developed, enabling a calculation of the steady-state fluxes for systems of enzymatic reactions controlled and gated by the arbitrary type stochastic dynamics of the enzymatic complex. For a single enzymatic reaction, it is shown that the phenomenological steady-state kinetics of Michaelis–Menten type remains essentially unaltered but the interpretation of its parameters needs substantial change. A possibility of dynamical rather then structural inhibition of enzymatic activity is supposed. Two coupled enzymatic cycles are studied in the context of the biologically important process of free energy transduction. The theoretical tools introduced are applied to elucidate the mechanism of mechanochemical coupling in actomyosin molecular motor. Relations were found between basic parameters of the flux-force dependences: the force stalling the motor, the degree of coupling between the ATPase and the mechanical cycles as well as the asymptotic turnover number, and the mean first-passage times in a random movement between the particular conformational substates of the myosin head. These times are to be determined within a definite model of conformational transition dynamics. The theory proposed, not contradicting the presently available experimental data, is capable to explain the recently demonstrated multiple stepping produced by a single myosin head during just one ATPase cycle.