Langevin equation differential equations

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

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... 

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. 

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. 

From (5.56) one can obtain an integro-differential equation for operator What we need is the mean particle position, <(Tz>, and in order to find it two approximations are made. First, in taking the bath averages we assume free bath dynamics. Second, we decouple the bath and pseudospin averages, guided by perturbation theory. The result is a Langevin-like equation for the expectation <(T2> [Dekker 1987a Meyer and Ernst 1987 Waxman 1985],... 

The exponential decay predicted by the Onsager-Machlup theory, and by the Langevin and similar stochastic differential equations, is not consistent with the conductivity data in Fig. 8. This and the earlier figures show a constant value for >.(x) at larger times, rather than an exponential decay. It may be that if the data were extended to significantly larger time scales it would exhibit exponential decay of the predicted type. 

Lagrange multipliers 255-256 Lagrange s moan-value theorem 30-32 Lagperre polynomials 140, 360 Lambert s law 11 Langevin function 61n Laplace transforms 279—286 convolution 283-284 delta function 285 derivative of a function 281-282 differential equation solutions 282-283... 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

The present analysis builds directly on three previous analyses of SDEs for constrained systems by Fixman [9], Hinch [10], and Ottinger [11]. Fixman and Hinch both considered an interpretation of the inertialess Langevin equation as a limit of an ordinary differential equation with a finite, continuous random force. Both authors found that, to obtain the correct drift velocity and equilibrium distribution, it was necessary to supplement forces arising from derivatives of C/eff = U — kT n by an additional corrective pseudoforce, but obtained inconsistent results for the form of the required correction force. Ottinger [11] based his analysis on an Ito interpretation of SDEs for both generalized and Cartesian coordinates, and thereby obtained results that... 

In either interpretation of the Langevin equation, the form of the required pseudoforce depends on the values of the mixed components of Zpy, and thus on the statistical properties of the hard components of the random forces. The definition of a pseudoforce given here is a generalization of the metric force found by both Fixman [9] and Hinch [10]. An apparent discrepancy between the results of Fixman, who considered the case of unprojected random forces, and those of Hinch, who was able to reproducd Fixman s expression for the pseudoforce only in the case of projected random forces, is traced here to an error in Fixman s use of differential geometry. 

In the traditional interpretation of the Fangevin equation for a constrained system, the overall drift velocity is insensitive to the presence or absence of hard components of the random forces, since these components are instantaneously canceled in the underlying ODF by constraint forces. This insensitivity to the presence of hard forces is obtained, however, only if both the projected divergence of the mobility and the force bias are retained in the expression for the drift velocity. The drift velocity for a kinetic interpretation of a constrained Langevin equation does not contain a force bias, and does depend on statistical properties of the hard random force components. Both Fixman and Hinch nominally considered the traditional interpretation of the Langevin equation for the Cartesian bead coordinates as a limit of an ordinary differential equation. Both authors, however, neglected the possible existence of a bias in the Cartesian random forces. As a result, both obtained a drift velocity that (after correcting the error in Fixman s expression for the pseudoforce) is actually the appropriate expression for a kinetic interpretation. 

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. 

The Langevin equation is the prototype of a stochastic differential equation, i.e., a differential equation whose coefficients are random functions of time with given stochastic properties, see chapter XVI. It defines V(t) as a stochastic process when an initial condition is supplied, say F(0) = F0. Then... 

The traditional derivation of the Fokker-Planck equation (1.5) or (VIII. 1.1) is based on Kolmogorov s mathematical proof, which assumes infinitely many infinitely small jumps. In nature, however, all jumps are of some finite size. Consequently W is never a differential operator, but always of the type (V.1.1). Usually it also has a suitable expansion parameter and has the canonical form (X.2.3). If it then happens that (1.1) holds, the expansion leads to the nonlinear Fokker-Planck equation (1.5) as the lowest approximation. There is no justification for attributing a more fundamental meaning to Fokker-Planck and Langevin equations than in this approximate sense. 

One example of a stochastic differential equation is the Langevin equation (IX.1.1). In fact, in most of the mathematical literature the name stochastic differential equation is restricted to this equation, which they call ltd equation. The emphasis on this special case is misdirected. If the equation... 

I. Linear differential equations in which only the inhomogeneous term is a random function, such as the Langevin equation. Such equations have been called additive and can be solved in principle. 

Setting W(t) = (2kBT )l/2W(t), the Langevin relation—describing the motion of the particle in the phase space—can be written as a two-variable stochastic differential equation system of the form of... 

The picture offered by the Fokker-Planck equation is, of course, in complete agreement with the Langevin equation and the assumptions made about the process. If we can solve the partial differential equation we can determine the probability density or eventually the transition probabilities at any time, and thereby determine any average value of functions of v by simple quadratures. 

B. Two-Dimensional Stochastic Differential Equation of the Langevin Type...66... 

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