Langevin dissipative forces

To provide an efficient simulation algorithm, the heat bath forces are assumed to be simple. They are represented as Langevin dissipative forces, proportional to the atomic velocities,... 

There are basically two ways of simulating a many-body system through a stochastic process, sueh as the Monte Carlo (MC) simulation, or through a deterministic process, such as a Molecular Dynamics (MD) simulation. Numerical simulations are also performed in a hybridized form, like the Langevin dynamics which is similar to MD except for the presence of a random dissipative force, or the Brownian dynamics, which is based on the condition that the acceleration is balanced out by drifting and random dissipative forces. 

The LB method and its improved versions are widely used for the effident treatment of polymer solution dynamics. In the application to polymer solution dynamics, the polymer itself is still treated on a partide-based CG level using, for example, a bead-spring model, while the solvent is treated on the level of a discretized Boltzmann equation. The two parts are coupled by a simple dissipative point-partide force, and the system is driven by Langevin stochastic forces added to both the fluid and the polymers. In this approach, the hydrodynamics of the low-molecular-weight solvent is correctly captured, and the HI between polymer segments, which is mediated by the hydrodynamic flow generated within the solvent through the motion of the polymer, is present in the simulation without explidt... 

Owing to its simplicity, the Langevin (LGV) thermostat is one of the most applied methods to perform simulations in the canonical ensemble. In this approach, the simulated particles are coupled to an external heat bath at constant temperature. The dissipative force acting on particle i is proportional to its velocity ... 

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 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),... 

A disadvantage of Langevin thermostats is that they require a (local) reference system. Dissipative particle dynamics (DPD) overcomes this problem by assuming that damping and random forces act on the center-of-mass system of a pair of atoms. The DPD equations of motion read as... 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

The Ohmic model memory kernel admits an infinitely short memory limit y(t) = 2y5(t), which is obtained by taking the limit a>c —> oo in the memory kernel y(t) = yG)ce c [this amounts to the use of the dissipation model as defined by Eq. (23) for any value of ]. Note that the corresponding limit must also be taken in the Langevin force correlation function (29). In this limit, Eq. (22) reduces to the nonretarded Langevin equation ... 

Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134], On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods to name just a few examples Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136,137], Colli-sional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. 

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. 

In order to improve the model further we are currently taking quantum effects in the lattice into account, i.e. treating the CH units not classically but on quantum mechanical basis. To this end we use an ansatz state similar to Davydov s so-called ID,> state [96] developed for the description of solitons in proteins. However, there vibrations are coupled to lattice phonons, while in tPA fermions (electrons) are coupled to the lattice phonons. The results of this study will be the subject of a forthcoming paper. Further we want to improve the description of the electrons by going to semiempirical all valence electron methods or even to density functional theories. Further we introduce temperature effects into the theory which can be done with the help of a Langevin equation (random force and dissipation terms) or by a thermal population of the lattice phonons. Starting then the simulations with an optimized soliton geometry in the center of the chain (equilibrium position) one can study the soliton mobility as function of temperature. Further in the same way the mobility of polarons can be... 

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