= Thermal transpiration
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<Thermal transpiration> is gas transport driven by an imposed <temperature> difference in a rarefied connection. Gas can move from cold to hot even when the two end <pressures> start equal. If the reservoirs are closed, redistribution builds a counteracting <pressure> difference until the net transfer stops. This final stationary state is not global <thermal equilibrium>: the two <temperatures> remain different. It explains why a heated pressure sensor can read a different <pressure> from the vessel to which it is connected. The correction depends on gas, connection geometry and molecular interaction with its surfaces. https://arxiv.org/html/1308.6794[Daudé, Elandaloussi and Janssen (2014), thermal-transpiration corrections for diaphragm gauges, introduction]{ref}
The simple molecular limit is beautiful: equal opposing particle rates need not mean equal <pressures>. Consider <effusion> through a thin hole connecting two locally equilibrated reservoirs of the same gas, at $T_h>T_c$. Equating their one-way rates gives
$$
\frac{p_c}{\sqrt{T_c}}=\frac{p_h}{\sqrt{T_h}},\qquad \frac{p_h}{p_c}=\sqrt{\frac{T_h}{T_c}}.
$$
At equal <pressure>, the hotter reservoir has lower <number density>; its faster molecules do not fully compensate for that reduction in outgoing particle rate. With $T_c=300\,\mathrm K$ and $T_h=351\,\mathrm K$, the ideal zero-transfer ratio is $\sqrt{351/300}\simeq1.0817$. The hydrodynamic, zero-flow limit instead has equal <pressures> when other forces are absent. These are limiting models, not an interpolation formula for a glass capillary. https://farside.ph.utexas.edu/teaching/355/Surveyhtml/node224.html[Fitzpatrick, Molecular Effusion, equations 5.286–5.287]{ref} In a real tube, incomplete exchange of molecular <energy> with the wall can alter the limiting ratio; a <Knudsen number> does not specify that exchange. https://arxiv.org/html/1308.6794[Daudé, Elandaloussi and Janssen, §I, limitations of the ideal Knudsen pressure ratio]{ref}
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