The random force term

[ "article:topic", "Author tag:Tuckerman", "showtoc:no" ]

Within the context of a harmonic bath, the term random force'' is something of a misnomer, since $$R (t)$$ is completely deterministic and not random at all!!! We will return to this point momentarily, however, let us examine particular features of $$R (t)$$ from its explicit expression from the harmonic bath dynamics. Note, first of all, that it does not depend on the dynamics of the system coordinate $$\underline {q}$$ (except for the appearance of $$q (0)$$). In this sense, it is independent or orthogonal'' to $$\underline {q}$$ within a phase space picture. From the explicit form of  $$R(t)$$, it is straightforward to see that the correlation function

$\langle \dot{q}(0)R(t)\rangle = 0$

i.e., the correlation function of the system velocity $$\underline {\dot {q} }$$ with the random force is 0. This can be seen by substituting in the expression for $$R (t)$$ and integrating over initial conditions with a canonical distribution weighting. For certain potentials $$\phi (q)$$ that are even in $$\underline {q}$$ (such as a harmonic oscillator), one can also show that

$\langle q(0)R(t)\rangle = 0$

Thus, $$R (t)$$ is completely uncorrelated from both $$\underline {q}$$ and $$\underline {\dot {q}}$$, which is a property we might expect from a truly random process. In fact, $$R(t)$$ is determined by the detailed dynamics of the bath. However, we are not particularly interested or able to follow these detailed dynamics for a large number of bath degrees of freedom. Thus, we could just as well model $$R(t)$$ by a completely random process (satisfying certain desirable features that are characteristic of a more general bath), and, in fact, this is often done. One could, for example, postulate that $$R (t)$$ act over a maximum time $$\underline {t_{max}}$$ at discrete points in time $$k \Delta t$$, giving $$N=t_{\rm max}/\Delta t$$ values of $$R_k=R(k\Delta t)$$, and assume that $$R_k$$ takes the form of a gaussian random process:

$R_k = \sum_{j=1}^N\left[a_je^{2\pi ijk/N} + b_je^{-2\pi ijk/N}\right]$

where the coefficients $$\{a_j\}$$ and $$\{b_j\}$$ are chosen at random from a gaussian distribution function. This might be expected to be suitable for a bath of high density, where strong collisions between the system and a bath particle are essentially nonexistent, but where the system only sees feels the relatively soft'' fluctuations of the less mobile bath. For a low density bath, one might try modeling $$R (t)$$ as a Poisson process of very strong collisions.

Whatever model is chosen for $$R (t)$$, if it is a truly random process that can only act at discrete points in time, then the GLE takes the form of a stochastic (based on random numbers) integro-differential equation. There is a whole body of mathematics devoted to the properties of such equations, where heavy use of an It calculus is made.