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

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

Michael te Vrugt Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany    Raphael Wittkowski raphael.wittkowski@uni-muenster.de Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany Center for Nonlinear Science, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany
Abstract

The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in a lot of scenarios like, for example, driven soft matter or nuclear magnetic resonance. In this article, we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.

I Introduction

The Mori-Zwanzig projection operator formalism Mori (1965); Zwanzig (1960); Grabert (1982); Zwanzig (2001); Forster (1989); Hansen and McDonald (2009) is a central tool of statistical physics. It is based on the observation that macroscopic systems are typically well described by a small number of relevant variables, even though they have a large number of microscopic degrees of freedom Grabert (1978). Typical examples include fluids, where the relevant variables are mass, momentum, and energy densities Forster (1989); Sasa (2014), or spin systems, where the relevant variable is the magnetization Kivelson and Ogan (1974). The key idea is to introduce a projection operator that projects the full microscopic dynamics of the system onto the subspace that depends only on the relevant variables. Thereby, one obtains closed equations of motion for the relevant variables, in which the irrelevant dynamics appears as a noise term Grabert (1982); Givon (2005).

This allows to derive mesoscopic and macroscopic theories based on known microscopic equations of motion in a systematic and rather compact way of coarse graining Hijón et al. (2010); Español (2004); Español and Donev (2015). Therefore, the Mori-Zwanzig formalism is a very useful method for a variety of fields, such as fluid mechanics Grabert (1982); Forster (1989), polymer physics Li et al. (2017); Hijón et al. (2010), classical dynamical density functional theory Español and Löwen (2009); Anero et al. (2013); Wittkowski et al. (2012, 2013); Camargo et al. (2018), solid-state theory Kakehashi and Fulde (2004), spin relaxation theory Kivelson and Ogan (1974); Bouchard (2007), dielectric relaxation theory Khamzin et al. (2012); Nigmatullin and Nelson (2006), spectroscopy Schmitt et al. (2006), calculation of correlation functions Hansen and McDonald (2009), plasma physics Diamond et al. (2010), and particle physics Huang et al. (2011). Due to its great importance, it is also studied in disciplines outside of physics such as mathematics Givon (2005); Chorin et al. (2000); Dominy and Venturi (2017) and philosophy Wallace (2015). A significant extension of the formalism is therefore likely to have a strong impact on a large number of areas.

The currently used form of the Mori-Zwanzig formalism faces the problem that it cannot be directly applied to systems with time-dependent Hamiltonians. Those, however, are relevant for a lot of scenarios including soft matter systems subject to time-dependent external driving forces Komura and Ohta (2012); Menzel (2015) or nuclear magnetic resonance (NMR) measurements with rapidly varying electromagnetic pulses Bouchard (2007).

If arising, time-dependent Hamiltonians can sometimes be treated as additional external perturbations Grabert (1982). This requires, however, that the perturbation is sufficiently small and couples to the macroscopic variables only. Generalizations of the projection operator method towards non-Hamiltonian dynamical systems have been developed by Chorin, Hald, and Kupferman Chorin et al. (2000, 2002). Xing and Kim use mappings between dissipative and Hamiltonian systems Xing (2010) to apply projection operators in the non-Hamiltonian case Xing and Kim (2011). The methods from Refs. Chorin et al. (2002); Xing (2010); Xing and Kim (2011), however, are not applicable to quantum-mechanical systems. Moreover, approximation methods commonly applied in the context of the Mori-Zwanzig formalism, such as the linearization around thermodynamic equilibrium Grabert (1982), rely on the existence of a Hamiltonian.

The Mori-Zwanzig formalism exists in a variety of forms. The original theory developed by Mori Mori (1965) can be applied to systems with time-independent Hamiltonians that are close to thermal equilibrium. A generalization towards systems far from equilibrium using time-dependent projection operators has been presented by Robertson Robertson (1966), Kawasaki and Gunton Kawasaki and Gunton (1973), and Grabert Grabert (1982, 1978). Recently, Bouchard has derived an extension of the Mori theory towards systems with time-dependent Hamiltonians Bouchard (2007) that can be applied close to equilibrium. What is still missing, however, is a general formalism for systems that have a time-dependent Hamiltonian and are far from equilibrium.

General discussions of projection operators used for deriving coarse-grained equations of motion also exist for time-dependent Hamiltonians in nonequilibrium statistical physics Uchiyama and Shibata (1999) and quantum field theory Koide and Maruyama (2000). These methods, like the Bouchard equation Bouchard (2007), are derived without explicitly assuming a close-to-equilibrium situation and therefore always formally valid. However, they use time-independent projection operators and therefore do not form generalizations of the full Mori-Zwanzig formalism presented by Grabert Grabert (1982). The problem with time-independent projection operators is that they are only useful if the relevant dynamics is linear, since nonlinear effects are projected out together with the irrelevant dynamics Grabert (1982); Zwanzig (2001). A useful description of far-from-equilibrium dynamics needs to be capable of representing nonlinear couplings. Moreover, these methods do not involve appropriate generalizations of the correlators, which are essential for the Mori-Zwanzig formalism.

In this article, we therefore derive an extension of the Mori-Zwanzig formalism for far-from-equilibrium systems with time-dependent Hamiltonians. For this purpose, we extend Grabert’s treatment by introducing suitable generalized correlators and equations of motion. We also discuss how observables with explicit time dependence can be treated within this framework. Furthermore, we develop approximation methods that can be used to simplify the resulting equations. These approximations include a generalized Markovian approximation for slow variables, a linearization around thermodynamic equilibrium which recovers the Bouchard equation, the Magnus expansion for time-ordered exponentials, and the classical limit. To demonstrate how the formalism can be used, we apply it in combination with the approximation methods to the important case of spin relaxation. In this context, we show how the Bloch equations for spin relaxation in the presence of a time-dependent external magnetic field can be derived.

In the classical case, mappings between dissipative and Hamiltonian systems as described by Xing and Kim Xing (2010); Xing and Kim (2011) can extend the applicability of our formalism towards non-Hamiltonian systems, since they allow for the construction of a corresponding Hamiltonian. Since our method is not restricted to time-independent Hamiltonians, this would even allow for the treatment of arbitrary nonautonomous dynamical systems. It is also possible to include stochastic equations in this way. The stochastic contributions can be modeled using a harmonic bath Hamiltonian Xing (2010); Xing and Kim (2011); Zwanzig (1973).

This article is organized as follows: In Section II, we derive the extended projection operator formalism. Approximations are discussed in Section III. In Section IV, the Bloch equations are obtained within our framework. Finally, we summarize our results in Section V.

Ii Derivation of the extended projection operator formalism

ii.1 Time-dependent Liouvillians

In the Heisenberg picture of quantum mechanics, a system is described by a set of observables corresponding to time-dependent Hermitian Hilbert space operators. Any operator obeys the Heisenberg equation of motion Münster (2010)

(1)

where is the imaginary unit, the reduced Planck constant, and a commutator. With , we denote the Heisenberg picture Hamiltonian of the system, which can differ from the Schrödinger picture Hamiltonian , if the Hamiltonian has explicit time dependence in the Schrödinger picture Münster (2010). In most cases, one assumes that the observables are not explicitly time-dependent. Defining the Heisenberg picture Liouvillian111Some authors include the imaginary unit in the definition of the Liouvillian, so that Eq. 3 reads . In this case, the form of all resulting equations has to be modified accordingly. This has no additional effect on any of the calculations.

(2)

we can write Eq. 1 as

(3)

The difference between Heisenberg picture Hamiltonian and Schrödinger picture Hamiltonian leads to a difference between the Heisenberg picture Liouvillian and a Schrödinger picture Liouvillian defined as

(4)

If the Hamiltonian does not depend on time, the Liouvillian also does not and we denote them by and , respectively. In the time-independent case, Schrödinger and Heisenberg picture Hamiltonians as well as Schrödinger and Heisenberg picture Liouvillians coincide and we do not need a subscript to distinguish between them. Equation (3) can then formally be solved as

(5)

where . However, this is no longer possible, if the Hamiltonian depends on time, since one usually222There are special situations in which, even though the Hamiltonian is time-dependent, the relevant Liouvillian is not, because the time-dependent part of the Hamiltonian commutes with the observable of interest, which is not true in general. has a time-dependent Liouvillian then. In this case, Eq. 3 has to be solved using time-ordered exponentials.

For convenience, we repeat here the standard derivation presented, e.g., by Peskin and Schroeder Peskin and Schroeder (1995). We assume . Integrating Eq. 3 gives

(6)

This can be solved by iteration:

(7)

Since the Liouvillians stand in time order, where the operators on the left correspond to later times, we can use the identity Peskin and Schroeder (1995)

(8)

with the left-time-ordering operator . That operator is, for a time-dependent operator , defined as

(9)

so that operators are always ordered in such a way that time increases from right to left. For a , we arrive at the solution

(10)

We also consider the backwards case, which we need further below. For a , we get

(11)

As the Liouvillians are also time-ordered here, now with earlier times on the left, we obtain, using the identity

(12)

where is the right-time-ordering operator that puts later times on the right, the solution

(13)

In particular, this means (replace and )

(14)

We introduce the abbreviations

(15)
(16)

to simplify our notation in the following.

An important difference between left- and right-time-ordered exponentials is their behavior under differentiation. For left-time-ordered exponentials one has Holian and Evans (1985)

(17)

i.e., the inner derivative stands on the left of the exponential. In contrast, for right-time-ordered exponentials one has

(18)

where the inner derivative is on the right. This is relevant, because usually and the time-ordered exponential do not commute due to the noncommutativity of the Hamiltonians at different points in time. For some parts of the derivation of the extended projection operator formalism, it is essential that inner derivatives stand on the right. The identity

(19)

allows to write Eq. 10 as

(20)

Using the Schrödinger picture Liouvillians therefore allows to take advantage of the properties of right-time-ordered exponentials.

Equation (19) can be proven in two ways. The first option is a direct calculation. Writing out the operator exponentials as expansions of commutators using the definitions (2) and (4) for Heisenberg and Schrödinger picture Liouvillians, respectively, and inserting the transformation rule

(21)

with the unitary operator Peskin and Schroeder (1995)

(22)

and a Hermitian adjoint denoted by , gives, after sorting terms, the identity (19). Here, we use the second option, which is more elegant and gives better insights into the physics behind Eq. 19. If we work in the Schrödinger instead of the Heisenberg picture, the operators are time-independent, while the wave functions are time-dependent. This leads to a time-dependent density operator Jensen and Mackintosh (1991), since the density operator is constructed from the wave functions. The time evolution of the density operator is, in the Schrödinger picture, given by the Liouville-von Neumann equation Grabert (1982)

(23)

which has the formal solution

(24)

The mean value of an operator in the Schrödinger picture is therefore obtained through Grabert (1982)

(25)

where denotes the trace. We now wish to transform Eq. 25 to the Heisenberg picture. This is important, because in the Mori-Zwanzig formalism is usually prescribed as an initial condition at , but not known for larger . Using Eqs. 10 and 8, we can write Eq. 25 as

(26)

We can now use the relation Grabert (1982)

(27)

which is based on . This relation is applied repeatedly to the right-hand side of Eq. 26 (once to the second term, twice to the third term,…). The result is

(28)

It shows that, if one transforms from the Schrödinger to the Heisenberg picture, the Schrödinger picture Liouvillians act on the operators in right time order. On the other hand, we also could have worked directly in the Heisenberg picture. In this case, the mean value is given by Grabert (1982)

(29)

where we have used Eq. 10. Comparing Eqs. 29 and 28 gives

(30)

Since we have made no assumptions about the form of , this also proves the identity (19). For extended discussions on transformations between Schrödinger and Heisenberg pictures, see Holian and Evans for the classical Holian and Evans (1985); Evans and Morriss (2008) and Uchiyama and Shibata Uchiyama and Shibata (1999) for the quantum mechanical case.

ii.2 Projection operator and correlator

As a starting point for our extension, we use the formalism presented by Grabert Grabert (1978, 1982). Since it works in the nonlinear regime arbitrarily far from thermal equilibrium, it is by now the most general projection operator theory for systems with not explicitly time-dependent Hamiltonians. Extending this formalism will therefore allow to obtain one that has a greater range of applicability than existing theories.

We start with introducing a relevant probability density . The complete microscopic state of the system is described by the actual probability density , which is typically not known. Supposing that the macroscopic thermodynamic state of the system is well described by a set of macroscopic observables with mean values , one constructs the relevant density in such a way that it is only a function of the and macroequivalent to the actual density in the sense that Grabert (1978)

(31)

Although any choice meeting those restrictions is formally possible, the resulting equations are particularly useful, if the relevant density is a good approximation for the actual density. This is facilitated, if one can assume – as we do for this derivation – that the system is initially prepared in the state Grabert (1982). Following Grabert Grabert (1978) and Anero et al. Anero et al. (2013), we use the microcanonical form333Throughout this article, summation over each index appearing twice in a term is assumed.

(32)

where is a normalization function ensuring . The conjugate variables are defined by the macroequivalence condition (31) and therefore functions of the . In the case of non-Hermitian operators , Eq. 32 has to be modified to ensure that the statistical operator is still Hermitian. Moreover, the macroequivalence condition (31) can, in particular cases, not be fulfilled for non-Hermitian operators, if the form (32) is chosen. However, this is unproblematic, since almost all applications are based on Hermitian operators. We will therefore, throughout this work, assume that all observables are Hermitian, if not stated otherwise. For a discussion of possible modifications of Eq. 32 in the case of certain important non-Hermitian operators, see Appendix B.

A motivation for choosing this form is that it maximizes the Gibbs entropy based on the available information Grabert (1978); Anero et al. (2013), which here is given by the . If one defines a coarse-grained dimensionless entropy as

(33)

the conjugate variables are given by Grabert (1978)

(34)

Typically, the use of the entropy as a thermodynamic potential is particularly appropriate for closed systems. It has, in the case of time-dependent Hamiltonians, the advantage that the Hamiltonian does not appear in its definition, so that it is not explicitly time-dependent. Alternatively, one can use a free energy combined with a canonical relevant density, which we discuss in Section III.2.

One might question whether a relevant probability density as in Eq. 32 is in fact a reasonable approximation for the actual probability density in the case of time-dependent Hamiltonians, since those systems are typically driven out of equilibrium, making arguments based on maximal-entropy principles seemingly less plausible. There are three reasons justifying to choose the form (32). First, the motivation for choosing the maximal Gibbs entropy form is that this is the “least biased” form regarding missing knowledge about the microscopic configuration, i.e., it is justified from an information theoretic point of view Anero et al. (2013). Second, the resulting equations of motion are exact regardless of the choice of as long as the macroequivalence condition (31) is satisfied Grabert (1978), and this choice allows to express this condition in a useful way as equations (34) for the conjugate variables . Third, we can assume that the time dependence (e.g., by an external field) is switched on at , which means that the initial condition is satisfied. For a detailed discussion of initial nonequilibrium states see Ref. Zwanzig (2001).

The aim is now to separate the dynamics into two parts: the organized motion, which is entirely determined by the macroscopic mean values, i.e., the relevant density , and the disorganized motion, which corresponds to deviations of the actual dynamics from the organized motion. For analyzing this, we decompose the operator as

(35)

where describes the fluctuations of around the mean value . The organized motion of the mean values is given by Grabert (1978)

(36)

This relation defines the organized drift Grabert (1978). Since the Liouvillian acts directly on the Schrödinger picture operator , we have to use the Schrödinger picture Liouvillian . Note that, for a time-dependent Liouvillian as assumed here, one cannot unambiguously write as it is done in the usual presentations. The deviation

(37)

is the disorganized drift. Based on the identity Grabert (1978)

(38)

and the fact that the time evolution of the relevant density is given by

(39)

Grabert shows that the organized motion of the fluctuations , which is defined by demanding that it does not lead to deviations of from , is given by Grabert (1978)

(40)

with the frequency matrix

(41)

Up to now, our treatment of the organized motion did not differ from Grabert’s treatment, since none of the steps done so far hangs on the fact that the time evolution is given by . New and interesting aspects come into play, however, if we introduce the projection operator . The aim of the projection operator is to extract the organized motion from the total dynamics, which means that the effect of the projection operator is determined by the previously derived relations for the organized motion. Grabert uses these results to write Grabert (1978)

(42)

where the projection operator is given by

(43)

If the relevant variables are not explicitly time-dependent, has the projection operator property444For explicitly time-dependent operators, this still holds for , so that in Eq. 43 is a projection operator. Grabert (1982, 1978)

(44)

It is important here that is a Schrödinger picture operator that is then propagated using to get the value of the projected observable at time . Therefore, in order to be able to directly apply the projection operator (43) used by Grabert, we need to use Schrödinger picture Liouvillians and, therefore, right-time-ordered exponentials.

An important part of the theory – and another point where there is a crucial difference between our formalism and the standard procedure – is the choice of a suitable correlator that is needed for specifying correlation functions. This correlator is of particular interest here as it provides a generalization of the scalar product in the space of dynamical variables. Grabert introduces the generalized canonical correlation Grabert (1978)

(45)

for two time-dependent operators and . This correlator is used to derive certain relations, in particular an equation for the frequency matrix Grabert (1978)

(46)

The generalized canonical correlation (45) reduces to Kubo’s correlation functions in thermal equilibrium and to the standard correlation functions of statistical mechanics for classical systems. Its main purpose in the Mori-Zwanzig formalism is that the projection operator can be written in terms of it. In the general case, however, Eq. 45 does not have the properties of a usual correlator. It has this specific form precisely because this allows to derive Eq. 46. An appropriate generalization of the canonical correlator towards time-dependent Liouvillians therefore has to be constructed in such a way that Eq. 46 still holds. We found that the best choice for the generalized correlator, i.e., the simplest form that meets the requirement (46) and reduces to Eq. 45 for time-independent Liouvillians, is

(47)

This correlator, together with the fact that is should be written using Schrödinger picture Liouvillians, constitutes the first main result of this section. In Appendix A, we explicitly prove that this correlator gives the desired result for the case of Hermitian operators, which correspond to physical observables. Since Eq. 47 reduces to Eq. 45 for a time-independent Liouvillian, it has the other properties of a correlator mentioned above.

The correlator (47) can be used to re-write the equation (43) for the projection operator . With Eqs. 236 and 231 (see Appendix A), one can write

(48)

Therefore, Eq. 43 can be re-written as

(49)

ii.3 Time evolution of relevant variables

To obtain an equation of motion for the relevant dynamics, we follow the familiar procedure of decomposing the time-evolution operator used, e.g., by Zwanzig Zwanzig (2001), Grabert Grabert (1982), and Bouchard Bouchard (2007). We use right-time-ordered exponentials due to the properties they have if they are acted upon by the time derivative (see Section II.1). To be able to do this, we also use Schrödinger picture Liouvillians.

We start with

(50)

where is the operator complementary to . From the term

(51)

we get the time derivative

(52)

The last term of Eq. 52 depends on macroscopic variables and can be considered a known inhomogeneity. Realizing that the homogeneous solution of Eq. 52 is with the time-ordered exponential

(53)

and using the initial condition

(54)

we thus find the result Grabert (1982)

(55)

This gives the operator identity

(56)

If we consider the special case and a time-independent projection operator , we can write this as555 is a propagator acting on the subspace orthogonal to the relevant variables, so it commutes with .

(57)

Comparing the prefactors of on both sides of the equation gives666Of course, does in general not prove , if is a projection operator, since and could in principle have different effects on the subspace orthogonal to . However, since the choice of relevant observables determining the projection operator is free, we consider the case , completing the proof. Alternatively, Eq. 58 can be easily confirmed by taking the time derivative on both sides of the equation Bouchard (2007).

(58)

This is the generalization of the Dyson decomposition derived by Holian and Evans Holian and Evans (1985); Evans and Morriss (2008). It reduces to the usual Dyson decomposition, if we assume that also the Liouvillian does not depend on time. In the case of time-independent projection operators, the generalized Dyson decomposition (58) can be used to derive the equation of motion more directly Bouchard (2007).

Applying the identity (56) to gives

(59)

Using Eq. 43 and Grabert (1978)

(60)

as well as introducing the after-effect function

(61)

the memory function

(62)

and the random force

(63)

we can write Eq. 59 as the equation of motion

(64)

As for the usual Mori-Zwanzig formalism, averaging of Eq. 64 allows to derive the dynamics of the mean values Grabert (1982)

(65)

with . The term can be dropped, if one assumes that the initial distribution is given by the relevant density Grabert (1982, 1978). Equations (64) and (65) also lead to Grabert (1982)

(66)

with .

Equation (64) is valid for any . The physical significance of is that the memory functions and are functionals of the mean paths . We get closed equations of motion (65) for the mean values, where depends only on the current macroscopic state given by , while the memory functions depend on the macroscopic state at previous times in the interval Grabert (1982). Note that the validity of Eq. 64 does not rely on choosing the form (32) for the relevant density as long as it has no explicit time dependence, but only on the form (43) of the projection operator.

ii.4 Time evolution of correlation functions

Using the previous results, it is possible to obtain very directly the rules for the time evolution of correlation functions defined via the generalized correlator (47). This is done by taking the time derivative of the equation for the correlation function one is interested in and applying the rules obtained so far.

It is particularly interesting to consider the time correlation of the fluctuations, since here a closed and elegant result can be found. This correlation function is defined as

(67)

Inserting Eq. 66 into the time derivative

(68)

results in

(69)

where we have used Grabert (1978)

(70)

ii.5 Explicitly time-dependent operators

An interesting scenario that receives very little attention in the literature is that of explicitly time-dependent operators, for which the partial derivative with respect to in Eq. 1 cannot be dropped. This scenario can occur, e.g., when the energy density is a relevant variable, since the corresponding operator adopts the explicit time dependence of the Hamiltonian. Therefore, we need to adapt our formalism to the full Heisenberg equation, if we want to derive equations of motion for explicitly time-dependent operators. We use the definition

(71)

for the projection operator to ensure that

(72)

holds, where is the explicitly time-dependent Schrödinger picture operator corresponding to in Eq. 43. The crucial question here is how to calculate the time evolution using Schrödinger picture Liouvillians. In the Schrödinger picture, the time evolution is given by

(73)

Equation (73) holds, because the explicit time dependence of has no influence on the time evolution of the density operator . Using the procedure described in Section II.1, a transformation to the Heisenberg picture gives

(74)

Therefore, the appropriate equation is

(75)

i.e., the partial derivative does not have to be taken into account when constructing the exponential. The time derivative is then

(76)

For deriving the equations of motion, we can apply the identity (56) for the time-ordered exponential in Eq. 76. A further difficulty here is that explicitly time-dependent observables also lead to explicitly time-dependent relevant densities and projection operators, so that

(77)

The result is the equation of motion

(78)

with the (modified) definitions

(79)
(80)
(81)
(82)
(83)
(84)

Interestingly, the explicitly time-dependent operators require an additional memory function . The general form (78) for the equation of motion is the second main result of this section. Note that here

(85)

as the derivation in Appendix A assumes .

ii.6 Discussion

The general form of our equations is very similar to Grabert’s results for time-independent Hamiltonians Grabert (1978, 1982). This is not surprising, since his treatment forms the starting point for our derivation. However, our discussion reveals some nontrivial new aspects. In particular, it is helpful to calculate the time evolution using Schrödinger picture Liouvillians, since this allows for the utilization of right-time-ordered exponentials. As discussed by Uchiyama and Shibata Uchiyama and Shibata (1999), right-time-ordered exponentials are a typical feature of Heisenberg picture projection operator formalisms. Moreover, we have constructed an equation for the generalization of the correlator, which also can be constructed based on Schrödinger picture Liouvillians, and explicitly calculated that it has the role it is supposed to have. What is completely new in treatments of the Mori-Zwanzig formalism is that we discuss the possibility of observables with explicit time dependence, to which we extend the formalism in a very natural way.

Iii Approximations for the extended formalism

The transport equations derived in Section II are, in practical applications, typically too complex to be solved exactly. In this section, we therefore derive approximation methods that can be used to simplify the results in the important cases of slow variables, close-to-equilibrium systems and classical dynamics. Throughout this section, we assume that the observables do not have explicit time dependence. A generalization using the methods described in Section II.5 is straightforward. Without loss of generality, we assume that , where is the reference time for the equation of motion (64) introduced in Section II.3.

iii.1 Generalized Markovian approximation

In many situations, one is interested in slowly varying macroscopic variables. A typical example is the case of physical quantities that follow a conservation law Forster (1989). Slow variables allow the use of certain approximations, which simplify the structure of the resulting equations and facilitate practical applications. This is known as Markovian approximation. In the following, we give a generalization of the corresponding discussion by Grabert Grabert (1982) towards the case of time-dependent Liouvillians.

Equations (27) and (38) allow to write the after-effect function (61) as

(86)

where we introduce the retardation matrix

(87)

Modifications of the form of can be necessary for the case of non-Hermitian operators, since Eq. (87) is based on Eq. (32). With Eq. 86, one can write Eq. 65 as

(88)

If we are interested in slow variables, we can drop terms that are of higher than second order in . Note that the assumption that is small does not imply that is slowly varying. Since the elements are of second order and variations of are of first order in , we can use the quasi-stationary path for in the last term of Eq. 88 Grabert (1982). The reason is that a Taylor expansion gives

(89)

i.e., any deviation of from is at least of first order in . For the same reason, we can replace by , by , and by , noting that the relevant density and the projection operator are functionals of the mean values. No such approximation is possible, however, where the time dependence does not arise through the mean values of the (slow) relevant variables but directly through the Hamiltonian, i.e., we cannot replace by . The time-ordered exponential can, in the case of a time-independent Hamiltonian, be approximated as

(90)

where one uses the quasi-stationary approximation and the fact that is of order , since

(91)

In our case, however, we have

(92)

Unlike in the usual case, the time-ordered exponential cannot be dropped here. We therefore get

(93)

One can then proceed by noticing that the quasi-stationary approximation requires that a clear separation of time scales between slow macroscopic and fast microscopic processes exists. This in turn requires that with the characteristic time scale of macroscopic variables , i.e., the have to decay on a much shorter relaxation time scale . Otherwise, higher-order terms in Eq. 89 would not be negligible for larger values of . But if the vanish for , we can extend the time integral in Eq. 88 to without changing the result. This allows in the case of a time-independent Liouvillian, where carries the only dependence that contributes to Eq. 93, to write

(94)

Then, the memory kernel (86) on the right-hand side of Eq. (88) has effectively no dependence on . Applying this procedure gives in our case

(95)

Unlike before, one cannot remove the dependence from the kernel. We therefore get

(96)

with the diffusion tensor

(97)

There is a very notable difference to the usual case discussed by Grabert Grabert (1982). In that case, the diffusion tensor does not explicitly depend on time, since the time dependence arises only through the or appearing in the reduced Hamiltonian and the projection operator. In our case, however, even the approximation corresponding to slow variables and time-scale separation still leaves us with an explicitly time-dependent diffusion tensor , which will have considerable effects on the phenomenology of transport equations that can be derived from the presented formalism. We can replace by in Eq. 88 as is of second order in Grabert (1982). This leads to

(98)

which is the generalized Markovian approximation of the equation of motion (88).

A similar line of argument can be used regarding the time evolution of the fluctuations, assuming that they are also slow. The memory function (62) is of second order in . We can therefore make the quasi-stationary approximation

(99)

where we have replaced by , by , by , and by . If we now consider the equation of motion for the fluctuations, which is given by Eq. 66, we can replace by . This corresponds to the assumption that also the variations of are negligible on the time scale on which decays.777The memory function decays on the same time scale as Grabert (1982). The integral over the memory function in Eq. 66 can be extended to an integral from to using the same series of substitutions as in Eq. 95. We can therefore write Eq. 66 as