Consistent implementation of non-zero-range terms into hydrodynamics

Non-zero-range interactions are often incorporated into mean field theories through gradient terms. Here, a formalism is developed to incorporate such terms into hydrodynamics. These terms alter expressions for the entropy, chemical potential, temperature, and the stress-energy tensor. The formalism respects local conservation of energy, charge, and entropy. The formalism leads to static solutions where the temperature, chemical potential, and hydrodynamic acceleration all vanish, even when the density profile might be nonuniform. Profiles for a phase boundary and for correlation functions are calculated to illustrate the gradient modifications for various thermodynamic quantities. Also, for hydrodynamic calculations that add thermal noise to generate density-density correlations of the desired strength, an additional noise term is derived so that, at equilibrium, correlations are generated with both the correct size and length scale.

Figure 1

In the upper panel (a), the density profile is displayed for the liquid-gas interface. Including gradient modifications, the temperature (b), chemical potential (c), and the stress-energy tensor element $T_{xx}$ (d) are all constant, as is required for an equilibrated system. The quantities evaluated as a function of the local energy density and local momentum density, $\overline{T}(\epsilon ,\rho ),\overline{\mu }(\epsilon ,\rho )$, and $\overline{P}(\epsilon ,\rho )$ all vary substantially across the interface. The element $T_{yy}$ (a) also varies, but because it does not depend on $y$ does not affect the constraint ${\partial }_i{T}_{ij}=0$. To make all quantities dimensionless, densities are in units of ${\rho }_s$, the temperature and chemical potential are in units of $a{\rho }_s$, and the pressure is in units of $a{\rho }_s^2$. The coordinate $x$ is in units of $\sqrt{\kappa /a}$.