Equilibrium Limit of Diffusion Equation for Interacting Particles:  Are Bifurcations and Multiplicity of Correlation Functions an Artifact or Real?

Abstract

The structure of a fluid is analyzed by taking the equilibrium limit of a diffusion equation including the Giacomin-Lebowitz term for intermolecular interactions. This equation represents the differential mass balance in fluids with the Metropolis algorithm for fluxes; it allows a new qualitative yet analytical approximation for the direct correlation function over the entire range of fluid densities and temperatures. This approximation is analogous to a classical Ono-Kondo model for adsorption if the distribution of molecules around a central molecule is viewed as the adsorption of molecules on a central molecule. While this model qualitatively predicts known behavior for both gas and liquid phases, approaching a phase transition (e.g., condensation of gas into liquid) results in a bifurcation and multiplicity of the direct correlation function. The model predicts a sequence in the transformation of correlation functions from that of a gas to that of a liquid. This sequence starts with the appearance of an isolated loop in the direct correlation function, indicating states that are stable but cannot be achieved without perturbation of the system. However, the system seems to sense its proximity to a phase transition and reflects the distance to the phase boundary by the size and shape of this isolated loop in the direct correlation function. At lower temperatures, this loop merges with the gas-phase peak, indicating that clusters can form spontaneously. Then, these clusters grow into a high-density (liquid) phase. The notion of bifurcations and multiplicity in correlation functions is an unusual and controversial concept. Certainly, it is unexpected and raises important questions: (a) if such a behavior is not real, why does the diffusion equation predict such behavior, i.e., is it a mathematical artifact or is it due to conflicting physical assumptions? (b) if this behavior is real, how does one interpret it at a molecular level? Here, we present some interpretations, but they are open for discussion.

Resolving the Inconsistency between Classical Diffusion and Adsorption

“…Though there are fundamental aspects of adsorption behavior which are understood well, − there are others that still remain unresolved. One of these problems is an inconsistency between the density distribution of fluids near surfaces and the solution of the phenomenological diffusion equation in the equilibrium limit. , The density distribution of adsorbate near a surface can be found from rigorous theories (such as density functional theory, DFT) or from simulations, and one might expect similar results from the solution of diffusion equations in the equilibrium limit. However, the classical diffusion theory fails to predict adsorption profiles.…”

Modification of classical approximations for diffusion in fluids with density gradients

“…where V is the characteristic velocity of molecules and l is the mean free path. Modern corrections (in particular, for systems with phase transitions [14][15][16][17] and for non-Fickian behavior [18][19][20] ) and rigorous molecular-statistical approaches (such as Green-Kubo and Chapman-Enskog modifications) indicate that D can be a complex function or functional of the free-path length distribution.…”