Chen Yng-Gwei, Weeks John D
Institute for Physical Science and Technology, and Department of Physics, University of Maryland, College Park, Maryland 20742, USA.
J Phys Chem B. 2005 Apr 14;109(14):6892-901. doi: 10.1021/jp0446884.
Percus showed that approximate theories for the structure of nonuniform hard sphere fluids can be generated by linear truncations of functional expansions of the nonuniform density rho(r) about that of an appropriately chosen uniform system. We consider the most general such truncation, which we refer to as the shifted linear response (SLR) equation, where the density response rho(r) to an external field phi(r) is expanded to linear order at each r about a different uniform system with a locally shifted chemical potential. Special cases include the Percus-Yevick (PY) approximation for nonuniform fluids, with no shift of the chemical potential, and the hydrostatic linear response (HLR) equation, where the chemical potential is shifted by the local value of phi(r). The HLR equation gives exact results for very slowly varying phi(r) and reduces to the PY approximation for hard core phi(r), where generally accurate results are found. We show that a truncated expansion about the bulk density (the PY approximation) also gives exact results for localized fields that are nonzero only in a "tiny" region whose volume V(phi) can accommodate at most one particle. The SLR equation can also exactly describe a limit where the fluid is confined by hard walls to a very narrow slit. This limit can be related to the localized field limit by a simple shift of the chemical potential, leading to an expansion about the ideal gas. We then try to develop a systematic way of choosing an optimal local shift in the SLR equation for general phi(r) by requiring that the predicted rho(r) is insensitive to small variations about the appropriate local shift, a property that the exact expansion to all orders would obey. The resulting insensitivity criterion (IC) gives a theory that reduces to the HLR equation for slowly varying phi(r) and is much more accurate than HLR both for very narrow slits, where the IC agrees with exact results, and for fields confined to "tiny" regions, where the IC gives very accurate (but not exact) results. However, the IC is significantly less accurate than the PY and HLR equations for single hard core fields. Only a small change in the predicted reference density is needed to correct this remaining limit.
珀库斯表明，非均匀硬球流体结构的近似理论可以通过对非均匀密度ρ(r)围绕适当选择的均匀系统的密度进行泛函展开的线性截断来生成。我们考虑最一般的这种截断，我们将其称为移位线性响应（SLR）方程，其中对外部场φ(r)的密度响应ρ(r)在每个r处围绕具有局部移位化学势的不同均匀系统展开到线性阶次。特殊情况包括非均匀流体的珀库斯 - 耶维克（PY）近似，其中化学势无移位，以及静水压线性响应（HLR）方程，其中化学势由φ(r)的局部值移位。HLR方程对于非常缓慢变化的φ(r)给出精确结果，并且对于硬核φ(r)简化为PY近似，在那里通常能找到准确结果。我们表明，围绕体密度的截断展开（PY近似）对于仅在体积V(φ)最多能容纳一个粒子的“微小”区域内非零的局部场也给出精确结果。SLR方程还可以精确描述流体被硬壁限制在非常窄的狭缝中的极限情况。通过化学势的简单移位，这个极限可以与局部场极限相关联，从而导致围绕理想气体的展开。然后，我们试图通过要求预测的ρ(r)对围绕适当局部移位的小变化不敏感来开发一种系统的方法，为一般的φ(r)在SLR方程中选择最优局部移位，这是精确展开到所有阶次都将遵循的一个性质。由此产生的不敏感准则（IC）给出了一种理论，该理论对于缓慢变化的φ(r)简化为HLR方程，并且对于非常窄的狭缝（其中IC与精确结果一致）以及对于局限于“微小”区域的场（其中IC给出非常准确但不精确的结果）都比HLR更准确。然而，对于单个硬核场，IC的准确性明显低于PY和HLR方程。只需对预测的参考密度进行小的改变就可以纠正这个剩余的局限性。
