  ariational implicit solvation models (VISM) have gained extensive popularity
in the molecular-level solvation analysis of biological systems due to their
cost-effectiveness and satisfactory accuracy. Central in the construction of
VISM is an interface separating the solute and the solvent. However,
traditional sharp-interface VISMs fall short in adequately representing the
inherent randomness of the solute-solvent interface, a consequence of
thermodynamic fluctuations within the solute-solvent system. Given that
experimentally observable quantities are ensemble-averaged, the computation of
the ensemble average solvation energy (EASE)-the averaged solvation energy
across all thermodynamic microscopic states-emerges as a key metric for
reflecting thermodynamic fluctuations during solvation processes. This study
introduces a novel approach to calculating the EASE. We devise two
diffuse-interface VISMs: one within the classic Poisson-Boltzmann (PB)
framework and another within the framework of size-modified PB theory,
accounting for the finite-size effects. The construction of these models relies
on a new diffuse interface definition $u(x)$, which represents the probability
of a point $x $ found in the solute phase among all microstates. Drawing upon
principles of statistical mechanics and geometric measure theory, we rigorously
demonstrate that the proposed models effectively capture EASE during the
solvation process. Moreover, preliminary analyses indicate that the
size-modified EASE functional surpasses its counterpart based on classic PB
theory across various analytic aspects. Our work is the first step towards
calculating EASE through the utilization of diffuse-interface VISM. energy by
using diffuse-interface VISMs.
