Spatial decay solution of the Boltzmann equation: converse properties of long time limiting behavior

We prove some converse properties of long time limiting behavior (along the particle paths) of a class of spatial decay solutions of the Boltzmann equation. It is shown that different initial data f0 determine different long time limit functions $f_{\infty}(x,v) = \lim_{t\rightarrow \infty}f(x+tv,v,t)$, and for any given function F(x,v) which belongs to a function set, there exists a solution f such that $f_{\infty}=F$. Existence of such spatial decay solutions are proven for the inverse power potentials with weak angular cut-off condition and for the initial data f0 satisfying $f_{0}(x,v)\leq C(1+|x|^2+|v|^2)^{-k}$, or $f_{0}(x,v) \leq C(1+|x-v|^2)^{-k}$ , etc. For the soft potentials, the solutions may have ``locally infinite particles," i.e., $\int_{{\bf R}^3}f(x,v,t)dv\equiv\infty.$