Langevin approach

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

The Langevin approach is widely used for the purpose of finding the effect of fluctuations in macroscopically known systems. The fluctuations are introduced by adding random terms to the equations of motion, called noise sources . This approach is popular because it gives a more concrete picture than the Fokker-Planck equation, but it is mathematically equivalent to it. In nonlinear cases it is subject to the same difficulties, and some additional ones. 

These three steps constitute what I call the Langevin approach . It is widely utilized, also when the fluctuations are not due to thermal motion or to discreteness of particles, and even when they are of unspecified or unknown provenance. The applications in many branches of physics, chemistry and biology are far too numerous to list. The approach has been highly successful whenever the deterministic equations are linear, but for nonlinear cases it leads to difficulties, which are analyzed in this section. The purpose is to shield the reader from the labyrinthine literature and to convince him of the necessity of the more firmly based treatment in the next chapter. 

More generally, suppose one has a physical system with a nonlinear equation of motion y = A(y) and, following the Langevin approach, one adds a Langevin term to describe the fluctuations in the system,... 

Although we have found that for internal noise the Ito-Stratonovich dilemma is undecidable for lack of a precise A(t) there are cases in which the Ito equation seems the more appropriate option. As an example we take the decay process defined in IV.6 the M-equation is (V.1.7) and the average obeys the radioactive decay law (V.1.9). As the jumps are relatively small one may hope to describe the process by means of a Langevin equation. Following the Langevin approach we guess... 

This Ansatz is the essential step. The -expansion is not just one out of a plethora of approximation schemes, to be judged by comparison with experimental or numerical results 0. It is a systematic expansion in and is the basis for the existence of a macroscopic deterministic description of systems that are intrinsically stochastic. It justifies as a first approximation the standard treatment in terms of a deterministic equation with noise added, as in the Langevin approach. It will appear that in the lowest approximation the noise is Gaussian, as is commonly postulated. In addition, however, it opens up the possibility of adding higher approximations. 

Exercise. The density fluctuations in diffusion were first calculated by van Vliet ) using the following Langevin approach. The density u and current J are assumed to obey... 

Here a is the differential cross-section, and depends only on Pi Pi = l/>3 Pa and on (/U - p2) p2 Pa)-The precise number of molecules in the cell fluctuates around the value given by the Boltzmann equation, because the collisions occur at random, and only their probability is given by the Stosszahlansatz. Our aim is to compute these fluctuations. If / differs little from the equilibrium distribution one may replace the Boltzmann equation by its linearized version. It is then possible to include the fluctuations by adding a Langevin term, whose strength is determined by means of the fluctuation-dissipation theorem.510 As demonstrated in IX.4, however, the Langevin approach is unreliable outside the linear domain. We shall therefore start from the master equation and use the -expansion. The whole procedure consists of four steps. 

Langevin approach. It may be skipped by the reader who is satisfied with the subsequent appeal to the mathematical foundation. 

Our answer to this difficulty is the following. Consider the equilibrium of the total system S + B, compute the fluctuations occurring in S and their time correlations. This provides a more consistent way for obtaining the information for which the Langevin approach was intended. The purpose of this section is to implement these general remarks by explicit calculations. Unfortunately they are rather laborious I apologize. 

Now we show that the validity of the cascade Langevin approach extends beyond the limits of validity of Boltzmann equation. Consider a frequency-dependent noise in a chaotic cavity, i.e. in a metallic island of irregular shape connected to the electrodes L, R via two quantum point contacts of conductances Gl,r e1 /h and arbitrary transparencies / /. . As the dwell time of... 

Fig. 15. Calculated values of d In W(r)/dr as a function of relative elongation r/nl for DGER molecules. 1 Langevin approach 2-4 Monte-Carlo analysis with energy difference between trans- and gauche-isomers E, = Eg (2) Et - E, = 0.5 kcal/mol (3) Et — Es = —0.5 kcal/mol...

IV. MOLECULAR DYNAMICS A CHALLENGE TO THE FOKKER-PLANCK AND LANGEVIN APPROACHES... 

The Langevin approach was also used for the oxidase system [38] and showed that in addition to an equilibrium distribution of on times also a... 

The quantitative agreement with simulations of this work is not found with existing models. For future developments the most important conclusion is that the influence of nonlinear coupling is accounted for by a correction factor that dep>ends only on the particle masses independent of the system energy. While a generalization of this factor will be required for polyatomics (. involves the masses of all three atoms for a diatomic in a bath and is not a property of atom-atom interactions), it indicates that a scaled generalized Langevin approach will have the necessary prerequisites for quantitative accuracy. 

The Langevin approach has been used by many authors in order to treat nonlinear systems. This is of importance to us since the equations of rotational motion are intrinsically nonlinear. The concept of a nonlinear Langevin equation is also subject to a number of criticisms. These have been discussed extensively by van Kampen [58] (Chapters 8 and 14). In our calculations, we shall encounter stochastic differential equations of the form... 

The Kirkwood method (or Fokker-Planck approach) has been compared with the Langevin approach by Zwanzig (80). 

According to Eq. 2, the magnetization of a ferrogel is directly proportional to the concentration of magnetic particles and their saturation magnetization. It is worth mentioning that the Langevin approach can be used either ... 

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