In this section we combine and rewrite the statistical conditions in order to obtain a single relation that will be useful in later sections .
We begin with the first statistical condition (6). The following calculation is very similar to the corresponding one-dimensional calculation reported in I; thus details may be omitted. We insert the definition (8) in (6) and replace the derivative of with respect to by the second term in the continuity equation (5). Using Gauss’ integral theorem and assuming that vanishes sufficiently rapidly for in order for the surface integral to vanish (we may even assume that it vanishes faster than an arbitrary finite power of ) we arrive at the following expression for the expectation value of the momentum
This relation agrees essentially with the corresponding one-dimensional relation obtained in I.
We study now the implications of the second statistical condition (7). We start by evaluating the left hand side of (7). Using the variables it is given by
Performing the derivative with respect to Eq. (14) takes the form
Note that each term in the integrand of (15) is single-valued but is not. As a consequence the order of two derivatives of (with respect to anyone of the variables ) must not be changed. We introduce the (single-valued) quantities
to describe the non-commuting derivatives.
Evaluating the first term in the integrand of (15) we replace the time derivative of by the divergence of the probability current according to the continuity equation (5) to obtain
Performing a partial integration and exchanging the derivatives with respect to and , Eq. (17) takes the form
Using the formula
and performing a second partial integration the first term in the integrand of (15) takes the form
Similar manipulations lead to the following expression for the second term in the integrand of (15):
Let us assume that the the macroscopic force entering the second statistical condition (7) can be written as a sum of two contributions, and ,
where takes the form of a negative gradient of a scalar function (mechanical potential). Since does (in contrast to ) not depend on , its average value can be calculated with the help of a known probability distribution, namely the dynamical variable . Performing a partial integration and collecting terms the second statistical condition (7) takes the form.
Comparing Eq. (23) with the corresponding formula obtained in I [see Eq. (24) of I] we see that two new terms appear now, namely the expectation value of the dependent force on the r.h.s. and the second term on the l.h.s. of Eq. (23). The latter is a direct consequence of our assumption of a multi-valued variable .
In the next section it will be shown that for vanishing multi-valuedness Eq. (23) has to agree with the three-dimensional generalization of the corresponding result [Eq. (24) of I] obtained in I. This means that the dependent term on the r.h.s. has to vanish too in this limit and indicates a relation between multi-valuedness of and dependence of the external force.