Stochastic equations of motion

Agreed what movements each should generate. 

Lucretius (c.99-e.55 bce The way things are translated by Rolfe Humphries, Indiana University Press, 1968 

We have already observed that the frill phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the deterministic Newton (or Sclrrbdinger) equations of motion, while the time evolution of a small subsystem is stochastic in nature. Focusing on the latter, we would like to derive or construct appropriate equations of motion that will describe this stochastic motion. This chapter discusses some methodologies used for this purpose, focusing on classical mechanics as the underlying dynamical theory. In Chapter 10 we will address similar issues in quantum mechanics. 

The time evolution of stochastic processes can be described in two ways  

rime evolution in probability space. In this approach we seek an equation (or equations) for the time evolution of relevant probability distributions. In the most general case we deal with an infinite hierarchy of functions, Piz t -,z - tn-U, ziti) as discussed in Section 7.4.1, but simpler cases exist, for example, for Markov processes the evolution of a single function, Piz, f,ZQtQ), fully characterizes the stochastic dynamics. Note that the stochastic variable z stands in general for all the variables that determine the state of our system. 

Not with wise intelligence imposed An order on themselves, nor in some pact Agreed what movements each should generate. 

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Depending on the desired level of accuracy, the equation of motion to be numerically solved may be the classical equation of motion (Newton s), a stochastic equation of motion (Langevin s), a Brownian equation of motion, or even a combination of quantum and classical mechanics (QM/MM, see Chapter 11). 

We will later further analyze the members of (3.7) as they stand, but it is useful for our subsequent discussion to now simply add a generalized dissipative term to the solvent equation of motion to obtain the stochastic equation of motion set... 

This calculation illustrates a point made in the introduction. We have been able to calculate a well-defined theoretical spectrum to a specified range of reliability. We used the rigorous mechanical equations of motion for a large spin system without actually undertaking the hopeless task of solving them, and without resorting to a phenomenological or stochastic equation of motion. 

When the constant field is weak and the fluctuating field is comparable to or even larger than the constant field, the above decomposition becomes meaningless. There is no way of distinguishing between the adiabatic and nonadiabatic effects. In order to obtain an understanding of this rather complex situation, we have examined a stochastic model,14 extending the theory in Section II. The stochastic equation of motion of a spin in a random local field is written as... 

Equation (57) is the stochastic equation of motion for r(/), in which the matrix element 2(/) is a random process. This is similar to Eq. (2). This may be written as a stochastic Liouville equation in the form... 

The stochastic equation of motion of v(t), Eq. (77), can be transformed into a stochastic Liouville equation of the type Eq. (7) if a Markovian process can be properly defined to generate the process of H(t). Then we again obtain Eq. (63) for the conditional expectation V(t) defined by Eq. (60). The line shape function is then given by... 

The GLE is a stochastic equation of motion for the coordinate z (see Figure 24). The left-hand side of Eq. (41) is the inertial force along z in terms of the effective polarization mass m of the solvent and the acceleration z. The term... 

In our approach [1, 2] termed the dynamic method the complex susceptibility x = x — ix" is determined by a law of undamped motion of a dipole in a given potential well and by dissipation mechanism often described as stosszahlansatz in the underlying kinetic or Boltzmann equation. In this review we shall refer to this (dynamic) method as the ACF method, since it is actually based on calculation of the spectrum of the dipolar autocorrelation function (ACF). Actually we use a one-particle approximation, in which the form of an employed potential well (being in many cases rectangular or close to it) is taken a priori. Correlation of the particles coordinates is characterized implicitly by the Kirkwood correlation factor g, its value being taken from the experimental data. The ACF method is simple and effective, because we do not employ the stochastic equations of motions. This feature distinguishes our method from other well-known approaches—for example, from those described in books [13, 14]. 

The system evolves according to deterministic or stochastic equations of motion. 

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. 

Two routes can be taken to obtain such stochastic equations of motions, of either kind ... 

As discussed in Section 8.1, we could attempt to derive these stochastic equations of motion from first principles, that is, from the full Hamiltonian of the system+environment. Alternatively we can attempt to construct the equation of motion using intuitive arguments and as much of the available physical information as possible. Again, this section takes the second route. As an example consider the equation of motion of a particle moving in a one-dimensional potential. 

The stochastic equation of motion (8.13) was introduced as a phenomenological model based on the combination of experience and intuition. We shall now attempt to derive such an equation from first principles, namely starting from the Newton... 

Equation (8.54) is a stochastic equation of motion similarto Eq. (8.13). However, we see an important difference Eq. (8.54) is an integro-differential equation in which the term yx of Eq. (8.13) is replaced by the integral /J drZ t — r)x(r). At the same time the relationship between the random force R t) and the damping, Eq. (8.20), is now replaced by (8.59). Equation (8.54) is in fact the non-Markovian generalization of Eq. (8.13), where the effect of the thermal environment on the system is not instantaneous but characterized by a memory—at time t it depends on the past interactions between them. These past interactions are important during a memory time, given by the lifetime of the memory kernel Z t). The Markovian limit is obtained when this kernel is instantaneous... 

Our aim is to find the corresponding equation for P x, t), the probability density to find the particle position at x the velocity distribution is assumed equilibrated on the timescale considered. Note that in Section 8.1 we have distinguished between stochastic equations of motion that describe the time evolution of a system in state space (here x), and those that describe this evolution in probability space. We now deal with the transformation between such two descriptions. 

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... 

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