Generalized Langevin Dynamics

Let us briefly summarize the numerical experiments of [220] using the SIN(R) method to simulate a system of 512 flexible water molecules using a fuUy flexible molecular model [294] and the smooth particle-mesh Ewald method (SPME) [123] (see Appendix A) to compute electrostatic forces. Initial simulations were conducted without a fast-slow force decomposition to demonstrate the effectiveness of the method as a thermostatting scheme. This technique was shown to allow accurate computation of the OH, HH and OO radial distribution functions when suitably small timesteps were used. 

As explained in Chap. 4, the resonances in multiple timestepping restrict the size of the large (outer) stepsize usable in simulation. In the case of fuUy flexible water the maximum outer stepsize that can be used for stable RESPA simulation is typically found to be less than 5 fs [245, 329]. A multiple timestepping method for the SPME method can be based on the decomposition 

We have already encountered Generalized Langevin dynamics in Chap. 6, where it appeared as an intermediate stage in the derivation of Langevin dynamics. In some cases, the properties of the heat bath and the resulting memory kernel are important 

The equations of Generalized Langevin dynamics are normally written in the form 

In the remainder of this discussion, for the purpose of simplified presentation, we confine ourselves to the simple case of a scalar memory kernel, writing 

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In general, Langevin dynamics simulations run much the same as molecular dynamics simulations. There are differences due to the presence of additional forces. Most of the earlier discussions (see pages 69-90 and p. 310-327 of this manual) on simulation parameters and strategies for molecular dynamics also apply to Langevin dynamics exceptions and additional considerations are noted below. 

The probability of a complete Brownian path is then obtained as the product of such single-time-step transition probabilities. For other types of dynamics, such as Newtonian dynamics, Monte Carlo dynamics or general Langevin dynamics, other appropriate short-time-step transition probabilities need to be used [5, 8]. 

Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134], On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods to name just a few examples Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136,137], Colli-sional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. 

Following the construction of the model is the calculation of a sequence of states (or a trajectory of the system). This step is usually referred to as the actual simulation. Simulations can be stochastic (Monte Carlo) or deterministic (Molecular Dynamics) or they can combine elements of both, like force-biased Monte Carlo, Brownian dynamics or general Langevin dynamics (see Ref. 16 for a discussion). It is usually assumed that the physical system can be adequately described by the laws of classical mechanics. This assumption will alsq be made throughout the present work. 

The DPD system may be viewed as a special case of the generalized Langevin dynamics with equations... 

Historically, Agl-based (soft framework, a situation similar to that in plastic acid sulphate) materials were the first solid electrolytes to be studied in detail. Their structural disorder is believed to correspond to Ag ion sublattice melting and the transport is governed by the immobile , counter ion lattice potential (low density approximation). This quasi-liquid state can be described by generalized Langevin dynamics... 

J. D. Doll and D. R. Dion, Generalized Langevin equation approach for atom/solid surface scattering Numerical techniques for Gaussian generalized Langevin dynamics, J. Chem. Phys. 65 3762 (1976). 

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) [ ]... 

Second, the classical dynamics of this model is governed by the generalized Langevin equation of motion in the adiabatic barrier [Zwanzig 1973 Hanggi et al. 1990 Schmid 1983],... 

C. C. Martens, Qualitative dynamics of generalized Langevin equations and the theory of chemical reaction rates, J. Chem. Phys. 116, 2516 (2002). 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

Keywords stochastic dynamics, generalized Langevin equation, nonstationary and colored friction... 

If Xe is somewhat larger, then there may arise an effective time scale Xr > Xe, with 5, < Xr sueh that the environment has some memory of the particle s previous history and therefore responds accordingly. This is the regime of the generalized Langevin equation (GLE) with colored friction. - In all these cases, the environment is sufficiently large that the particle is unable to affect the environment s equilibrium properties. Likewise, the environment is noninteracting with the rest of the universe such that its properties are independent of the absolute time. All of these systems, therefore, describe the dynamics of a stochastic particle in a stationary —albeit possibly colored— environment. 

Use of this v (t) as the friction (t) in the generalized Langevin equation provides a complete specification of a nonlocal stationary stochastic dynamics with the exponential friction jo. 

The irreversible Generalized Langevin Equation (iGLE) described in Sec. II. is capable of modeling some of the nonstationary folding dynamics motivated in this section.Such an application is the subject of present work, but it has been mentioned here in order to further motivate the reader to assess the ubiquity of nonstationary phenomenon in physical problems. 

The rate theory of Grote and Hynes [149] included the non-Markovian (memory) effects by considering the following generalized Langevin equation (GLE) for the dynamics along the reaction coordinate ... 

In the Kramers approach the friction models collisions between the particle and the surrounding medium, and it is assumed that the collisions occur instantaneously. There is a time-scale separation between the reactive mode and its thermal bath. The dynamics are described by the Langevin equation (4.141). The situation where the collisions do not occur instantaneously but take place on a time scale characterizing the interactions between the particle and its surrounding can be described by a generalized Langevin equation (GLE),158,187... 

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