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 )               =   pk  .
                             ∂ 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.