Langevin equation basic equations

The basic equation of motion for stochastic dynamics is the Langevin equation. 

We will present the equation of motion for a classical spin (the magnetic moment of a ferromagnetic single-domain particle) in the context of the theory of stochastic processes. The basic Langevin equation is the stochastic Landau-Lifshitz(-Gilbert) equation [5,45]. More details on this subject and various techniques to solve this equation can be found in the reviews by Coffey et al. [46] and Garcia-Palacios [8]. 

This discrepancy can be due to a breakdown in the hydrodynamics friction law, Eq. (11.45), i.e., Stokes law, and/or a breakdown of the basic assumptions of Kramers theory. As we will see in the following section, a problem with Kramers theory is that the Langevin equation does not provide a sufficiently accurate description of the dynamics associated with the reaction coordinate. 

The basic BD algorithm developed by Ermak and McCam-mon (64) provides an approximate solution to the Langevin equation in the highly damped diffusive limit by using the positions of a solute particle at time t, together with the forces acting on them, to estimate the particle s new position at time, t + At. The translational behavior of the particle is described by ... 

In the Fokker-Planck equation these quantities (which express the fact that in small times in the process under consideration the space coordinate can only change by a small amount, which is the central idea underlying the theory of the Brownian motion) are to be calculated from the Langevin equation. Thus that equation is the basic equation of the theory of the Brownian movement. We have assumed in writing down the... 

All of the simulation approaches, other than harmonic dynamics, include the basic elements that we have outlined. They differ in the equations of motion that are solved (Newton s equations, Langevin equations, etc.), the specific treatment of the solvent, and/or the procedures used to take account of the time scale associated with a particular process of interest (molecular dynamics, activated dynamics, etc.). For example, the first application of molecular dynamics to proteins considered the molecule in vacuum.15 These calculations, while ignoring solvent effects, provided key insights into the important role of flexibility in biological function. Many of the results described in Chapts. VI-VIII were obtained from such vacuum simulations. Because of the importance of the solvent to the structure and other properties of biomolecules, much effort is now concentrated on systems in which the macromolecule is surrounded by solvent or other many-body environments, such as a crystal. 

The basic method utilized to treat the friction term in (3.62) can be illustrated by the following example. If we suppose for the moment that 3(t) in a simple onedimensional generalized Langevin equation is given by... 

We will first show how one can obtain the time-correlation function expression for the susceptibility in a classical statistical ensemble of particles which exhibits a linear response to an externally applied perturbation. This will be followed by an outline of the argument that leads to the generalized Langevin equation for the time-dependence of an arbitrary function of the molecular canonical coordinates. In both cases, derivations with minor modifications have been presented previously in numerous reviews, monographs, etc. However, the results are employed in a large fraction of current descriptions of dynamical processes in dense phases and thus, it seems worthwhile to again show the basic ideas underlying the formalism. 

The outlay of this chapter is as follows In Section 9.2, we introduce the system/ spin-bath model and derive the operator Langevin equation for the particle. This is followed by a discussion on stochastic dynamics in the presence of c-number noise, highlighting the role of the spectral density function in the high- and low-temperature regimes. A scheme for the generation of spin-bath noise as a superposition of several Ornstein-Uhlenbeck noise processes and its implementation in numerical simulation of the quantum Langevin equation are described in Section 9.3. Two examples have been worked out in Section 9.4 to illustrate the basic theoretical issues. This chapter is concluded in Section 9.5. 

Another inportant aspect of the Kramers treatment is the fact that the differential equation describing the probability density w (x,t) for a particle A to be at position x at time t is derived from the basic Langevin equation ... 

Quantum mechanics and statistical mechanisms did not change. Starting from the basic equations - Schrodinger, Langevin, Fokker-Planck - one has developped very good methods of solution as the simulation methods which take into account most of the observed features and, perhaps, some imaginary ones. 

Another popular thermostat used in molecular dynamics simulations is the Langevin thermostat. It covers the heat-bath coupling part of the Langevin equation by friction and Gaussian random forces f. The Langevin equation basically describes the dynamics of a Brownian particle in solvent under the influence of external forces F. Its simplest form therefore reads ... 

The remainder of this section is devoted to the derivation of Eq.[54]. Besides the mathematics we also define the range of applicability of simulations based on the Nernst-Planck equation. The starting point for deriving the Nernst-Planck equation is Langevin s equation (Eq. [45]). A solution of this stochastic differential equation can be obtained by finding the probability that the solution in phase space is r, v at time t, starting from an initial condition ro, Vo at time = 0. This probability is described by the probability density function p r, v, t). The basic idea is to find the phase-space probability density function that is a solution to the appropriate partial differential equation, rather than to track the individual Brownian trajectories in phase space. This last point is important, because it defines the difference between particle-based and flux-based simulation strategies. 

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