Statistical conditions

3 Statistical conditions

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 $ρ$ with respect to $t$ by the second term in the continuity equation (5). Using Gauss’ integral theorem and assuming that $ρ$ vanishes sufficiently rapidly for $|x | → ∞$ in order for the surface integral to vanish (we may even assume that it vanishes faster than an arbitrary finite power of $- 1 x k$ ) we arrive at the following expression for the expectation value of the momentum
$∫ ∞ ˜ 3 ∂--S-(x,---t-) d x ρ (x, t ) = p¯k . ∂ x - ∞ k$ (13)

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 $ρ, S˜$ it is given by
$∫ ∞ ˜ d ---- d 3 ∂ S (x, t) ----p = ---- d x ρ (x, t )------------- . dt k dt ∂ x - ∞ k$ (14)

Performing the derivative with respect to $t$ Eq. (14) takes the form
$[ ] ∫ ∞ d ---- 3 ∂ ρ (x, t) ∂ S˜ (x, t ) ∂ ∂ ˜S (x, t) ----p = d x -------------------------- + ρ (x, t )----------------- . dt k ∂ t ∂ x ∂ t ∂ x - ∞ k k$ (15)

Note that each term in the integrand of (15) is single-valued but $˜ S$ is not. As a consequence the order of two derivatives of $˜ S$ (with respect to anyone of the variables $xk , t$ ) must not be changed. We introduce the (single-valued) quantities
$[ ] [ ] 2 ˜ 2 ˜ 2 ˜ 2 ˜ ˜ ---∂---S----- ---∂---S----- ˜ -∂---S---- --∂---S--- S [j,k ] = - , S [0,k ] = - ∂ xj ∂ xk ∂ xk ∂ xj ∂ t∂ xk ∂ xk ∂ t$ (16)

to describe the non-commuting derivatives.

Evaluating the first term in the integrand of (15) we replace the time derivative of $ρ$ by the divergence of the probability current according to the continuity equation (5) to obtain
$∫ ∫ ∞ ∂ ρ ∂ S˜ ∞ ∂ S˜ ∂ ρ ∂ S˜ d3x ------------= - d3x -----------------------. ∂ t ∂ x ∂ x ∂ x m ∂ x - ∞ k - ∞ k j j$ (17)

Performing a partial integration and exchanging the derivatives with respect to $xk$ and $xj$ , Eq. (17) takes the form
$[ ] ∫ ∞ ˜ ∫ ∞ ˜ ˜ 3 ∂ ρ ∂ S 3 ρ ∂ S ∂ ∂ S d x ----------- = d x ----------- --------------+ S˜[j,k ] . ∂ t ∂ x m ∂ x ∂ x ∂ x - ∞ k - ∞ j k j$ (18)

Using the formula
$( ) 2 ∂ S˜ ∂ 2 ˜S ∂ ∑ ∂ ˜S 2 -------------------- = ------- ------- , ∂ xj ∂ xk ∂ xj ∂ xk ∂ xj j$ (19)

and performing a second partial integration the first term in the integrand of (15) takes the form
$∫ ∞ ∂ ρ ∂ S˜ d3x ------------ = - ∞ ∂ t ∂ xk ∫ ( )2 ∫ , ∞ ∑ ˜ ∞ ˜ 3 -∂--ρ----1--- -∂-S--- 3 --ρ----∂-S---˜ - d x + d x S [j,k ] - ∞ ∂ xk 2m ∂ xj - ∞ 2m ∂ xj j$ (20)

Similar manipulations lead to the following expression for the second term in the integrand of (15):
$∫ ∫ ∫ ∞ ∂ ∂ ˜S ∞ ∂ ρ ∂ S˜ ∞ 3 ----------- 3 ------------ 3 ˜ d x ρ = - d x + d x ρ S [0,k ], - ∞ ∂ t ∂ xk - ∞ ∂ xk ∂ t - ∞$ (21)

Let us assume that the the macroscopic force $F (x, p, t ) k$ entering the second statistical condition (7) can be written as a sum of two contributions, $(m ) F (x, t) k$ and $(e ) F (x, p, t) k$ ,
$(m ) (e) F (x, p, t) = F (x, t ) + F (x, p, t), k k k$ (22)

where $(m ) F (x, t ) k$ takes the form of a negative gradient of a scalar function $V (x, t)$ (mechanical potential). Since $(m ) F k$ does (in contrast to $(e ) F k$ ) not depend on $p$ , its average value can be calculated with the help of a known probability distribution, namely the dynamical variable $ρ$ . Performing a partial integration and collecting terms the second statistical condition (7) takes the form.
$⌊ ( ) ⌋ ∫ ∞ 2 3 ∂ ρ ∂ ˜S 1 ∑ ∂ S˜ - d x ------- ⌈ ----- + ------ ------- + V ⌉ ∂ x ∂ t 2m ∂ x - ∞ k j j [ ] , ∫ ∞ --------------------- 1 ∂ S˜ (e ) + d3x ρ ----------S˜ + S˜ = F (x, p, t ) m ∂ x [j,k ] [0,k ] k - ∞ j$ (23)

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 $p -$ 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 $S˜$ .

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 $p -$ dependent term on the r.h.s. has to vanish too in this limit and indicates a relation between multi-valuedness of $S˜$ and $p -$ dependence of the external force.

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