Stochastic, equations variables

Huller and Baetz [1988] have undertaken a numerical study of the role played by shaking vibrations. The vibration was supposed to change the phase of the rotational potential V (p — a(t)). The phase a(t) was a stochastic classical variable subject to the Langevin equation... 

Stochastic equation (A8.7) is linear over SP and contains the operators La and V.co of differentiation over time-independent variables Q and co. Therefore, if we assume that the time fluctuations of the liquid cage axis orientation Z(t) are Markovian, then the method used in Chapter 7 yields a kinetic equation for the partially averaged distribution function P(Q, co, t, E). The latter allows us to calculate the searched averaged distribution function... 

In a stochastic approach, one replaces the difficult mechanical equations by stochastic equations, such as a diffusion equation, Langevin equation, master equation, or Fokker-Planck equations.5 These stochastic equations have fewer variables and are generally much easier to solve than the mechanical equations, One then hopes that the stochastic equations include the significant aspects of the physical equations of motion, so that their solutions will display the relevant features of the physical motion. 

The set of stochastic equations given by (3.37) is equivalent (in the linear case) to equations (3.11) with the memory functions defined in Section 3.3, but, in contrast to equations (3.11), set (3.37) is written as a set of Markov stochastic equations. This enables us to determine the variables that describe the collective motion of the set of macromolecules. In this particular approximation, the interaction between neighbouring macromolecules ensures that the phase variables of the elementary motion are co-ordinates, velocities, and some other vector variables - the extra forces. This set of phase variables describes the dynamics of the entire set of entangled macromolecules. Note that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approximations of the memory functions. We are considering the case with a single relaxation time, but generalisation for a case with a few relaxation times is possible. 

The system described by Eqs. (5.7) is equivalent to the stochastic equation in the variable x alone. 

The subject of multiplicative fluctuations (in linear and especially nonlinear systems) is still deeply fraught with ambiguity. The authors of Chapter X set up an experiment that simulates the corresponding nonlinear stochastic equations by means of electric circuits. This allows them to shed light on several aspects of external multiplicative fluctuation. The results of Chapter X clearly illustrate the advantages resulting from the introduction of auxiliary variables, as recommended by the reduced model theory. It is shown that external multiplicative fluctuations keep the system in a stationary state distinct from canonical equilibrium, thereby opening new perspectives for the interpretation of phenomena that can be identified as due to the influence of multiplicative fluctuations. 

We perform concrete calculations in the complex P-representation [Drummond 1980 McNeil 1983] in the frame of both probability distribution functions and stochastic equations for the complex c-number variables. We follow the standard procedures of quantum optics to eliminate the reservoir operators and to obtain a master equation for the density operator of the modes. The master equation is then transformed into a Fokker-Planck equation for the P-quasiprobability distribution function. In particular, for an ordinary NOPO and in the case of high cavity losses for the pump mode (73 7), if in the operational regime the pump depletion effects are involved, this approach yields... 

In Secs. 3 and 4 we will use the stochastic equations in the positive P-representation for the complex c-number variables o2 and fk corresponding to the operators a and a. For the generalized model of NOPO they read as ... 

The stochastic equations for two groups of independent complex c-number variables a 2) and Pi(2) corresponding to operators a1(2) and aj(2) for the case of zero detunings have the form ... 

The relevance of stochastic descriptions brings out the issue of their theoretical and numerical evaluation. Instead of solving the equations of motion for 6x102 degrees of freedom we now face the much less demanding, but still challenging need to construct and to solve stochastic equations of motion for the few relevant variables. The next section describes a particular example. 

As discussed in Section 8.2.1, the Langevin equation (8.13) describes a Markovian stochastic process The evolution of the stochastic system variable x(Z) is determined by the state of the system and the bath at the same time t. The instantaneous response of the bath is expressed by the appearance of a constant damping coefficient y and by the white-noise character of the random force 7 (Z). 

When the relaxation is not overdamped we need to consider the full Kramers equation (14.41) or, using Eqs (14.42) and (14.43), Eq. (14.44) forf. In contrast to Eq. (14.45) that describes the overdamped limit in terms of the stochastic position variable x, we now need to consider two stochastic variables, x and v, and their probability distribution. The solution of this more difficult problem is facilitated by invoking another simplification procedure, based on the observation that if the... 

To generate the trajectories that result from stochastic equations of motion (14.39) and (14.40) one needs to be able to properly address the stochastic input. For Eqs (14.39) and (14.40) we have to move the particle Linder the influence of the potential T(.v), the friction force—yvm and a time-dependent random force R(t). The latter is obtained by generating a Gaussian random variable at each time step. Algorithms for generating realizations of such variables are available in the applied mathematics or numerical methods hterature. The needed input for these algorithms are the two moments, (2J) and In our case (7 ) = 0, and (cf. Eq. (8.19)) = liiiyk/jT/At. where Ai is the time interval... 

Set (7.1) consists of two vector and one scalar stochastic equation. This set is helpful in examining statistical properties of two vector (v and w ) and one scalar (p ) random unknown variable as functions of 1) the statistical properties of random variable 2) the macroscopic characteristics of suspension flow, and 3) physical parameters. Since Equations 7.1 are linear, it is natural to use the correlation theory of random processes when investigating these vector and scalar variables in terms of those functions [35], Particulars of necessary calculation are described at considerable length in reference [14,25). Here, we confine ourselves to only a brief enumeration of the major logical steps of this calculation. 

Let a limit cycle oscillator be exposed to some weak random forces which may depend on the state variable X. The governing equation is a nonlinear stochastic equation ... 

One is the time-dependent Ginzburg-Landau equation which is described by a complex order parameter and vector potential. The other is the Langevin-type stochastic equation of motion for magnetic vortices in two and three dimensions, which is described in terms of vortex position variables. 

The phase a(r) was a stochastic classical variable subject to the Langevin equation... 

Starting from the master equation (3.14) and after the introduction of scaled variables (3.25) the stochastic equation (3.29) and the Fokker-Planck equation... 

Now the density matrix does not inelude stochastic lattice variables and depends only on spin variables. As a result, in the case of the nitroxide spin label of the electron spin coupled to a nuclear spin of N, the SLE is redueed to a system of only nine eoupled differential equations. The size of the system thus does not depend on the nature of the... 

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... 

A single experiment consists of the measurement of each of the m response variables for a given set of values of the n independent variables. For each experiment, the measured output vector which can be viewed as a random variable is comprised of the deterministic part calculated by the model (Equation 2.1) and the stochastic part represented by the error term, i.e.,... 

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