Langevin methods

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. 

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In a purely formal sense, Langevin and Fokker-Planck approaches to a problem are equivalent but, as is often the case, one approach or the other may be preferable for practical reasons. In the Fokker-Planck method, one has to solve a partial differential equation in many variables. In the Langevin method, one has to solve coupled equations of motion for the same variables, under the influence of a random force. It is likely that this second approach will be most useful for performing computer experiments to simulate the actual motion of individual polymer molecules. 

Because of the formal equivalence of Fokker-Planck and Langevin methods, there is no intrinsic difficulty in translating Kirkwood s theory into Langevin terms. As far as 1 am aware, this has not yet been done. The main purpose of this article is to perform the translation. 

Summary. The semiclassical Boltzmann-Langevin method is extended to calculations of higher cumulants of current. Rs efficiency is demonstrated for mesoscopic diffusive contacts and chaotic cavities. We show that in addition to a dispersion at the inverse RC time characteristic of charge relaxation, higher cumulants of noise have a low-frequency dispersion at the inverse dwell time of electrons in the system. 

Langevin method, in simulations, 388 Large component, in relativistic wave function, 207... 

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... 

Only recently has the theory of e for systems in the presence of applied fields reached a level at which one can compute with satisfactory accuracy for nontrivial Hamiltonian models at liquid-state densities by direct extension of the original Debye-Langevin method. We touch on this extension in Section VI, but for the most part we treat only systems in the absence of external fields. 

The sticking of atoms and molecules on Ar(s) using Langevin methods has been investigated (188). 

Nos6-Hoover/Nos6-Hoover-Langevin Methods. 393... 

Of particular interest for sampling the canonical distribution are symmetric Langevin methods. We believe these are likely to be the most useful class of methods for practitioners, as by symmetrizing the expansion the odd order terms in (7.16) vanish identically using the Jacobi identity in the BCH expansions. This implies that a symmetric scheme gives a second order error in computed averages. Many symmetric methods can be constructed that require only one evaluation of the force per iteration (effectively making them as inexpensive as a first order method). 

Let us first demonstrate that the Nos6-Hoover-Langevin method, is, in fact, a thermostat. We observe that we have already shown that the Nos6-Hoover method preserves the extended canonical distribution pp /2). Therefore, by the... 

For the Nose-Hoover-Langevin method, we observe that, while small values of IX increase the high frequency oscillations observed in the kinetic energy, the rate of convergence is essentially determined by y and die average kinetic energy indeed converges in the timescale of these simulations. The presence of the stochastic... 

Observations in Chap. 7 and our experience with the Nos6-Hoover-Langevin method lead us to propose splitting methods to separate the Ornstein-Uhlenbeck processes... 

Extended Variable Momentum-Conserving Thermostats Pairwise Nose-Hoover/Nose-Hoover-Langevin Methods... 

To improve ergodicity, it is intuitive to employ the same stochastic device used in the Nos6-Hoover-Langevin method, namely linear (OU-type) stochastic terms introduced in the equation for the auxiliary variable If the OU process is properly tuned the extended canonical distribution with density will still be preserved. 

If the parameter yp can be kept small, the Langevin method not only generates the correct canonical distribution but also dynamical properties are accurately described [60]. In order to achieve this goal, the error of the forces can be consistently reduced by noting that the last self-consistent cycle can be interpreted as a Harris functiOTial. The missing force Fbo — F c can then be approximated to a high degree by... 

Using the Langevin method, Kiihne et al. were able to accelerated their benchmark calculations on liquid Si02 by one to two orders of magnitude. In 2009, Kiihne et al. simulated liquid water with the new method [60]. Oxygen-oxygen radial distribution functions agreed well with other approaches. Because of the acceleration they were able to estimate for the first time reliably the diffusion coefficient and shear viscosity of liquid water. 

If the noise term is turned off, the system is driven towards the nearest saddle point. Therefore, the same set of equations can be used to find and test mean-field solutions. The complex Langevin method was first applied to dense melts of copolymers [74], and later to mixtures of homopolymers and copolymers [80] and to diluted polymers confined in a slit under good solvent conditions [77]. Figure 2 shows examples of average density configurations (p ) for a ternary block copolymer/homopolymer system above and below the order/disorder transition. 

Compared to the Complex Langevin method, the Monte Carlo method has the advantage of being well foimded theoretically. However, it can become very inefficient when spreads over a wide range and the reweighting factor oscillates strongly, hi practice, it relies on the fact that the integral is indeed dominated by one (or several) saddle points. 

We should note that the Monte-Carlo simulation with tw = 0 effectively samples the EP Hamiltonian. This version of field-theoretic Monte Carlo is equivalent to the real Langevin method (EPD), and can be used as an alternative. Monte Carlo methods are more versatile than Langevin methods, because an almost unlimited number of moves can be invented and implemented. In our applications, the W and tw-moves simply consisted of random increments of the local field values, within ranges that were chosen such that the Metropolis acceptance rate was roughly 35%. In principle, much more sophisticated moves are conceivable, e.g., collective moves or combined EPD/Monte Carlo moves (hybrid moves [84]). On the other hand, EPD is clearly superior to Monte Carlo when dynamic properties are studied. This will be discussed in the next section. 

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