Local Hydrodynamic Pressure for Strongly Inhomogeneous States

Continuum descriptions of complex systems, both classical and quantum, are increasingly adopted as alternatives first principles many-body analyses. The macroscopic momentum balance equation has its origin in the underlying exact conservation law, with a precisely defined operator for the associated momentum flux. Its average in a local equilibrium ensemble provides the basis for a hydrodynamic description. In particular, a local pressure is defined in terms of the trace of the average momentum flux. Other definitions for a local pressure are possible, such as a local virial pressure or a thermodynamic pressure obtained from the grand potential for local equilibrium. It is shown that all three local pressures have the same global pressure but differ locally, except for weakly inhomogeneous states. For uniform temperature, the trace of the momentum flux agrees with the local thermodynamic pressure even for strong density inhomogeneity.
This provides an important connection to density functional methods for hydrodynamic applications. For non-uniform temperatures and strong inhomogeneities there is no simple relationship among the different definitions of the local pressure, so the proper choice is that from the average momentum flux.