  It has been revealed recently that, in the scale free range, i.e. from the
scale of the onset of nonlinear evolution to the scale of dissipation, the
velocity and mass density fields of cosmic baryon fluid are extremely well
described by the self-similar log-Poisson hierarchy. As a consequence of this
evolution, the relations among various physical quantities of cosmic baryon
fluid should be scale invariant, if the physical quantities are measured in
cells on scales larger than the dissipation scale, regardless the baryon fluid
is in virialized dark halo, or in pre-virialized state. We examine this
property with the relation between the Compton parameter of the thermal
Sunyaev-Zel'dovich effect, $y(r)$, and X-ray luminosity, $L_{\rm x}(r)$, where
$r$ being the scale of regions in which $y$ and $L_{\rm x}$ are measured.
According to the self-similar hierarchical scenario of nonlinear evolution, one
should expect that 1.) in the $y(r)$-$L_x(r)$ relation, $y(r)=10^{A(r)}[L_{\rm
x}(r)]^{\alpha(r)}$, the coefficients $A(r)$ and $\alpha(r)$ are
scale-invariant; 2.) The relation $y(r)=10^{A(r)}[L_{\rm x}(r)]^{\alpha(r)}$
given by cells containing collapsed objects is also available for cells without
collapsed objects, only if $r$ is larger than the dissipation scale. These two
predictions are well established with a scale decomposition analysis of
observed data, and a comparison of observed $y(r)$-$L_x(r)$ relation with
hydrodynamic simulation samples. The implication of this result on the
characteristic scales of non-gravitational heating is also addressed.
