Mori-Zwanzig formalism
The Mori–Zwanzig formalism, named after the physicists Hajime Mori 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. 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 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 irrelevant part appears in the equations as noise. 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. The projection operator then projects the dynamics onto the subspace spanned by the relevant variables.[1] 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,[2] which is why the formalism can also be used for analyzing the dynamics of correlation functions.[3]
Derivation
A not explicitly time-dependent observable[note 1] obeys the Heisenberg equation of motion
where the Liouville operator is defined using the commutator in the quantum case and using the Poisson bracket 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. [4] This equation is formally solved by
The projection operator acting on an observable is defined as
where is the relevant variable (which can also be a vector of various observables), and 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 , it is defined as[5]
where is the inverse temperature, Tr is the trace (corresponding to an integral over phase space in the classical case) and is the Hamiltonian. 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.[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 , which depends on its value at the current time , the value at previous times (memory term) and the random force (noise, depends on the part of the dynamics that is orthogonal to ).
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, this has the effect of integrating over an infinite time limit while disregarding the lag in the convolution. We see this by expanding the equation to second order in , to obtain[7]
- ,
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. [8] In this case, the Hamiltonian has explicit time-dependence.[note2 1] In this case, the transport equation for a variable
- ,
where is the mean value and 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. [2] Further generalizations can be used to apply the formalism to time-dependent Hamiltonians,[4][10] general relativity,[11] and arbitrary dynamical systems [12]
See also
Notes
- ^ An analogous derivation can be found in, e.g., Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed., Oxford University Press, New York, 2001, S.149 ff.
- ^ For a detailed derivation of the generalized equations of motion see Hermann Grabert Nonlinear Transport and Dynamics of Fluctuations Journal of Statistical Physics, Vol. 19, No. 5, 1978 and Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, Band 95, 1982
References
- ^ Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed., Oxford University Press, New York, 2001, S.144 ff.
- ^ a b Hermann Grabert Nonlinear Transport and Dynamics of Fluctuations Journal of Statistical Physics, Vol. 19, No. 5, 1978
- ^ Jean-Pierre Hansen und Ian R. McDonald, Theory of Simple Liquids: with Applications to Soft Matter 4th ed. (Elsevier Academic Press, Oxford, 2009), S.363 ff.
- ^ a b M. te Vrugt and R. Wittkowski Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians Physical Review E 99, 062118 (2019)
- ^ Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, Band 95, 1982, S.37
- ^ Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, Band 95, 1982, S.13
- ^ Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed., Oxford University Press, New York, 2001, S.165 ff.
- ^ Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, Band 95, 1982, S.36
- ^ Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, Band 95, 1982, S.18
- ^ Hugues Meyer, Thomas Voigtmann and Tanja Schilling On the dynamics of reaction coordinates in classical, time-dependent, many-body processes J. Chem. Phys. 150, 174118 (2019)
- ^ M. te Vrugt, S. Hossenfelder and R. Wittkowski, Mori-Zwanzig formalism for general relativity: a new approach to the averaging problem Phys. Rev. Lett. 127, 231101 (2021)
- ^ A. J. Chorin, O. H. Hald und R. Kupferman Optimal prediction with memory Physica D: Nonlinear Phenomena 166, 239{257 (2002)
- Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, Band 95, 1982
- Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed., Oxford University Press, New York, 2001