The Zwanzig projection operator is a mathematical device used in statistical mechanics.[1] It operates in the linear space of phase space functions and projects onto the linear subspace of "slow" phase space functions. It was introduced by Robert Zwanzig to derive a generic master equation. It is mostly used in this or similar context in a formal way to derive equations of motion for some "slow" collective variables.[2]

The Zwanzig projection operator operates on functions in the 6 N {\displaystyle 6N} -dimensional phase space Γ = { q i , p i } {\displaystyle \Gamma =\{\mathbf {q} _{i},\mathbf {p} _{i}\}} of N {\displaystyle N} point particles with coordinates q i {\displaystyle \mathbf {q} _{i}} and momenta p i {\displaystyle \mathbf {p} _{i}} . A special subset of these functions is an enumerable set of "slow variables" A ( Γ ) = { A n ( Γ ) } {\displaystyle A(\Gamma )=\{A_{n}(\Gamma )\}} . Candidates for some of these variables might be the long-wavelength Fourier components ρ k ( Γ ) {\displaystyle \rho _{k}(\Gamma )} of the mass density and the long-wavelength Fourier components π k ( Γ ) {\displaystyle \mathbf {\pi } _{\mathbf {k} }(\Gamma )} of the momentum density with the wave vector k {\displaystyle \mathbf {k} } identified with n {\displaystyle n} . The Zwanzig projection operator relies on these functions but does not tell how to find the slow variables of a given Hamiltonian H ( Γ ) {\displaystyle H(\Gamma )} .

A scalar product[3] between two arbitrary phase space functions f 1 ( Γ ) {\displaystyle f_{1}(\Gamma )} and f 2 ( Γ ) {\displaystyle f_{2}(\Gamma )} is defined by the equilibrium correlation