  We provide sufficient conditions for the solution of the classical inverse
problem in the canonical distribution for multi-particle densities.
Specifically, we show that there exists a unique potential in the form of a sum
of m-particle (m greater then 1) interactions producing a given m-particle
density. The existence and uniqueness of the solution to the multi-particle
inverse problem is essential for the numerical simulations of matter using
effective potentials derived from structural data. Such potentials are often
employed in coarse- grained modeling. The validity of the multi-particle
inverse conjecture also has implications for liquid state theory. For example,
it provides the first step in proving the existence of the hierarchy of
generalized Ornstein-Zernike relations. For the grand canonical distribution,
the multi-particle inverse problem has been solved by Chayes and Chayes [J.
Stat. Physics 36, 471-488 (1984)]. However, the setting of the canonical
ensemble presents unique challenges arising from the impossibility of
uncoupling interactions when the number of particles is fixed.
