The Zwanzig projection operator is a mathematical device used in statistical mechanics.
[1] This projection operator acts 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.
A special subset of these functions is an enumerable set of "slow variables"
Candidates for some of these variables might be the long-wavelength Fourier components
of the mass density and the long-wavelength Fourier components
of the momentum density with the wave vector
The Zwanzig projection operator relies on these functions but does not tell how to find the slow variables of a given Hamiltonian
A scalar product[3] between two arbitrary phase space functions
"Fast" variables, by definition, are orthogonal to all functions
This definition states that fluctuations of fast and slow variables are uncorrelated, and according to the ergodic hypothesis this also is true for time averages.
is correlated with some slow variables, then one may subtract functions of slow variables until there remains the uncorrelated fast part of
take the scalar product with the slow function
The Zwanzig projection operator fulfills
The space of slow variables thus is an algebra.
as given above is that it allows to derive a master equation for the time dependent probability distribution
denote the time-dependent probability distribution in phase space.
) is a solution of the Liouville equation The crucial step then is to write
denotes the equilibrium distribution of the slow variables.
The starting point for the standard derivation of a Langevin equation is the identity
Consider discrete small time steps
generates expressions which are fast variables at every time step.
The expectation is that fast variables isolated in this way can be represented by some model data, for instance by a Gaussian white noise.
directly leads to the operator identity of Kawasaki[2] A generic Langevin equation is obtained by applying this equation to the time derivative of a slow variable
is the fluctuating force (it only depends on fast variables).
If these functions constitute a complete orthonormal function set then the projection operator simply reads A special choice for
are orthonormalized linear combinations of the slow variables
This leads to the Mori projection operator.
[3] However, the set of linear functions is not complete, and the orthogonal variables are not fast or random if nonlinearity in