Equation Langevin

The same result as eqn (3.80) is obtained from die Langevin equation. For the potential (eqn (3.71)), the Langevin equation is written as 

Finally we derive an explicit form of the Green fimction G(x,x t). Again we use the same argument as used in deriving eqn (3.36). Since x(t) is a linear combination of f(t), the distribution of x is Gaussian, which is generally written as 

To calculate these quantities, we solve the Langevin equation under the initial condition x(0) =x  

An equivalent approach to the Fokker-Planck equation for determining the time evolution of a stochastic process is based on the Langevin equation. 

In Chapter 11, we discussed how Paul Langevin devised an equation that describes the motion of a Brownian particle (Eq. 11.33). Similar equations can be devised to describe the time evolution of any stochastic process X t). 

Function f(X, t) is called the drift term and function g X, t) is called the diffusion term. The Langevin force F(0 is generally assumed to be 

Practically, T is defined as a normal random variable with zero mean and variance ofl/dt. Concisely, we write 

Let x t) be the temporal variable of the excitable system which is also subject to external noise. A corresponding differential equation for the time evolution of x t) is called a Langevin-equation and includes random parts. We specify to the situation where randonmess is added linearly to modify the time derivative of x t), i.e. 

The solution of eq. 1.4 depends on the sample of (I). Formally one can interprete the latter as a time dependent parameter and the variable x t) is found by integration over time. We note that integrals over need a stochastic definition and are defined via the existence of the moments [50]. For this purpose the moments of (t) have to be given. 

In practical applications one uses Gaussian sources (I) or the so called Markovian random telegraph process. For both the formulation of the mean and the correlation function is sufficient to define the stochastic process. Later on we will define the value support of t) (Gaussian or dichotomic) and will give the mean and the correlation function, i.e. 

Here we reduced to stationary noise sources. Without loss of generality the mean is set to zero. For the later on considered types of noise this formulation is sufficient to obtain general answers for ensembles and their averages of the stochastic excitable system. Thus we can formulate evolution laws for the probability densities and the other moments. We note that the generalization to cases with more than one noise sources is straightforward and crosscorrelations between the noise source have to be defined. 

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If we now average the Langevin equation, (A3.1.56). we obtam a very simple equation for (v(0), whose solution is clearly... 

With the fomi of free energy fiinctional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3.48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

T) I c)t for model B. In temis of these variables the model B Langevin equation can be written as... 

Some features of late-stage interface dynamics are understood for model B and also for model A. We now proceed to discuss essential aspects of tiiis interface dynamics. Consider tlie Langevin equations without noise. Equation (A3.3.57) can be written in a more general fonn ... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

Zwanzig R 1973 Nonlinear generalized langevin equations J. Stat. Phys. 9 215-20... 

In the limit of a very rapidly fluctuating force, the above equation can sometimes be approximated by the simpler Langevin equation... 

Poliak E 1990 Variational transition state theory for activated rate processes J. Chem. Phys. 93 1116 Poliak E 1991 Variational transition state theory for reactions in condensed phases J. Phys. Chem. 95 533 Frishman A and Poliak E 1992 Canonical variational transition state theory for dissipative systems application to generalized Langevin equations J. Chem. Phys. 96 8877... 

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

An appropriate value of 7 for a system modeled by the simple Langevin equation can also be determined so as to reproduce observed experimental translation diffusion constants, Dt in the diffusive limit, Dt is related to y hy Dt = kgTmy. See [22, 36], for example. 

Fig. 7. Writhe distributions for closed circular DNA as obtained by LI (see Section 4.1) versus explicit integration of the Langevin equations. Data are from [36].

Discretizing the Langevin equation (2,3) by IE produces the following system which implicitly, rather than explicitly, defines in terms of quantities... 

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

We recently received a preprint from Dellago et al. [9] that proposed an algorithm for path sampling, which is based on the Langevin equation (and is therefore in the spirit of approach (A) [8]). They further derive formulas to compute rate constants that are based on correlation functions. Their method of computing rate constants is an alternative approach to the formula for the state conditional probability derived in the present manuscript. 

We further note that the Langevin equation (which will not be discussed in detail here) is an intermediate between the Newton s equations and the Brownian dynamics. It includes in addition to an inertial part also a friction and a random force term ... 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

To improve the accuracy of the solution, the size of the time step may be decreased. The smaller is the time step, the smaller are the assumed errors in the trajectory. Hence, in contrast (for example) to the Langevin equation that includes the friction as a phenomenological parameter, we have here a systematic way of approaching a microscopic solution. Nevertheless, some problems remain. For a very large time step, it is not clear how relevant is the optimal trajectory to the reality, since the path variance also becomes large. Further-... 

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) ... 

To integrate the Langevin equation, HyperChem uses the method of M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987 Ch.9, page 261 ... 

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. 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

The basic equation of motion for stochastic dynamics is the Langevin equation. 

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. 

The classical motion of a particle interacting with its environment can be phenomenologically described by the Langevin equation... 

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),... 

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],... 

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