We introduce a combinatorial version Mori-Zwanzig theory and develop from it
a family of self-consistent evolution equations for the correlation function or
Green’s function of interactive many-body systems. The core idea is to use an
ansatz to rewrite the memory kernel (self-energy) of the regular Mori-Zwanzig
equation as a function composition of the correlation (Green’s) function. Then
a series of algebraic combinatorial tools, especially the commutative and
noncommutative Bell polynomials, are used to determine the exact Taylor series
expansion of the composition function. The resulting combinatorial Mori-Zwanzig
equation (CMZE) yields novel non-perturbative expansions of the equation of
motion for the correlation (Green’s) function. The structural equation for
deriving such a combinatorial expansion resembles the combinatorial
Dyson-Schwinger equation and may be viewed as its temporal-domain analogue.
After introducing the abstract word and tree representation of the CMZE, we
show its wide-range application in classical, stochastic, and quantum many-body
systems. In all these examples, the new self-consistent expansions we obtained
with the CMZE are similar to the diagrammatic skeleton expansions used in
quantum many-body theory and lattice statistical field theory. We expect such a
new framework can be used to calculate the correlation (Green’s) function for
strongly correlated/interactive many-body systems.

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