Mori-twenty formalism

The Mori-twenty formalism , named after the physicists Hajime Mori and Robert Zwanzig , is a method of statistical physics. It enables the dynamics of a system to be broken down into a relevant and an irrelevant part using projection operators, whereby closed equations of motion can be found for the relevant part.

idea

Macroscopic systems with a large number of microscopic degrees of freedom are typically very well described by a small number of relevant variables, such as the total magnetization in spin systems. The Mori-Twenty formalism enables macroscopic equations to be obtained based on known microscopic equations of motion of a system - typically based on the Hamilton function of classical mechanics or the quantum mechanical Hamilton operator - which only depend on the relevant variables. The irrelevant part appears as noise in these equations. The formalism does not allow any statements about what the relevant variables are; these typically result from the properties of the system under investigation.

The observables that describe the system form a Hilbert space . The projection operator then corresponds to a mapping onto the subspace , which is only spanned by the relevant variables. The irrelevant part of the dynamics depends on observables orthogonal to the relevant variables. A correlation function serves as the scalar product in the space of the dynamic variables. As a result, the Mori twenty formalism can also be used to treat correlation functions.

Derivation

An observable that is not explicitly time-dependent obeys the Hamiltonian equation of motion in the Heisenberg picture ${\displaystyle A}$

${\displaystyle {\frac {d}{dt}}A=\mathrm {i} LA,}$

where the Liouville operator in the quantum mechanical case is defined by the commutator and in the classical case by the Poisson bracket (it is assumed here that the Hamilton operator does not depend on time, generalizations for the time-dependent case also exist). This equation is formally solved by ${\displaystyle L}$ ${\displaystyle L(\cdot )={\frac {1}{\hbar }}[H,\cdot ]}$ ${\displaystyle L(\cdot )=-\mathrm {i} \{H,\cdot \}}$ ${\displaystyle H}$

${\displaystyle A(t)=e^{\mathrm {i} Lt}A.}$

The projection operator that acts on an observable is defined by ${\displaystyle X}$

${\displaystyle PX=(A,A)^{-1}(X,A)A,}$

where is the relevant observable (this can also be a vector of several relevant quantities). The Mori product, a generalization of the usual correlation function, is typically used as the scalar product - noted here with round brackets. For observables this is defined by ${\displaystyle A}$ ${\displaystyle X,Y}$

${\displaystyle (X,Y)={\frac {1}{\beta }}\int _{0}^{\beta }\mathrm {d} \alpha \,\operatorname {Tr} ({\bar {\rho }}Xe^{-\alpha H}Ye^{\alpha H}).}$

Here is the inverse temperature, is the trace (in the classical case the phase space integral) and the Hamilton operator or the Hamilton function. is the relevant probability density (or the density operator in quantum mechanics). This is chosen so that it can only be written as a function of the relevant observables, but at the same time approximates the (usually unknown) actual probability density as well as possible, i.e. H. in particular delivers the same expected values ​​for the relevant variables as these. ${\displaystyle \beta =(k_{B}T)^{-1}}$ ${\displaystyle \operatorname {Tr} }$ ${\displaystyle H}$ ${\displaystyle {\bar {\rho }}}$

Now you apply the operator identity

${\displaystyle e^{\mathrm {i} Lt}=e^{\mathrm {i} (1-P)Lt}+\int _{0}^{t}\mathrm {d} s\,e^{\mathrm {i} L(t-s)}PLe^{\mathrm {i} (1-P)Ls}}$

on the size

${\displaystyle (1-P)\mathrm {i} LA}$

on. Taking advantage of the projection operator definition and definitions above

${\displaystyle \Omega =(\mathrm {i} LA,A)(A,A)^{-1}}$

(Frequency matrix),

${\displaystyle F(t)=e^{t(1-P)L}(1-P)\mathrm {i} LA}$

(stochastic force) and

${\displaystyle K(t)=(\mathrm {i} LF(t),A)(A,A)^{-1}}$

(Memory function) this can be written as

${\displaystyle {\dot {A}}(t)=\Omega A(t)+\int _{0}^{t}\mathrm {d} s\,K(s)A(t-s)+F(t).}$

This is an equation of motion for the relevant observable , which depends on the value of the observable at the point in time , the value at earlier points in time (memory term) and the stochastic force (noise, corresponds to the influence of the dynamics that are too orthogonal). ${\displaystyle A(t)}$ ${\displaystyle t}$ ${\displaystyle A(t)}$

Markov approximation

Due to the convolution term, the above equation is usually difficult to solve. Typically, one is interested in slow (macroscopic) variables, the change of which takes place on larger time scales than the (microscopic) noise. If one expands the equation to the second order in , one obtains ${\displaystyle \mathrm {i} LA(t)}$

${\displaystyle {\dot {A}}(t)\approx \Omega A(t)+\int _{0}^{\infty }\mathrm {d} s\,K(s)A(t)+F(t)}$,

in which

${\displaystyle K(t)=-(e^{\mathrm {i} Lt}(1-P)\mathrm {i} LA,(1-P)\mathrm {i} LA)(A,A)^{-1}}$

is.

Generalizations

In the event of any deviations from the thermodynamic equilibrium, the more general form of the Mori-twenty formalism is used, from which the above equations result as a linearization. In this case the projection operator depends explicitly on time. In this case the equation can be for a relevant observable

${\displaystyle A(t)=a(t)-\delta A(t)}$

(where is the mean and fluctuation) are written as (use index notation with summation convention) ${\displaystyle a(t)}$ ${\displaystyle \delta A(t)}$

${\displaystyle {\dot {A}}_{i}(t)=v_{i}(t)+\Omega _{ij}(t)\delta A_{j}(t)+\int _{0}^{t}\mathrm {d} s\,K_{i}(t,s)+\phi _{ij}(t,s)\delta A_{j}(t)+F_{i}(t,0)}$,

in which

${\displaystyle v_{i}(t)=\operatorname {Tr} \left({\bar {\rho }}(t)A_{i}\right)}$,

${\displaystyle \Omega _{ij}(t)=\operatorname {Tr} \left({\frac {\partial {\bar {\rho }}(t)}{\partial a_{j}(t)}}{\dot {A}}_{i}\right)}$,

${\displaystyle K_{i}(t,s)=\operatorname {Tr} \left({\bar {\rho }}(s)\mathrm {i} L(1-P(s))G(s,t){\dot {A}}_{i}\right),}$

and

${\displaystyle \phi _{ij}(t,s)=\operatorname {Tr} \left({\frac {\partial {\bar {\rho }}(t)}{\partial a_{j}(t)}}\mathrm {i} L(1-P(s))G(s,t){\dot {A}}_{i}\right)-{\dot {a}}_{k}(t)\operatorname {Tr} \left({\frac {\partial ^{2}{\bar {\rho }}(t)}{\partial a_{k}(t)\partial a_{j}(t)}}G(s,t){\dot {A}}_{i}\right)}$.

Here were the time-ordered exponentials

${\displaystyle G(s,t)=T_{-}\exp \left(\int _{s}^{t}\mathrm {d} u\,\mathrm {i} L(1-P(u))\right)}$

as well as the time-dependent projection operator

${\displaystyle P(t)X=\operatorname {Tr} \left({\bar {\rho }}(t)X)+(A_{i}-a_{i}(t)\right)\operatorname {Tr} \left({\frac {\partial {\bar {\rho }}(t)}{\partial a_{i}(t)}}X\right)}$

used. These equations can also be represented by means of a generalization of the Mori product using correlation functions. Further variants of the Mori-twenty formalism are used to describe systems with time-dependent Hamilton operators or general dynamic systems.

See also

• Nakajima twenty equation

Remarks

1. ↑ An analogous derivation can be found, e.g. B., in Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed. , Oxford University Press, New York, 2001, pp. 149 ff.
2. ↑ For a more detailed description of the derivation of the generalized equations 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, Volume 95, 1982

literature

• Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics , Springer Tracts in Modern Physics, Volume 95, 1982
• Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed. , Oxford University Press, New York, 2001

Individual evidence

1. ^ Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed. , Oxford University Press, New York, 2001, pp. 144 ff.
2. ^ ^{A b} Hermann Grabert Nonlinear Transport and Dynamics of Fluctuations Journal of Statistical Physics, Vol. 19, No. 5, 1978
3. ^ Jean-Pierre Hansen and Ian R. McDonald, Theory of Simple Liquids: with Applications to Soft Matter 4th ed. (Elsevier Academic Press, Oxford, 2009), p. 363 ff.
4. ↑ ^{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)
5. ^ Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics , Springer Tracts in Modern Physics, Volume 95, 1982, p. 37
6. ^ Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics , Springer Tracts in Modern Physics, Volume 95, 1982, p. 13
7. ^ Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed. , Oxford University Press, New York, 2001, pp. 165 ff.
8. ^ Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics , Springer Tracts in Modern Physics, Volume 95, 1982, p. 36
9. ^ Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics , Springer Tracts in Modern Physics, Volume 95, 1982, p. 18
10. ↑ 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)
11. ^ AJ Chorin, OH Hald and R. Kupferman Optimal prediction with memory Physica D: Nonlinear Phenomena 166, 239 {257 (2002)