  Taking place naturally in a gas subject to a given wall temperature
distribution [Maxwell1879], the ``ghost effect'' exhibits a rare kinetic effect
beyond the prediction of classical fluid theory and Fourier law in such a
classical problem in physics. As the Knudsen number $\varepsilon$ goes to zero,
the finite variation of temperature in the bulk is determined by an
$\varepsilon$ infinitesimal, ghost-like velocity field, created by a given
finite variation of the tangential wall temperature as predicted by Maxwell's
slip boundary condition. Mathematically, such a finite variation leads to the
presence of a severe $\varepsilon^{-1}$ singularity and a Knudsen layer
approximation in the fundamental energy estimate. Neither difficulty is within
the reach of any existing PDE theory on the steady Boltzmann equation in a
general 3D bounded domain. Consequently, in spite of the discovery of such a
ghost effect from temperature variation in as early as 1960's, its mathematical
validity has been a challenging and intriguing open question, causing confusion
and suspicion. We settle this open question in affirmative if the temperature
variation is small but finite, by developing a new $L^2-L^6-L^{\infty}$
framework with four major innovations: 1) a key $\mathscr{A}$-Hodge
decomposition and its corresponding local $\mathscr{A}$-conservation law
eliminate the severe $\varepsilon^{-1}$ bulk singularity, leading to a reduced
energy estimate; 2) A surprising $\varepsilon^{\frac{1}{2}}$ gain in $L^2$ via
momentum conservation and a dual Stokes solution; 3) the
$\mathscr{A}$-conservation, energy conservation and a coupled dual
Stokes-Poisson solution reduces to an $\varepsilon^{-\frac{1}{2}}$ boundary
singularity; 4) a crucial construction of $\varepsilon$-cutoff boundary layer
eliminates such boundary singularity via new Hardy and BV estimates.
