Global and Local Properties of Mixtures - Fluctuation Theory of Solutions - page 41

In the one component system, we use an ordering of particles, that is, 0 < X 1 < X 2
< X 3 < ⋯ < X N < L , to label the particles according to their relative locations. But, in
our case, since there are c different species, we have to distinguish between a specific
ordering of the species (SOOS) and a specific ordering of the particles (SOOP). The
configurational integral on the right-hand side of Equation 2.19 can be rewritten by
first fixing the order of the species and then summing over all orderings of the spe-
cies. Thus, we have

LL L L L X

2
∫

∑ ∑ ∏

∫∫ ∫ ∫ ∫

= = N ! dX d X (2.21)

i N 1

SOOS

SOOP

allorderings
of species with
fixed,

allorderings
of

i

0 0 0 0 0 0

species with
fixed, NN

1

… , NN

1

… ,

c c

where in the first step, we have a SOOS, and in the second step, we have a SOOP,
and we multiplied by the factor i ∏ N i !
From Equation 2.19 and Equation 2.21, we have,

L X

2

1

∑

∫ ∫

( ) = ( )

QTLN ,,{} dX dX exp − β U {{} N (2.22)

∏

N 1

N i

Λ

SOOP

all orderings
of species with
fixed

i

0 0

i

NN , … ,

1 c

Next, we open the system with respect to all particles. The corresponding grand PF is,

∑∏

( ) =

( ) =

( )

N i

Ξ TL , {} λ λ Ξ TL N ,,{} exp β pL (2.23)

i

{} N i

where p = ( T , L , {λ}) is the thermodynamic pressure, given as a function of the
variables T , L , {λ}, and λ i is the absolute activity of the species i . We now take
the Laplace transform of Equation 2.23 with respect to the variable L by introduc-
ing the new unspecified variable p * ,

∞

( )

∫

( )

() =

*

Ψ exp − β pL -,,{ } Ξ TL λ dL (2.24)

0

Applying the convolution theorem to Equation 2.23, we obtain

∫ ∞

∑

* N

ΨΞ () = exp( − β pL ) ( λ / Λ )

i

i i

0

N

(2.25)

∞ ∞

∑ ∑ ∑ ∑

∫ ∫

N

OOP = M SS MM = Tr ( M )

SS SS

12 23 N 1

S

allorderings
of species

N = 0 S N = 0

with

fixed N 1 ,, … N

c