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<Passive transport> is net movement down the transported substance's <electrochemical gradient> without coupling that movement to another source of <free energy>. <Facilitated diffusion> is <passive transport> through an <ion channel> or a passive <carrier protein>. Providing such a route can speed passage, but does not turn an uphill transfer into a downhill one. For an uncharged <solute> under the ideal conditions above, net <passive transport> runs from higher to lower <concentration>; for an <ion>, the <membrane potential> also matters. See https://www.ncbi.nlm.nih.gov/books/NBK9847/[Cooper, The Cell: transport of small molecules].

In the figure, stipulate eight neutral <solute> tracers in A and two in B, with equal fixed <volumes>, identical surroundings, and no imposed <electric potential> difference. The <concentration> ratio is $c_{\mathrm A}/c_{\mathrm B}=8/2=4$, so A-to-B transfer has $\Delta\widetilde\mu=-RT\ln4\simeq-1.38629RT$, or $-3.458\,\mathrm{kJ\,mol^{-1}}$ at the assumed <temperature> $T=300\,\mathrm K$. If this <solute> can cross, <thermodynamic equilibrium> has equal expected <concentrations>: five tracers per side on average. Individual crossings occur in both directions; an observed ten-particle snapshot need not contain exactly five on each side. These stipulated counts illustrate a driving force, not a measured flow rate or a prediction of the time to equilibrate.

For a separate illustrative <ion> calculation, keep $c_{\mathrm A}/c_{\mathrm B}=4$, but assign $z=+1$ and $V_{\mathrm A}-V_{\mathrm B}=-60\,\mathrm{mV}$ at $300\,\mathrm K$. Using $R\simeq8.31446\,\mathrm{J\,mol^{-1}\,K^{-1}}$ and $F\simeq96485.33\,\mathrm{C\,mol^{-1}}$, B-to-A transfer gives $RT\ln4-0.060F\simeq3.458-5.789=-2.331\,\mathrm{kJ\,mol^{-1}}$. Thus an appropriate open <ion channel> can permit <passive transport> toward the higher <concentration>: the electrical contribution outweighs the opposing <chemical potential> contribution. This stipulated <membrane potential> is not inferred from the tracer counts.


 Passive transport (source code)