Constrained Langevin Dynamics

We may use a splitting approach to develop discretization algorithms for other forms of dynamics. In Chap.4, we included holonomic constraints to the ODEs defining constant-energy Hamiltonian dynamics, as a way to increase the stability of our algorithms and increase the usable timestep. We can add constraints to the Langevin dynamics SDKs similarly as 

One approach for algorithms is simply to evolve unconstrained Langevin dynamics using any algorithm of choice, then follow each step by a step of the SHAKE algorithm (see Chapter 4 or [322]) to pull the system back on to the manifold satisfying the holonomic constraints. 

We could instead consider splitting the dynamics up and performing the integration using a splitting method. Such a splitting strategy is considered in [232] where one uses 

The vectors and R +i/2 are vectors ofAfCO, 1) i.i.d. random numbers, with y 0 the usual Langevin dynamics friction parameter. Setting y = 0 reduces the scheme to the usual RATTLE scheme for solving holonomically constrained Hamiltonian equations of motion, whereas if y is chosen large then this will be expected to cause instability in the scheme, as we are not solving the OU process exactly. The 

Lagrange multiplier A +i/4 (and likewise A +i) can be obtained by solving the linear system 

---

In this section, we introduce generalized definitions of sets of reciprocal basis vectors, and of corresponding projection tensors, which include the dynamical reciprocal vectors and the dynamical projection tensor introduced in Section VI as special cases. These definitions play an essential role in the analysis of the constrained Langevin equation given in Section IX. 

Due to the size constrains of the simulations special boundary conditions are used in many cases to dissipate the energy produced by the high-energy particle. The method employed in most of the simulations is a thermal bath at the boundary of the simulation box. This thermal bath is forced to keep a constant temperature, and several techniques can be used for such purpose, for example, rescaling the velocities of those atoms in the thermal bath, or applying Langevin dynamics to those atoms at the boundary. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

Lequeux et al. [74] and other authors [43,52] have shown that the nanocomposite contains a phase at the filler-polymer interface exhibiting dynamics substantially slower in comparison to the neat polymer above Tg. Davis and coworkers [43] have shown that the time constant Tj (determined by the NMR method) belonging to the immobilized phase is comparable to the T2 value of the neat matrix below its Tg. Thus, it seems necessary to extend the Langevin effect [71] by the dynamics effects, which are constrained to the diffuse shell of nanometer thickness surrounding each particle. 

---