  We consider chip-firing dynamics defined by arbitrary M-matrices. M-matrices
generalize graph Laplacians and were shown by Gabrielov to yield avalanche
finite systems. Building on the work of Baker and Shokrieh, we extend the
concept of energy minimizing chip configurations. Given an M-matrix, we show
that there exists a unique energy minimizing configuration in each equivalence
class defined by the matrix.
  We define the class of $z$-superstable configurations which satisfy a
strictly stronger stability requirement than superstable configurations
(equivalently $G$-parking functions or reduced divisors). We prove that for any
M-matrix, the $z$-superstable configurations coincide with the energy
minimizing configurations. Moreover, we prove that the $z$-superstable
configurations are in simple duality with critical configurations. Thus for all
avalanche-finite systems (including all directed graphs with a global sink)
there exist unique critical, energy minimizing and $z$-superstable
configurations. The critical configurations are in simple duality with energy
minimizers which coincide with $z$-superstable configurations.
