Langevin dynamics molecular modeling

Langevin dynamics simulates the effect of molecular collisions and the resulting dissipation of energy that occur in real solvents, without explicitly including solvent molecules. This is accomplished by adding a random force (to model the effect of collisions) and a frictional force (to model dissipative losses) to each atom at each time step. Mathematically, this is expressed by the Langevin equation of motion (compare to Equation (22) in the previous chapter) ... 

Other recent microscopic approaches are based on the Langevin dipoles solvent model or on the all-atom solvent model, using a standard force field with van der Waals and electrostatic terms as well as intramolecular terms, and molecular dynamics simulations of the fluctuation of the solvent and the solute, incorporating the potential from the permanent and induced solvent dipoles in the solute Hamiltonian in a self-consistent way (Luzhkov and Warshel, 1991). 

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. 

In most cases, the DNA CG models presented here are used in Langevin-based molecular dynamics (MD) simulations that propagate the positions and velocities of the CG sites in time and space. As described in the examples, when the appropriate sampling strategies are used, these simulations provide detailed information about the conformation space, in particular, appropriately describing the thermodynamics associated with the different conformations. Moving beyond equilibrium structural distributions, much less attention has been paid to the dynamical behavior of molecular systems in coarse-grained simulations, which should be addressed in future studies. 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

In this section, we discuss briefly how the Langevin equation, which is a stochastic equation, can be derived from the molecular equations of motion. The stochastic model described by the Langevin equation has been of great use in interpreting a large number of experiments and physical systems. The stochastic model is extremely simple but, as always, its ultimate justification rests on the molecular dynamical laws. 

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