A symmetry-field theoretical framework for quasi-stationary interfaces in fully miscible fluids

Long-lived sharp interfaces in fully miscible fluids exhibit a transient quartic excess free-energy landscape and a Fermi-Dirac logistic concentration profile, whose microscopic statistical-mechanical origin has remained unresolved. Here, we develop an independent statistical-mechanical theory based on a microscopic pair Hamiltonian and the corresponding grand partition function. Exchange symmetry naturally introduces the invariant p=c1-c, whose regular local expansion uniquely generates the quartic excess free-energy density as the leading nonvanishing contribution after coarse-graining the microscopic Hamiltonian. The resulting free-energy functional completely determines the Euler-Lagrange equation of the quasi-stationary interface, whose exact solution is the Fermi-Dirac logistic profile. The independently derived microscopic formulation converges to the same quartic free-energy functional obtained previously from a continuum variational approach, thereby establishing a unified microscopic statistical-mechanical framework for quasi-stationary interfaces.