Mori-Zwanzig formalism

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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

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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

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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

2124: 3143: 634: 70:. 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 2832:{\displaystyle \phi _{ij}(t,s)={\text{Tr}}({\frac {\partial {\bar {\rho }}(t)}{\partial a_{j}(t)}}iL(1-P(s))G(s,t){\dot {A}}_{i})-{\dot {a}}_{k}(t){\text{Tr}}({\frac {\partial ^{2}{\bar {\rho }}(t)}{\partial a_{k}(t)\partial a_{j}(t)}}G(s,t){\dot {A}}_{i})} 2340: 2489: 3148:

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, general relativity, and arbitrary dynamical systems

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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.

1597: 908: 1350: 1885: 2966: 42:. 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 2212: 2955: 1737: 512: 63:. 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. 2219: 1216: 1033: 1123: 1812: 224: 153: 689: 427: 268: 738: 329: 1877: 958: 480:

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

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does not have explicit time-dependence. The derivation can also be generalized towards time-dependent Hamiltonians. This equation is formally solved by

754: 1227: 2119:{\displaystyle {\dot {A}}_{i}(t)=v_{i}(t)+\Omega _{ij}(t)\delta A_{j}(t)+\int _{0}^{t}dsK_{i}(t,s)+\phi _{ij}(t,s)\delta A_{j}(t)+F_{i}(t,0)} 24: 3138:{\displaystyle P(t)X={\text{Tr}}({\bar {\rho }}(t)X)+(A_{i}-a_{i}(t)){\text{Tr}}({\frac {\partial {\bar {\rho }}(t)}{\partial a_{i}(t)}}X).} 1404:, the value at previous times (memory term) and the random force (noise, depends on the part of the dynamics that is orthogonal to 2136: 2849: 1609: 629:{\displaystyle (X,Y)={\frac {1}{\beta }}\int _{0}^{\beta }d\alpha {\text{Tr}}({\bar {\rho }}Xe^{-\alpha H}Ye^{\alpha H}),} 691:

is the inverse temperature, Tr is the trace (corresponding to an integral over phase space in the classical case) and

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Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

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variables based on microscopic equations of motion of a system, which are usually determined by the

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is the fluctuation, be written as (use index notation with summation over repeated indices)

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On the dynamics of reaction coordinates in classical, time-dependent, many-body processes

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Mori-Zwanzig formalism for general relativity: a new approach to the averaging problem

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For a detailed derivation of the generalized equations of motion see Hermann Grabert

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is the relevant variable (which can also be a vector of various observables), and

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in the convolution. We see this by expanding the equation to second order in

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Journal of Statistical Physics, Vol. 19, No. 5, 1978 and Hermann Grabert

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