Dynamics, Langevin

Paul Langevin (1872-1946), French physicist, professor at the College de France. His main achievements are in the theory of magnetism and in relativity theory. His PhD student Louis de Broglie made a breakthrough in quantum theory. 

In the MD we solve Newton equations of motion for all atoms of the qrstem. Imagine we have a large molecule in an aqueous solution (biologr offers us important examples). We have no chance to solve Newton equations because there are too many of them (a lot of water molecules). What do we do then Let us recall that we are interested in the macromolecule, the water molecules are interesting only 

A reasonable part of this problem may be incorporated into the Langevin equation of motion  

In Langevin dynamics, we simulate the effect of a solvent by making two modifications to equation 15.1. First of all, we take account of random collisions between the solute and the solvent by adding a random force R. It is usual to assume that there is no correlation between this random force and the particle velocities and positions, and it is often taken to obey a Gaussian distribution with zero mean. 

Second, we take account of the frictional drag as the solute molecule moves through the solvent. The frictional force is taken to be proportional to the velocity of the particle, with a proportionality constant called the friction coefficient %  

The quantity y 1 is sometimes called the velocity relaxation time it can be considered to be the time taken for the particle to forget its initial velocity. The Eangevin equation of motion for a particle is therefore 

If the timestep is short relative to the velocity relaxation time, the solvent does not play much part in the motion. Indeed, if y = 0, there are no solvent effects at all. A simple algorithm for advancing the position vector r and velocity v has been given by van Gunsteren (van Gunsteren, Berendsen and Rullmann, 1981)  

The random force is taken from a Gaussian distribution with zero mean and variance 

In Chapter 2, I gave you a brief introduction to molecular dynamics. The idea is quite simple we study the time evolution of our system according to classical mechanics. To do this, we calculate the force on each particle (by differentiating the potential) and then numerically solve Newton s second law 

The calculation is advanced by a suitable timestep, typically a femtosecond, and statistical data is collected for comparison with experiment. 

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This time development of the order parameter is completely detenninistic when the equilibrium p(r) = const is reached the dynamics comes to rest. Noise can be added to capture the effect of themial fluctuations. This leads to a Langevin dynamics for the order parameter. 

An alternative framework to Newtonian dynamics, namely Langevin dynamics, can be used to mask mild instabilities of certain long-timestep approaches. The Langevin model is phenomenological [21] — adding friction and random... 

Fig. 5. Langevin trajectories for a harmonic oscillator of angular frequency u = 1 and unit mass simulated by a Verlet-like method (extended to Langevin dynamics) at a timestep of 0.1 (about 1/60 the period) for various 7. Shown for each 7 are plots for position versus time and phase-space diagrams.

R. W. Pastor. Techniques and applications of Langevin dynamics simulations. In G. R. Luckhurst and C. A. Veracini, editors. The Molecular Dynamics of Liquid Crystals, pages 85-138. Kluwer Academic, Dordrecht, The Netherlands, 1994. 

R. J. Loncharich, B. R. Brooks, and R. W. Pastor. Langevin dynamics of peptides The frictional dependence of isomerization rates of N-acetylalanyl-N -methylamide. Biopolymers, 32 523-535, 1992. 

G. Ramachandran and T. Schlick. Solvent effects on supercoiled DNA dynamics explored by Langevin dynamics simulations. Phys. Rev. E, 51 6188-6203, 1995. 

G. Zhang and T. Schlick. Implicit discretization schemes for Langevin dynamics. Mol. Phys., 84 1077-1098, 1995. 

E. Barth and T. Schlick. Overcoming stability limitations in biomolecular dynamics I. combining force splitting via extrapolation with Langevin dynamics in LN. J. Chem. Phys., 109 1617-1632, 1998. 

The errors in the present stochastic path formalism reflect short time information rather than long time information. Short time data are easier to extract from atomically detailed simulations. We set the second moment of the errors in the trajectory - [Pg.274]

The molecular dynamics method is useful for calculating the time-dependent properties of an isolated molecule. However, more often, one is interested in the properties of a molecule that is interacting with other molecules. With HyperChem, you can add solvent molecules to the simulation explicitly, but the addition of many solvent molecules will make the simulation much slower. A faster solution is to simulate the motion of the molecule of interest using Langevin dynamics. 

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) ... 

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. 

Temperature is handled the same way in Langevin dynamics as it its in molecular dynamics. High temperature runs may be used to overcome potential energy barriers. Cooling a system to a low temperature in steps may result in a different stable conformation than would be found by direct geometry optimization. 

Btamp/e Conformations of molecules like n-decane can be globally characterized by the end-to-end distance, R. In a comparison of single-molecule Brownian (Langevin) dynamics to molecular dynamics, the average end-to-end distance for n-decane from a 600 ps single-molecule Langevin dynamics run was almost identical to results from 19 ps of a 27-molecule molecular dynamics run. Both simulations were at 481K the time step and friction coeffi-... 

Monte Carlo calculations are somewhat similar to the molecular (or Langevin) dynamics calculations discussed earlier. All function by repeated application of a computational algorithm that generates a new configuration from the current configuration. The... 

Thus, unlike molecular dynamics or Langevin dynamics, which calculate ensemble averages by calculating averages over time, Monte Carlo calculations evaluate ensemble averages directly by sampling configurations from the statistical ensemble. 

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