When the number density is sufficiently small (but still positive), the structure factor forms in (II.5) obey the second nonnegativity condition I.2. However, as increases, a terminal density _t is reached at which that necessary condition no longer is satisfied. The violation first occurs at k = 0 for D = 1, 2, and 3. Expanding the expressions displayed in eq II.5 around the origin shows that

Although this establishes an upper terminal density above which the step-function g(r) cannot exist, it does not guarantee that this simple pair correlation function is actually achievable up to that density. In other words, sufficiency of constraints I.1 and I.2 for this case (eq II.4) mathematically remains an open question (although some limited numerical evidence supporting the proposition is available¹⁵). The step-function g(r) is not special in this regard; any g(r) for which the integral term in its S(k) has a negative region leads to a qualitatively similar situation. The objective of the following section III is to develop a simple testing ground to aid in deciding whether the two conditions I.1 and I.2 are indeed sufficient.

III. Lattice Model

To permit an exact and complete analysis, we now restrict attention to the case of a linear array of M equally spaced sites. This array will be subject to periodic boundary conditions, so the first and the Mth sites topologically are nearest neighbors. The sites in principle can act as single occupancy locations for 0 N M point particles. The configurations of particles in this array can conveniently be specified by a binary string of occupancy variables _j = 0,1 (1 j M), for empty and filled sites, respectively, so that

The number of distinct particle configurations in the array, with first-neighbor exclusion, is

The primary objective will be to determine, for given M and N, what sets of weights (if any) will produce a flat site-pair distribution function for distances beyond the excluded first-neighbor:

Collective density variables (k) can be defined for any pattern of particles on the lattice

For nonzero k, the last expression attains its minimum value at k = 2/M, the points closest to the origin. Because (k) cannot be negative, it is necessary that N not exceed a terminal upper limit N_t(M) derivable from eq III.14, specifically the following:

IV. Enumeration Details

The (M,N) configurations of N particles distributed on M sites, subject to nearest-neighbor exclusion, need to be separated into pattern classes. Each class collects all configurations that differ only by the symmetry operations of translation and/or inversion. The weights W({_i}) will be the same for all members of the same class, and can be denoted as w_j for the jth class. If m_j is the number of configurations belonging to class j, then the normalization eq III.3 can be restated as

For the sake of illustration, consider the case M = 11, N = 3. Five distinct patterns for the three particles are possible. They correspond to the following binary strings {_j} with respective weights w₁...w₅:

Table 1 presents the values of C(M,N) for all cases with M 17. The central objective is to search for sets of class weights w_j which produce a flat pair distribution function. No such set can exist if N exceeds the terminal value N_t(M).

To examine the possibility of attaining a flat pair distribution function, it is necessary to evaluate the contribution of individual classes to each <_ii+n> <_1n+1>. This requires identifying how many members of each class exhibit simultaneous occupancy of sites 1 and n, and attributing a weight w_j to each. For the specific M = 11, N = 3 case considered above, this process yields the following expressions:

V. Results

The four linear equations in eq IV.3 for the illustrative example M = 11, N = 3 can formally be solved in terms of w₁:

Similar considerations apply to other M, N choices, although the outcomes can be quite different. Table 2 describes the results qualitatively for all cases with M 15. Those cases M, N for which a fixed (i.e. unique) set of weights produces the desired flat pair correlation are designated by "f" in Table 2; in particular this is true whenever N = 2, the lowest density situation for which pair correlation is meaningfully defined. If no solution with nonnegative weights exists, the corresponding entry in Table 2 is "n", and this is necessarily true whenever N > N_t(M). The remaining cases have multiple solution sets that constitute bounded convex manifolds with some positive dimension i; these cases have been identified in Table 2 by the symbol "m(i)". For some of the cases in this last category, numerical exploration was used to identify exact positive rational values for the w_j satisfying the flat pair distribution constraint, i of which could then be independently varied while still satisfying that constraint.

Examination of Table 2 reveals the presence of two cases for which no solution exists ("n"), although N < N_t(M). These are M, N = 8, 3 and 15, 5, and for both the N value is just below the N_t(M) boundary by less than unity. The reason for nonexistence of acceptable solutions is not the same for both of these anomalous cases. In the former, only two distinct configuration types exist [C(8,3) = 2], while three pair distribution constraints need to be imposed; thus the two weights are overconstrained. In the latter, more than enough distinct configurations are available [C(15,5) = 16] to satisfy the six pair distribution flat constraints; however no solution with nonnegative weights for those configurations exists.

Because eq III.15 indicates for large M that N_t(M) ~ M/3, the number of particle configurations available at this boundary can be estimated by applying Stirling's asymptotic approximation:

Yamada⁷ has derived a necessary condition for pair correlation functions that concerns number variance in subregions ("windows") of the full system.¹⁴ For the present context, the fluctuating quantity of interest is N_a, the number of particles occurring in 1 a M contiguous sites:

VI. Concluding Remarks

The principal focus of this paper has been the pair distribution function realizability problem, illustrated by study of finite-size lattice systems in one dimension. These lattice systems involve single-occupancy lattice sites, nearest-neighbor exclusion, and periodic boundary conditions. The objective was to determine what configurational probabilities, if any, would lead to a pre-assigned pair distribution function. The specific situation examined involved a "flat" pair distribution function, i.e., one which is independent of distance beyond the excluded nearest-neighbor separation. Table 2 summarizes the results obtained (without approximation) for various modest values of the number of sites M and of the number of particles N. These results include cases in which (a) no solution is possible, (b) an unique set of configurational probabilities produces the desired flat pair distribution, and (c) a multidimensional manifold of configurational-weight sets exists yielding the desired flat pair distribution.

Although the flat pair distribution is perhaps the simplest example that might be chosen to illustrate the realizability issue, alternatives could equally well have been analyzed using the same basic enumeration approach. One such alternative could have been a pair distribution function with a maximum value at the second-neighbor separation, then a smaller constant value beyond. One would expect that the corresponding entries analogous to those shown in Table 2 would be somewhat altered, but still would present a qualitatively similar pattern.

For any M, N case that admits of at least an unique set of configurational weights, those weights can always be expressed as normalized Boltzmann factors:

The overall pattern of configuration-weight solution types presented by Table 2 has some significant implications. In particular, the reader will notice that for a given system size M > 10, the midrange values of the particle number N lead to solutions that are not unique, and that the maximum dimensionality of the solution sets appears to rise with increasing system size M. Although the pair distribution function remains invariant over these multidimensional solution sets, the higher-order distribution functions will not. This is clear from the fact that the potential energy function in eq VI.1 will vary across the solution sets involved. To paraphrase, fixing the pair distribution function generally does not fix the higher-order distribution functions, and specifically, it does not determine the triplet distribution function. This last observation points out an explicit violation of the Kirkwood superposition approximation.^16,17

Another area of relevance for the present analysis is the so-called "reverse Monte Carlo" method.^18-21 This method attempts to infer typical full-system configurations using only measured pair correlation functions as input, and utilizes a stochastic procedure to move particles spatially so as to conform closely to that input. However, as just stressed in connection with the Kirkwood superposition approximation, fixing the pair distribution generally leaves higher-order distribution functions undetermined. The final pattern of particle positions produced by a reverse Monte Carlo computation implicitly depends on details of that computation in a way that will bias that result in an unanticipated, perhaps even unphysical, manner. Examples such as M, N = 15, 4 above directly illustrate the underlying difficulty, because the 7 pair constraints that would be applied by the reverse Monte Carlo method cannot resolve the 14-dimensional degeneracy [m(14) in Table 2] presented by this case. Consequently, one may legitimately question the logical basis of the reverse Monte Carlo approach to the extent that it presumes uniquely to produce valid many-particle spatial patterns.

Of course it would be desirable to extend the present calculations in a variety of directions, not the least of which would be increasing M substantially above the present upper limit 15. It seems unlikely to the authors that the presently revealed trends would be qualitatively overturned by that substantial increase. Specifically, in the large-system limit (whether lattice or continuum models are involved) it appears that pair distribution realizability is feasible over a nonvanishing density range that includes = 0. The appearance of the two anomalous cases M, N = 8, 3 and 15, 5 with no solution, with N < N_t, illustrates the fact that the two necessary conditions I.1 and I.2 generally cannot be sufficient. In particular, it has been pointed out^7,10 that local number-fluctuation constraints exist that are not implied by (I.1) and (I.2). Furthermore, the specific failure of the M, N = 15, 5 case to possess any solution hints that at least one additional necessary condition beyond those already known has yet to be articulated. However, whether a finite set of necessary conditions will ultimately suffice to ensure pair correlation function realizability remains a challenging open question.

Acknowledgment

The authors thank Prof. Joel L. Lebowitz for the benefit of probing discussions concerning the realizability problem. In addition, the authors acknowledge support of the Office of Basic Energy Sciences, DOE, under Grant No. DE-FG02-04ER46108.

Appendix

The conventional density expansion for the pair correlation function g(r₁₂,), in an infinite system of spherically symmetric particles, can be expressed in the following way:²²

The objective of the iso-g problem is to find a -dependent pair interaction v*(r₁₂,) to stand in place of v(r), which causes g(r₁₂) to be independent of , at least for some interval of this variable including the origin. Suppose that the required effective pair interaction has at least a formal density expansion:

Let the series A.3 for v* be inserted into each of the Mayer f functions in eq A.2 in place of v, followed by density expansion of each of those f functions. As a result, each of the _n likewise becomes a power series in :