Stratonovich integral

Next, unlike in the case of the Riemann sums, it turns out that the result depends on where we place the points tk. Using the properties of the Wiener process we may easily show that the choice 4 = kSt (left endpoint) and the choice tk = (k+ l/2)St yield different formulas for the integral. The first choice is commonly referred to as ltd integration, and the second as Stratonovich integration. As an example, consider the case where g(t) = W(t). We suppose this to be approximated by a sum... 

Plainly this is different than the Ito version of the integral, but in the cases we consider it is possible to reformulate Stratonovich integrals as Ito integrals. In the remainder of this book we will adopt the Ito form of stochastic integration. 

A proper explanation of Stratonovich integration is somewhat outside the scope of this book. 

The function a(x, t) appearing in the FPE is called the drift coefficient, which, due to Stratonovich s definition of stochastic integral, has the form [2]... 

The present analysis follows the approach taken by aU three of these authors, in which SDEs are constructed by choosing the drift and diffusivity coefficients so as to yield a desired diffusion equation. Peters [13] has pioneered an alternative approach, in which expressions for the drift and diffusivity are derived from a direct, but rather subtle, analysis of the underlying inertial equations of motion, in which (for rigid systems) he integrates the instantaneous equations of motion over time intervals much greater than the autocorrelation time of the particle velocities. Peters has expressed his results both as standard Ito SDEs and in a nonstandard interpretation that he describes heuristically as a mixture of Stratonovich and Ito interpretations. Peters mixed Ito—Stratonovich interpretation is equivalent to the kinetic interpretation discussed here. Here, we recover several of Peters results, but do not imitate his method. 

We now instead calculate the drift velocity and diffusivity by directly integrating the traditional formulation of the Langevin equation in terms of random forces, and compare the results to those obtained above by rewriting the Langevin equation as a standard Stratonovich SDE. As in the analysis of the Stratonovich SDE, we calculate the first and second moments of an increment AX (f) = Z (f) — X (0) by integrating Eq. (2.262) from a known initial condition at f = 0. 

Using a Langevin dynamics approach, the stochastic LLG equation [Eq. (3.46)] can be integrated numerically, in the context of the Stratonovich stochastic calculus, by choosing an appropriate numerical integration scheme [51]. This method was first applied to the dynamics of noninteracting particles [51] and later also to interacting particle systems [13] (see Fig. 3.5). 

Here a = a t - 1), t = T/T°P, and a, b, c are either phenomenological parameters or else they can be calculated from mean-field calculations [48] or better yet, by using the Hubbard-Stratonovich transformation to convert the partition function into a functional integral [39]. In the latter case, one obtains around t = 1 the values... 

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too irregular for Riemann integrals to be applied. Application of Stieltjes integration yields a dependence of the moments on how the limit to white noise is taken. If t) is the limit of the Ornstein-Uhlenbeck -process with r —> 0 (Stratonovich sense) the coefficients read [50]... 

In contrast, the Stratonovich (midpoint) integral (denoted by odW(f)) can be calculated as... 

The integral with respect to the Wiener process is taken using the Ito approach that has an important advantage over the Stratonovich approach, namely the integral in this case will be a martingale. Definitions and... 

The partition function Eq. 6 describes a system of mutually interacting chains. Introducing auxiliary fields, U and W, via a Hubbard-Stratonovich transformation, one can decouple the interaction between the chains and rewrite the Hamiltonian in terms of independent chains in fluctuating fields. Then, one can integrate over the chain conformations and obtain a Hamiltonian which only depends on the auxiliary fields. Thermodynamic averages like density or structure factors can be expressed as averages over the fields, U and W, without approximation. 

Historically, the two major interpretations are from Ito and Stratonovich. In both these formulations, (x) = 5(x) l However, (x) = A(x) in Ito interpretation while (x) = A(x)-jd B(x) in the Stratonovich interpretation. From the practical perspective, Ito interpretation allows one to simulate the SDE using the usual forward Euler scheme. However, special differentiation and integration rules are required for analytical calculations. On the other hand, Stratonovich interpretation allows using the regular rules of calculus but has to be simulated using implicit schemes. We emphasize that the FPE does not suffer from such ambiguity of interpretation SDEs corresponding to different interpretations of the same FPE lead to the same physical results [3, 7]. 

A number of attempts have been made to incorporate the effect of composition fluctuations [86-88] in theories involving neutral polymers. Here, we present systematic one-loop expansion to go beyond the saddle-point approximation described in the previous section. In order to carry out the loop expansion, it is advantageous to use Hubbard-Stratonovich transformation to get rid of redundant functional integrals over collective density variables (q in Eq. (6.85)) and use Eq. (6.81) as the starting point for the partition function with the explicitly known normalization constants except A,. Saddle-point approximation within this formalism now requires taking functional derivatives with respect to fields only. 

Now, using the methods of collective variables (cf. Section 6.4.2.1) for decoupling all the interactions except the electrostatics and the Hubbard-Stratonovich transformation [14, 55] (cf. Section 6.4.2.2) for the electrostatic part in Eq. (6.124), the partition function can be written as integrals over the collective densities and corresponding fields so that Eq. (6.124) becomes... 

Virtually all of the successful path integral simulations of 2-d models for electronic systems have been carried out by the auxiliary field MC method, sometimes called the determinantal method. The only thing that complicates the computation of the fermion partition in equation (8) is the interaction action 5i. As explained in Section 5.3, without 5i, the sum over exchanges can be performed analytically. Therefore, if the two-electron interaction term can be eliminated or at least decoupled, the fermion sign problem could be partially removed. This can be accomplished by a so-called Hubbard-Stratonovich transformation. The details can be found in the original paper. Briefly, two electrons (of opposite spin) on the same site i experience a repulsion of strength U and add a term —eUni ni to the action Si, where = 0, 1 is the occupation number of an f-spin electron on site i, and n, is the same for a -spin electron. To decouple the two-electron interaction, the following transformation (correct up to a multiplicative constant) can be used. 

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