ABSTRACT
To the lowest approx imat ion , the part ic les move i ndependent ly a long straigh t - l i ne orb i t s . Hence , the zero-order posi t ion of parti c le i at t ime tl/ == nil ! i s
( I )
and the Four ier transformed number densi ty i s
where w I( - 27T / Il t and we have i n troduced the periodic del ta-funct ion comb
which replaces the ord inary de l ta funct ion of the cont i nuum transform. We now consider an example of systems such tha t i ts averages are
i ndependent of where and when they are taken , i. e. , a un i form and stat i on ary ensemble . Th i s means tha t the ensemble average of the ne t charge den s i ty , say , w i l l be zero, but the average of products need no t van i sh . We find the ensemble average
from which the fluctuat ions of other quant i t i es may eas i ly be found . First , consider i ng non i nteract i ng part ic les , we u se ( 2 ) su bst i t uted i n to 8 -
9 ( 8 ) , I n performing t he ensemble average we u se t he fol lowing i n format ion : the zero-order part ic le pos i t ions and veloci t ies a re i ndependent , n o i s the average part ic le density, and the veloc i ty d istr i but ion i s fo (v) normal i zed to un i ty . I n the double sum over part ic les , terms correspond ing to pa i rs of d ifferi ng part i cles cancel terms due to the mean neutral i z i ng charge dens i ty of other species . I n the remai n i ng terms ,
( 5 )
(b) L i m i t i ng Cases In many-parameter regimes i t i s poss ib le to obtain addi t ional i nformation
analyt ica l ly _ Efficient numerical evaluat ion of the mul t ipl e sums i n the gen eral case has been discussed in Langdon (I 979b) .
