Effect Langevin

Analogous to the derivation of the effective Langevin equation for the chain in dilute solutions, we get in semidilute solutions... 

Carmeli and Nitzan have shown that the dynamics of the Hamiltonian Eq. (36) is equivalent to that of the effective Langevin equation... 

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. 

The Langevin model has been employed extensively in the literature for various numerical and physical reasons. For example, the Langevin framework has been used to eliminate explicit representation of water molecules [22], treat droplet surface effects [23, 24], represent hydration shell models in large systems [25, 26, 27], or enhance sampling [28, 29, 30]. See Pastor s comprehensive review [22]. 

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

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

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) ... 

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

Langevin dynamics a technique to reduce the total number of equations of motion that are solved. Utilize the Coupled Heat Bath, wherein the method models the solvent effect by incorporating a friction constant into the overall expression for the force. 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

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 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. 

Approximating the intermoleculai interactions to only include two-body effects, e.g. electrostatic forces are only calculated between pairs of fixed atomic chai ges in force field techniques. Or the discrete interactions between molecules may be treated only in an average fashion, by using Langevin dynamics instead of molecular dynamics. 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

Langevin dipoles, 52, 53, 125 Langevin dipoles model, 49-53, 50. See also Protein dipoles-Langevin dipole model for catalytic effect of carbonic anhydrase, 199... 

Intramolecular Isotope Effects. The data in Figure 2 clearly illustrate the failure of the experimental results in following the predicted velocity dependence of the Langevin cross-section. The remark has been frequently made that in the reactions of complex ions with molecules, hydrocarbon systems etc., experimental cross-sections correlate better with an E l than E 112 dependence on reactant ion kinetic energy (14, 24). This energy dependence of reaction presents a fundamental problem with respect to the nature of the ion-molecule interaction potential. So far no theory has been proposed which quantitatively predicts the E l dependence, and under these circumstances interpreting the experiment in these terms is questionable. 

Dran J-C, Langevin Y, Petit J-C (1988) Uraruum isotopic disequilibrium reappraisal of the alpha-recoil effect. Chem Geol 70 126... 

Many solvents do not possess the simple structure that allows their effects to be modeled by the Langevin equation or generalized Langevin equation used earlier to calculate the TS trajectory [58, 111, 112]. Instead, they must be described in atomistic detail if their effects on the effective free energies (i.e., the time-independent properties) and the solvent response (i.e., the nonequilibrium or time-dependent properties) associated with the... 

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