Langevin equation harmonic oscillators

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... 

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 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. 

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. 

Proteins are very anharmonic systems, so what is the justification of modeling the protein environment as a set of harmonic oscillators in Equation (10) The real approximation we have made is modeling the proton transfer by the Langevin... 

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. 

Generalized Langevin Equation (Application to the Harmonic Oscillator)... 

The original Langevin equation considers a Markovian stochastic process [6, 7] with simple constant friction H in the field of an external fluctuation force F f). For a harmonic oscillator this equation has the form ... 

From the spectral density of R t) we can find the spectral density of stochastic observables that are related to R via linear Langevin equations. For example, consider the Langevin equation (8.13) with V(x) = /2)mo (the so called Brownian harmonic oscillator)... 

This paper describes some new developments and applications based on the reaction path model. In section II the original form of the reaction path Hamiltonian [of. equation (1) below] is transformed to a new representation that has a more desirable structure for some applications. Section III shows how the reaction path model makes application of the unified statistical model for chemical reactions especially simple, and a generalized version of the unified statistical model is also developed there. Finally, in section IV the fact that the reaction path model consists of one special degree of freedom — i.e., the reaction coordinate — coupled to a number of harmonic oscillators is exploited to derive a generalized Langevin equation (GLE) for motion along the reaction path. This is a reduced equation of motion for only the reaction coordinate, but it experiences friction" and a "random force" because of coupling to the transverse vibrational modes. 

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