The translational invariance properties of a one‐dimensional fluid with finite range forces are investigated. For N particles in the interval [0, L], with a two‐body interaction potential w(x) = 0 for x ≥ R, we find the following: (a) If w(x) has a hard core of diameter d and R ≤ 2d, each n‐particle distribution function Dn(x1, …, xn) is translationally invariant if and only if L > 2(N − n)R and x1, …, xn lie in [(N − n)R, L − (N − n)R]. (b) For arbitrary finite values of R, with or without a hard core, the above conditions are sufficient for translational invariance of the Dn. These conditions hold for all temperatures.

Topics
Invariance properties, Particle distribution functions
1.
H. S.
Leff
 and
M. H.
Coopersmith
,
J. Math. Phys.
8
,
306
(
1967
).
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2.
This notation is slightly more detailed than that in paper I, indicating both N and L dependence.
3.
The concept of nearest‐neighbor forces is meaningful only if the potential has a hard‐core part. The requirement of a hard core is dropped in Sec. V.
4.
Here, we are explicitly using the fact that Z̃(x,n)≡0 for x<nd.
5.
Strictly speaking, the upper limit of Eq. (21) should be R+ε where ε is an arbitrarily small positive number. Then, for the pure hard‐core system R = d and the delta function peak is within the domain of integration. For this case, Eq. (21) reduces to Eq. (14).
6.
Here, it is convenient to explicitly use the form
(∂/∂x) exp[−βw(x)] = −βw′(x) exp[−βw(x)]
. In the preceding sections this was not the case.
7.
For the case of nearest‐neighbor forces, this equation becomes a recursion relation for D1(N+1) being identical with Eq. (21). This can be seen using Eq. (3) applied to D2(N+1)(x−z,x|L) with 0⩽z⩽2d. Since Z̄(z,n) =  exp[−βw(z)]δ0,n, Eq. (3) yields
D2(N+1)(x−z,x|L) = (N+1)Z(L−z,N) exp[−βw(z)]×D1(N)(x−z|L−z)|Z(L,N+1).
8.
Recall that if one or more of the {xi} lies outside [0, L], gN+1(N+1) vanishes.
9.
Equation (26) may be interpreted as signifying a translation of the container, holding the particles fixed. The fact that the ε− dependence can be removed from the integrand and isolated in the limits of integration is crucial to the present method.
10.
If the set XN lies to the right of R, gN+1(N+1)(Xn,0|L) = (N+1)gN(N)(XN−R|L−R). If the set XN lies to the left of L−R, gN+1(N+1)(XN,L|L−R) = (N+1)gN(N)(XN|L−R). These expressions are equal since gN(N) is manifestly translationally invariant. Since XN is restricted to [R,L−R] one could denote the domain in the gN(N) functions by L−2R, rather than L−R. The precise label is irrelevant for our purposes.
11.
Note that the intervals [0, jR] and [L−jR,L] are both denoted simply as jR in the gk+1(k+1) functions. The specific interval involved is clear from its context.
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© 1967 The American Institute of Physics.
1967
The American Institute of Physics
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