Equation Langevine

In a stochastic approach, one replaces the difficult mechanical equations by stochastic equations, such as a diffusion equation, Langevin equation, master equation, or Fokker-Planck equations.5 These stochastic equations have fewer variables and are generally much easier to solve than the mechanical equations, One then hopes that the stochastic equations include the significant aspects of the physical equations of motion, so that their solutions will display the relevant features of the physical motion. 

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. 

This is a verbal version of the old Langevin equation (Langevin, 1908). According to a standard further assumption the random term is a linear function of a white noise . White noise is considered as a stationary Gaussian process with = 0 and ] = 8 / - t, where 5 is the... 

We consider the motion of a large particle in a fluid composed of lighter, smaller particles. We also suppose that the mean free path of the particles in the fluid, X, is much smaller than a characteristic size, R, of the large particle. The analysis of the motion of the large particle is based upon a method due to Langevin. Consider the equation of motion of the large particle. We write it in the fonn... 

If we now average the Langevin equation, (A3.1.56). we obtam a very simple equation for (v(0), whose solution is clearly... 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

With the fomi of free energy fiinctional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3.48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

T) I c)t for model B. In temis of these variables the model B Langevin equation can be written as... 

Some features of late-stage interface dynamics are understood for model B and also for model A. We now proceed to discuss essential aspects of tiiis interface dynamics. Consider tlie Langevin equations without noise. Equation (A3.3.57) can be written in a more general fonn ... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

Zwanzig R 1973 Nonlinear generalized langevin equations J. Stat. Phys. 9 215-20... 

In the limit of a very rapidly fluctuating force, the above equation can sometimes be approximated by the simpler Langevin equation... 

Poliak E 1990 Variational transition state theory for activated rate processes J. Chem. Phys. 93 1116 Poliak E 1991 Variational transition state theory for reactions in condensed phases J. Phys. Chem. 95 533 Frishman A and Poliak E 1992 Canonical variational transition state theory for dissipative systems application to generalized Langevin equations J. Chem. Phys. 96 8877... 

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... 

An appropriate value of 7 for a system modeled by the simple Langevin equation can also be determined so as to reproduce observed experimental translation diffusion constants, Dt in the diffusive limit, Dt is related to y hy Dt = kgTmy. See [22, 36], for example. 

Fig. 7. Writhe distributions for closed circular DNA as obtained by LI (see Section 4.1) versus explicit integration of the Langevin equations. Data are from [36].

Discretizing the Langevin equation (2,3) by IE produces the following system which implicitly, rather than explicitly, defines in terms of quantities... 

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

We recently received a preprint from Dellago et al. [9] that proposed an algorithm for path sampling, which is based on the Langevin equation (and is therefore in the spirit of approach (A) [8]). They further derive formulas to compute rate constants that are based on correlation functions. Their method of computing rate constants is an alternative approach to the formula for the state conditional probability derived in the present manuscript. 

We further note that the Langevin equation (which will not be discussed in detail here) is an intermediate between the Newton s equations and the Brownian dynamics. It includes in addition to an inertial part also a friction and a random force term ... 

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