The Mori–Zwanzig formalism, named after the physicists Hajime Mori [de] and Robert Zwanzig, is a method of statistical physics.
It allows the splitting of the dynamics of a system into a relevant and an irrelevant part using projection operators, which helps to find closed equations of motion for the relevant part.
Macroscopic systems with a large number of microscopic degrees of freedom are often well described by a small number of relevant variables, for example the magnetization in a system of spins.
The Mori–Zwanzig formalism allows the finding of macroscopic equations that only depend on the relevant variables based on microscopic equations of motion of a system, which are usually determined by the Hamiltonian.
The formalism does not determine what the relevant variables are, these can typically be obtained from the properties of the system.
The observables describing the system form a Hilbert space.
[1] The irrelevant part of the dynamics then depends on the observables that are orthogonal to the relevant variables.
obeys the Heisenberg equation of motion where the Liouville operator
[4] This equation is formally solved by The projection operator acting on an observable
is the inverse temperature, Tr is the trace (corresponding to an integral over phase space in the classical case) and
It is chosen in such a way that it can be written as a function of the relevant variables only, but is a good approximation for the actual density, in particular such that it gives the correct mean values.
[6] Now, we apply the operator identity to Using the projection operator introduced above and the definitions (frequency matrix), (random force) and (memory function), the result can be written as This is an equation of motion for the observable
, the value at previous times (memory term) and the random force (noise, depends on the part of the dynamics that is orthogonal to
The equation derived above is typically difficult to solve due to the convolution term.
Since we are typically interested in slow macroscopic variables changing timescales much larger than the microscopic noise, this has the effect of integrating over an infinite time limit while disregarding the lag in the convolution.
is the fluctuation, be written as (use index notation with summation over repeated indices)[9] where and We have used the time-ordered exponential and the time-dependent projection operator These equations can also be re-written using a generalization of the Mori product.