The Zwanzig projection operator operates on functions in the -dimensional phase space of point particles with coordinates and momenta. 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 identified with. 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 between two arbitrary phase space functions and is defined by the equilibrium correlation where denotes the microcanonical equilibrium distribution. "Fast" variables, by definition, are orthogonal to all functions of under this scalar product. 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. If a generic function is correlated with some slow variables, then one may subtract functions of slow variables until there remains the uncorrelated fast part of. The product of a slow and a fast variable is a fast variable.
The projection operator
Consider the continuous set of functions with constant. Any phase space function depending on only through is a function of the, namely A generic phase space function decomposes according to where is the fast part of. To get an expression for the slow part of take the scalar product with the slow function, This gives an expression for, and thus for the operator projecting an arbitrary function to its "slow" part depending on only through, This expression agrees with the expression given by Zwanzig, except that Zwanzig subsumes in the slow variables. The Zwanzig projection operator fulfills and. The fast part of is. Functions of slow variables and in particular products of slow variables are slow variables. The space of slow variables thus is an algebra. The algebra in general is not closed under the Poisson bracket, including the Poisson bracket with the Hamiltonian.
The ultimate justification for the definition of as given above is that it allows to derive a master equation for the time dependent probability distribution of the slow variables. To sketch the typical steps, let denote the time-dependent probability distributionin phase space. The phase space density is a solution of the Liouville equation The crucial step then is to write, and to project the Liouville equation onto the slow and the fast subspace, Solving the second equation for and inserting into the first equation gives a closed equation for . The latter equation finally gives an equation for where denotes the equilibrium distribution of the slow variables.
The starting point for the standard derivation of a Langevin equation is the identity, where projects onto the fast subspace. Consider discrete small time steps with evolution operator, where is the Liouville operator. The goal is to express in terms of and. The motivation is that is a functional of slow variables and that 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. The decomposition is achieved by multiplying from the left with, except for the last term, which is multiplied with. Iteration gives The last line can also be proved by induction. Assuming and performing the limit directly leads to the operator identity of Kawasaki A generic Langevin equation is obtained by applying this equation to the time derivative of a slow variable,, Here is the fluctuating force. Mode coupling term and damping term are functionals of and and can be simplified considerably.
Discrete set of functions, relation to the Mori projection operator
Instead of expanding the slow part of in the continuous set of functions one also might use some enumerable set of functions. 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. However, the set of linear functions is not complete, and the orthogonal variables are not fast or random if nonlinearity in comes into play.
