Langevin equation of motion

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) ... 

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

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],... 

The concept of the TS trajectory was first introduced [37] in the context of stochastically driven dynamics described by the Langevin equation of motion... 

The CMDS method is based on the Langevin equations of motion of classical PO4 dipoles ... 

The Langevin equation, Eq. (11.5), that was used in Kramers calculation of the dynamical effects on the rate constant, is only valid in the limit of long times, where an equilibrium situation may be established. The reaction coordinate undergoes many collisions with the atoms in the solvent due to thermal agitation. From the Langevin equation of motion and Eq. (11.9), we obtained an expression for the autocorrelation function of the velocity ... 

We begin with an abstract of the physics that underlies the kinetics of bond dissociation and structural transitions in a liquid environment. Developed from Einstein s theory of Brownian motion, these well-known concepts take advantage of the huge gap in time scale that separates rapid thermal impulses in liquids (< 10 s) from slow processes in laboratory measurements (e.g. from 10 s to min in the case of force probe tests). Three equivalent formulations describe molecular kinetics in an overdamped liquid environment. The first is a microscopic perspective where molecules behave as particles with instantaneous positions or states x(t) governed by an overdamped Langevin equation of motion,... 

Two of more sophisticated and commonly used approaches are the Nose-Hoover thermostat [79,80] and Langevin method [81]. In the Langevin method, additional terms are added to the equations of motion corresponding to a fiiction term and a random force. The Langevin equation of motion is given by... 

The Langevin Equation of Motion and the Spectrum of the Relaxation Times... 

The FPE has its genesis in the Langevin equations of motion of the particles, in which the influence of the bath particles is characterized by a friction and a fluctuating random force. Exact treatments lead to generalized Langevin equations when the solvent degrees of freedom are projected out from the classical equations of motion for the full particle-bath system In this case a frequency-dependent friction, or time-dependent memory kernel,... 

The first objective of this review is to describe a method of solution of the Langevin equations of motion of the itinerant oscillator model for rotation about a fixed axis in the massive cage limit, discarding the small oscillation approximation in the context of dielectric relaxation of polar molecules, this solution may be obtained using a matrix continued fraction method. The second... 

The dynamics simulation is limited to the atoms in the reaction zone. Atoms in the reaction region are treated by ordinary molecular dynamics and their motions are governed by Newton s equations of motion. Atoms in the buffer region, as indicated above, obey a Langevin equation of motion. Thus we have a set of simultaneous equations... 

The third contribution to the force on the particle is due to random fluctuations caused by interactions with solvent molecules. We will write this force as R(f). The Langevin equation of motion for a particle i can therefore be written" ... 

Eq. (14.17) is called the Langevin equation of motion, and gives rise to stochastic or Brownian dynamics " The magnitude of the friction coefficient determines the importance of the intramolecular forces compared with the friction term, and large values of C, lead to the Brownian dynamics limit. 

Macromolecules are modeled as Gaussian strings of beads and described using the same path-integral formalism as in SCFT. For a multicomponent system, the time-dependent distribution of bead density (r,t) is assumed to obey the following set of functional Langevin equations of motion ... 

The dynamics of the GGS are described by a set of linearized Langevin equations of motion. In addition to the friction and elastic forces, one has also stochastic forces due to the random collisions of the solvent with the solute (the GGS beads) and, in general, a superimposed field due to forces external to the polymer system. We note that at the time-scales usually treated in the study of polymers, the inertial term is rather unimportant thus we neglect it. Taking all this into account, the Langevin equation of motion for the Ith bead of the GGS reads ... 

Let us consider the influence of a flow (or velocity) field in the solvent, hi terms of the Langevin equations of motion, Eq. 2, we then have [2,22] ... 

Let us consider a situation in which a time-dependent external force is applied to the GGS beads see the Langevin equation of motion, Eq. 2. Now, averaging this equation over the thermal noise components we have for Ri t)) ... 

Let US consider a GGS that has a linear topology and that consists of Ntot = N beads connected by elastic springs, see Fig. lA. The Langevin equation of motion, Eq. 2, can be rewritten for the inner chain beads, 1 < < in the form [1,2,40,41] ... 

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