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

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