Langevin equation oscillator

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

Figure 3. Phase portrait of the noiseless dynamics (43) corresponding to the linear Langevin equation (15) (a) in the unstable reactive degree of freedom, (b) in a stable oscillating bath mode, and (c) in an overdamped bath mode. (From Ref. 37.)...

Niunerical algorithms for solving the GLE are readily available. Only recently, Hershkovitz has developed a fast and efficient 4th order Runge-Kutta algorithm. Memory friction does not present any special problem, especially when expanded in terms of exponentials, since then the GLE can be represented as a finite set of memoiy-less coupled Langevin equations. " Alternatively (see also the next subsection), one can represent the GLE in terms of its Hamiltonian equivalent and use a suitable discretization such that the problem becomes equivalent to that of motion of the reaction coordinate coupled to a finite discrete bath of harmonic oscillators. ... 

Exercise. A damped harmonic oscillator with delta-correlated fluctuations in the frequency is given by the Langevin equation ... 

In order to compute Eq. (158), write the Langevin equations governing the dynamics of the Brownian oscillator. In the present situation that leads us to consider three time-dependent stochastic variables S(t), Q(f), and v(t), described by the following three equations ... 

Let us consider the two usual Langevin equations (161) and (162), dealing with the Brownian oscillator together with the definition of the stochastic coordinate S... 

In Section II, motivated by the fact that in typical experiments an aging system is not isolated, but coupled to an environment which acts as a source of dissipation, we recall the general features of the widely used Caldeira-Leggett model of dissipative classical or quantum systems. In this description, the system of interest is coupled linearly to an environment constituted by an infinite ensemble of harmonic oscillators in thermal equilibrium. The resulting equation of motion of the system can be derived exactly. It can be given, under suitable conditions, the form of a generalized classical or quantal Langevin equation. 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

As will be seen below, the kernel y(f) decreases on a characteristic time of the order of mf. 1, the angular frequency ooc characterizing the bandwidth of the bath oscillators effectively coupled to the particle. The quantity y(t — t,) is thus negligible if (i)c(t - U) 1. Mathematically, this condition can be realized at any time t by referring the initial time f, to -oo. Then, the initial particle position becomes irrelevant in Eq. (10), which takes the form of the generalized Langevin equation, namely,... 

Brownian motion in an oscillator potential. We consider a Brownian motion of Langevin type with a damping constant /S independent of position, so that the residual random acceleration behaves as white noise, independent of both position and velocity. The Langevin equation is... 

It is important to emphasize, however, that our model is different from the Langevin equation, which is a stochastic differential equation. Our model has no noise in the limit of small time steps in which the numerical errors approach zero. The noise we introduce is numerical. Once we filter the rapid oscillations, it is impossible for us to recover the tme trajectory using only the low-frequency modes. The noise in the SDE approach is introduced when we approximate a differential equation by a finite difference formula and filter out high-frequency motions. 

The first objective of this review is to describe a method of solution of the Langevin equations of motion of the itinerant oscillator model for rotation about a fixed axis in the massive cage limit, discarding the small oscillation approximation in the context of dielectric relaxation of polar molecules, this solution may be obtained using a matrix continued fraction method. The second... 

The Langevin equation (8.31), with R t) taken to be a Gaussian random force that satisfies (7 ) = 0 and (7 (0)7 (Z)) = ImykRTh t), is a model for the effect of a thermal environment on the motion of a classical harmonic oscillator, for example, the nuclear motion of the internal coordinate of a diatomic molecule in solution. 

The torsional potential of mean force (Fig. 24) and the correlation function for the torsional motions of the Tyr-21 ring in BPTI suggest that the time dependence of A can be described by the Langevin equation for a damped harmonic oscillator (see Chapt. IV.C and D). 

Eq. (A.52) is identical to the Langevin equation for a set of n—p coupled harmonic oscillators each of unit mass with coordinates y(t), dynamical matrix and friction matrix p. 

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