Langevin expansion

In the case where A, 17, (VI Vo) are unity, N = and only the first term of the Langevin expansion is used, the whole expression reduces to ... 

By making the above approximations (preaveraging and uniform expansion), the Langevin equation for the polymer segment can be written as... 

Internal noise is described by a master equation. When this equation cannot be solved exactly it is necessary to have a systematic approximation method - rather than the naive Fokker-Planck and Langevin approximations. Such a method will now be developed in the form of a power series expansion in a parameter Q. In lowest order it reproduces the macroscopic equation and thereby demonstrates how a deterministic equation emerges from the stochastic description. 

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. 

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. 

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. 

Applying these methods to systems in the vicinity of the non-equilibrium critical points, the conclusion was drawn [72] that the mesoscopic approach contains excess information about spatial particle distribution the details of how the whole system s volume is divided into cells become unimportant as — oo. The possibility to employ expansion in inverse powers of vo -similarly to a complete mixing case - was also discussed. Asymptotically it leads to the Focker-Planck equation equivalent to the Langevin-like equation. 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

U(0) is the orientation dependent local potential energy and U (6) accounts for any anisotropy due to the local environment. For instance, if the sample were liquid crystalline, U (O) and U (rc) would represent the potential well associated with nematic or smectic director. In addition to local potentials it is clear that f(0) depends on the experimentally controllable quantities p, E, and T. The solution to equation 16 is the series expansion referred to as the third-order Langevin function L,3(p)... 

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

This expression may determine the relationship between ea as a function of Aa in intense fields, e.g., for x > 3, when the approximation L(x) A = 1 - l/xA applies. However, it assumes classical statistics, so it will require a quantum correction for x values in this range, assuming that the Langevin function is still applicable. For smaller values of xA, Eq. (48) may be solved graphically. A more transparent expression for small xA values would be useful. Equation (42) with m = 1/2 leads to a quartic equation in ea which may be simplified by expansion to a cubic expression. A solution is the use of Booth s approximation ea and e0 n2, with substitution ofEq. (47) into Eq. (42). This gives a simple and useful approximation ... 

For a network of Gaussian chains having the same number n of links, uniaxially stretched by an amount L/Lo = A., the assumptions of affine displacement of jimction points and initial Gaussiein distribution of end-to-end vectors allows one to calculate the optical anisotropy of the network by integrating Eq.lO over the distribution of end-to-end vectors in the stretched state. By taking Treloar s expansion [11] for the inverse Langevin function, the orientation distribution function for the network can be put into the form of a power series of the number of Unks per chain ... 

In the range of weak fields, if the first two terms in the series expansion of the Langevin function = f/3 - f /45 + 2 f /945 -. .. are used and the effect of the internal field is taken into account according to Lorenz, Eq. (84) becomes... 

D.2. The Kramers-Moyal Expansion Coefficients for Nonlinear Langevin Equations... 

This appendix is an expansion of the discussion of Section IIB into a short critical review of the standard Langevin, Onsager, Mori [1-5] slow variable model of irreversible processes. Specifically, we will outline here some basic topics in statistical and thermal physics [37], focusing on Onsager s slow variable theory of irreversible thermodynamic processes [1,3,4]. While a number of useful discussions of Onsager s theory exist in the literature [37,38], ours differs from most others in that it scrutinizes some of the theory s less frequently examined assumptions and thus identifies some of its important but not commonly discussed limitations. 

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