Local Pressure for Strongly Inhomogeneous states

Continuum descriptions of complex classical and quantum systems increasingly are alternatives to first-principles many-body analyses. The macroscopic momentum balance equation originates 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. A local pressure can be defined in terms of the trace of the local equilibrium average momentum flux, or alternatively as a thermodynamic pressure obtained from the grand potential for the local equilibrium ensemble. Both have the same global pressure but can differ locally. For uniform temperature (equilibrium), the momentum flux trace agrees with the local thermodynamic pressure even for strong density inhomogeneity, providing a connection to density functional methods for hydrodynamic applications. For non-uniform temperatures (local equilibrium) and strong inhomogeneities, no simple relationship exists among the different local pressure definitions. However, agreement can be restored by adding a suitable divergence-free contribution to the local equilibrium average momentum flux*.
*J Dufty, J Wrighton, K Luo. AIChe J. doi:10.1002/aic.17037