Zwanzig projection operator

The Zwanzig projection operator is a mathematical device used in statistical mechanics. 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. The Zwanzig projection operator is a mathematical device used in statistical mechanics. 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. 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 ho _{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 doesn't tell how to find the slow variables of a given Hamiltonian H ( Γ ) {displaystyle H(Gamma )} . A scalar product 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