Mori-Zwanzig formalism

The Mori-Zwanzig formalism, named after the physicists Hajime Mori and Robert Zwanzig, is a method of statistical physics. It allows to split the dynamics of a system in a relevant and an irrelevant part using projection operators, which allows to find closed equations of motion for the relevant part. It is used e.g. in fluid mechanics or condensed matter physics.
Idea.
Macroscopic systems with a large number of microscopic degrees of freedom are usually well described by a small number of relevant variables, e.g. the magnetization in a system of spins. The Mori-Zwanzig formalism allows to find 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 irrelevant part appears in the equations as noise. The formalism does not allow to 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. The projection operator then projects the dynamics onto the subspace spanned by the relevant variables.. The irrelevant part of the dynamics then depends on the observables that are orthogonal to the relevant variables. A correlation function is used as a scalar product, which is why the formalism can also be used for analyzing the dynamics of correlation functions.
Derivation.
A not explicitly time-dependent observable formula_1 obeys the Heisenberg equation of motion
where the Liouville operator formula_3 is defined using the commutator formula_4 in the quantum case and using the Poisson bracket formula_5 in the classical case. We assume here that the Hamiltonian does not have explicit time-dependence. The derivation can also be generalized towards time-dependent Hamiltonians. This equation is formally solved by
The projection operator acting on an observable formula_7 is defined as
where formula_1 is the relevant variable (which can also be a vector of various observables), and formula_10 is some scalar product of operators. The Mori product, a generalization of the usual correlation function, is typically used for this scalar product. For observables formula_11, it is defined as
where formula_13 is the inverse temperature, Tr is the trace (corresponding to an integral over phase space in the classical case) and formula_14 is the Hamiltonian. formula_15 is the relevant probability operator (or density operator for quantum systems). 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.
Now, we apply the operator identity
to
Using the projection operator introduced above and the definitions
(frequency matrix),
(random force) und
(memory function), the result can be written as
This is an equation of motion for the observable formula_22, which depends on its value at the current time formula_23, the value at previous times (memory term) and the random force (noise, depends on the part of the dynamics that is orthogonal to formula_24).
Markovian approximation.
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. Expanding the equation to second order in formula_25, we obtain
where
Generalizations.
For larger deviations from thermodynamic equilibrium, the more general form of the Mori-Zwanzig formalism is used, from which the previous results can be obtained through a linearization. In this case, the Hamiltonian has explicit time-dependence. In this case, the transport equation for a variable
where formula_29 is the mean value and formula_30 is the fluctuation, be written as (use index notation with summation over repeated indices)
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. Further generalizations can be used to apply the formalism to time-dependent Hamiltonians and arbitrary dynamical systems 