Fig. 2: Illustration of the sum rules (14), (15), and (16).
From: Hyperforce balance via thermal Noether invariance of any observable
The simulation results were obtained from adaptive BD sampling of the LJ fluid confined between two parallel planar LJ walls. The profiles are shown as a function of scaled distance z/σ across the planar slit. a The density profile ρ(r) of the confined system is shown as a reference. b Comparison of the correlator \(\beta \langle {\hat{{{{{{{{\bf{F}}}}}}}}}}_{{{{{{{{\rm{ext}}}}}}}}}^{0}\hat{\rho }({{{{{{{\bf{r}}}}}}}})\rangle\) and the density gradient ∇ρ(r), see Eq. (14); the zoomed inset demonstrates the respective noise levels and it also shows the scaled force density sum βFU(r) = βFint(r) + βFext(r) which equals ∇ρ(r) due to the local force balance. c Comparison of \({\beta }^{2}\langle {\hat{{{{{{{{\bf{F}}}}}}}}}}_{{{{{{{{\rm{ext}}}}}}}}}^{0}{\hat{{{{{{{{\bf{F}}}}}}}}}}_{{{{{{{{\rm{int}}}}}}}}}({{{{{{{\bf{r}}}}}}}})\rangle\) and β ∇Fint(r), see Eq. (15). The former route carries less statisical noise and hence can serve as a starting point for a force sampling scheme. d Comparison of \({\beta }^{2}\langle {\hat{{{{{{{{\bf{F}}}}}}}}}}_{{{{{{{{\rm{ext}}}}}}}}}^{0}\hat{{{{{{{{\bf{F}}}}}}}}}({{{{{{{\bf{r}}}}}}}})\rangle\) and the local external potential curvature density βρ(r) ∇∇Vext(r), see Eq. (16).
