Temporal coarse-graining and elimination of slow dynamics with the generalized Langevin equation for time-filtered observables

We consider a general time-independent Hamiltonian for a system of $N$ interacting particles or atoms in three-dimensional space

$\displaystyle H(\omega)$

$\displaystyle=\sum_{n=1}^{N}\frac{\mathbf{p}_{n}^{2}}{2m_{n}}+V(\{\mathbf{r}_{N}\}),$

(1)

where a point in the 6$N$-dimensional phase space is denoted by $\omega=(\{\mathbf{r}_{N}\},\{\mathbf{p}_{N}\})$, which is a $6N$-dimensional vector consisting of the Cartesian particle positions $\{\mathbf{r}_{N}\}$ and the conjugate momenta $\{\mathbf{p}_{N}\}$ and fully specifies the microstate of the system. The Hamiltonian splits into a kinetic and a potential part and $m_{n}$ is the mass of particle $n$. The potential $V(\{\mathbf{r}_{N}\})$ contains all interactions between the particles and includes possible external potentials.

Using the Liouville operator

$\displaystyle{\cal L}(\omega)$

$\displaystyle=\sum_{n=1}^{N}\sum_{\alpha=x,y,z}\left(\frac{\partial H}{\partial p_{n}^{\alpha}}\frac{\partial}{\partial r_{n}^{\alpha}}-\frac{\partial H}{\partial r_{n}^{\alpha}}\frac{\partial}{\partial p_{n}^{\alpha}}\right),$

(2)

the $6N$-dimensional Hamilton equation of motion can be compactly written as $\dot{\omega}(t)={\cal L}(\omega)\omega(t)$, where $\omega(t)$ is the phase-space location of the system at time $t$ and $\dot{\omega}(t)={\rm d}\omega(t)/{\rm d}t$ is the corresponding phase-space velocity. From the Hamilton equation and since the Liouville operator is time-independent, it follows that the phase space position is propagated in time by the operator exponential $e^{(t-t_{0}){\cal L}(\omega)}$, i.e., $e^{(t-t_{0}){\cal L}(\omega)}\omega(t_{0})=\omega(t)$ [44]. Instead of following microstate trajectories in phase space, which is the Lagrangian description of the system dynamics, it is convenient to switch to the Eulerian description and consider the time-dependent probability density distribution as a function of phase space, $\rho(\omega,t)$. The density obeys the Liouville equation $\dot{\rho}(\omega,t)=-{\cal L}(\omega)\rho(\omega,t)$, which has the formal solution $\rho(t)=e^{-(t-t_{0}){\cal L}(\omega)}\rho(\omega,t_{0})$ with some initial distribution $\rho(\omega,t_{0})$, which need not be a stationary distribution [44]. A system observable can be generally written as a Schrödinger-type (i.e. time-independent) phase-space function $B_{S}(\omega)$, it can for example represent the position of one particle, its momentum, the center-of-mass position of a group of particles, or the reaction coordinate describing a chemical reaction or the folding of a protein. Since the phase space can be defined as including the entire universe, so that no external time-dependent external perturbation needs to be taken into account [15, 45], $B_{S}(\omega)$ can also represent more complex observables such as the position or state of a living organism, a meteorological or an economical observable. To simplify the notation, we consider a scalar observable but note that our formalism can be straightforwardly extended to multidimensional observables. Using the probability density, the time-dependent expectation value (or mean) of the observable $B_{S}(\omega)$ can be written as

	$\displaystyle b(t)$	$\displaystyle\equiv\int{\rm d}\omega\,B_{S}(\omega)\rho(\omega,t)$
		$\displaystyle=\int{\rm d}\omega\,B_{S}(\omega)e^{-(t-t_{0}){\cal L}(\omega)}\rho(\omega,t_{0}).$		(3)

Since the Liouville operator is anti-self adjoint, it follows that [44]

	$\displaystyle b(t)$	$\displaystyle=\int{\rm d}\omega\,\rho(\omega,t_{0})e^{(t-t_{0}){\cal L}(\omega)}B_{S}(\omega)$
		$\displaystyle=\int{\rm d}\omega\,\rho(\omega,t_{0})B(\omega,t).$		(4)

In the last step we have defined the Heisenberg observable as

$\displaystyle B(\omega,t)$

$\displaystyle\equiv e^{(t-t_{0}){\cal L}(\omega)}B_{S}(\omega),$

(5)

which is the central object of the projection formalism and of GLEs. Obviously, as follows from Eq. (5), it satisfies the equation of motion

$\displaystyle\dot{B}(\omega,t)$

$\displaystyle={\cal L}(\omega)B(\omega,t)$

(6)

with the initial condition $B(\omega,t_{0})=B_{S}(\omega)$. To understand the meaning of a Heisenberg observable, we for the moment consider the density distribution $\rho(\omega,t_{0})=\delta(\omega-\omega_{0})$, which describes a system that at time $t_{0}$ is in the microstate $\omega_{0}$. Inserting this into Eq. (II.1), we obtain $b(t)=B(\omega_{0},t)$. In other words, $B(\omega_{0},t)$ describes the dynamics of an observable for a system that at time $t=t_{0}$ was in the microstate $\omega_{0}$, i.e., it describes the temporal evolution of the conditional mean of the observable $B_{S}(\omega)$. It transpires that if we derive an equation of motion for $B(\omega,t)$, we have an equation for how this conditional mean changes in time. This is the central idea of projection and of GLEs.

In all what follows, we will omit the phase-space argument of the Liouville operator and simply denote it as ${\cal L}$. Introducing the general convolutional filter function $f(t)$, we define the filtered mean of an observable as

$\displaystyle b_{f}(t)$

$\displaystyle\equiv\int_{-\infty}^{\infty}{\rm d}s\,f(s)b(t-s).$

(7)

Filtering necessarily occurs whenever an experimental measurement is done, but it can also be used purposely to decompose data into separate contributions, to remove unwanted components of time series data or to enhance certain interesting features. Using Eq. (II.1), the time filtering of the observable mean can be written in terms of a phase-space average as

	$\displaystyle b_{f}(t)$	$\displaystyle=\int_{-\infty}^{\infty}{\rm d}s\,f(s)\int{\rm d}\omega\,\rho(\omega,t_{0})B(\omega,t-s)$
		$\displaystyle=\int{\rm d}\omega\,\rho(\omega,t_{0})\int_{-\infty}^{\infty}{\rm d}sf(s)B(\omega,t-s)$
		$\displaystyle=\int{\rm d}\omega\,\rho(\omega,t_{0})B_{f}(\omega,t).$		(8)

In the last line we defined the filtered Heisenberg observable

	$\displaystyle B_{f}(\omega,t)$	$\displaystyle\equiv\int_{-\infty}^{\infty}{\rm d}s\,f(s)B(\omega,t-s)$		(9)
		$\displaystyle=\int_{-\infty}^{\infty}{\rm d}s\,f(s)e^{(t-s-t_{0}){\cal L}}B_{S}(\omega),$		(10)

which by construction has the same properties as the unfiltered Heisenberg observable in Eqs. (5) and (6): it is time propagated according to

	$\displaystyle e^{t^{\prime}{\cal L}}B_{f}(\omega,t)$	$\displaystyle=e^{t^{\prime}{\cal L}}\int_{-\infty}^{\infty}{\rm d}s\,f(s)e^{(t-s-t_{0}){\cal L}}B_{S}(\omega)$
		$\displaystyle=\int_{-\infty}^{\infty}{\rm d}s\,f(s)e^{(t^{\prime}+t-s-t_{0}){\cal L}}B_{S}(\omega)$
		$\displaystyle=B_{f}(\omega,t^{\prime}+t)$		(11)

and it satisfies the equation of motion

$\displaystyle\dot{B}_{f}(\omega,t)$

$\displaystyle={\cal L}B_{f}(\omega,t).$

(12)

Thus, although $B_{f}(\omega,t)$ involves a weighted integral over time and thus is not an ordinary observable, i.e., an instantaneously defined phase-space function, its dynamics is identical to that of an ordinary Heisenberg observable. This simple but profound property, which is one of the main results of this paper, will allow us to derive a GLE for the filtered Heisenberg observable $B_{f}(\omega,t)$ in exactly the same way as it is commonly done for the ordinary (unfiltered) Heisenberg observable $B(\omega,t)$.

In principle any function $f(t)$ can be employed as a temporal filter, but there are a few particularly useful filter functions. In Fourier space Eq. (9) reads

$\displaystyle\tilde{B}_{f}(\omega,\nu)$

$\displaystyle=\tilde{f}(\nu)\tilde{B}(\omega,\nu),$

(13)

where we define Fourier transforms as $\tilde{f}(\nu)=\int{\rm d}te^{-\imath\nu t} f(t)$. The normalized Gaussian filter with a temporal width $\lambda$,

$\displaystyle f_{\rm LP}(t)$

$\displaystyle=e^{-t^{2}/(2\lambda^{2})}(2\pi\lambda^{2})^{-1/2},$

(14)

is a low-pass filter that is commonly used for smoothing experimental and simulation data. Its Fourier transform is given by a Gaussian as well,

$\displaystyle\tilde{f}_{\rm LP}(\nu)$

$\displaystyle=e^{-\lambda^{2}\nu^{2}/2}.$

(15)

The Gaussian high-pass filter is the complement of the Gaussian low-pass filter and is given by

$\displaystyle f_{\rm HP}(t)$

$\displaystyle=\delta(t)-f_{\rm LP}(t)$

(16)

with the Fourier transform

$\displaystyle\tilde{f}_{\rm HP}(\nu)$

$\displaystyle=1-e^{-\lambda^{2}\nu^{2}/2}.$

(17)

It can be used to eliminate the slow or transient dynamics of a trajectory. The normalized Gaussian band-pass filter

$\displaystyle f_{\rm BP}(t,\nu_{0})$

$\displaystyle=\frac{\cos(\nu_{0}t)e^{-t^{2}/(2\lambda^{2})}}{(2\pi\lambda^{2})^{1/2}e^{-\lambda^{2}\nu_{0}^{2}/2}}$

(18)

with its Fourier transform

$\displaystyle\tilde{f}_{\rm BP}(\nu,\nu_{0})$

$\displaystyle=\frac{e^{-\lambda^{2}(\nu-\nu_{0})^{2}/2}+e^{-\lambda^{2}(\nu+\nu_{0})^{2}/2}}{2e^{-\lambda^{2}\nu_{0}^{2}/2}}$

(19)

can be used to perform convolutional Fourier analysis. To illustrate its properties, we consider as an example a cosine function with a finite phase $\phi$,

$\displaystyle B(\omega,t)=B_{0}(\omega)\cos(\nu^{\prime}_{0}t+\phi),$

(20)

where $B_{0}(\omega)$ is an arbitrary phase-space function. $B(\omega,t)$ has the Fourier transform

$\displaystyle\tilde{B}(\omega,\nu)=\pi B_{0}(\omega)\left(e^{\imath\phi} \delta(\nu-\nu^{\prime}_{0})+e^{-\imath\phi} \delta(\nu+\nu^{\prime}_{0})\right).$

(21)

According to Eqs. (13) and (18) and after a few intermediate steps, the filtered function can be written in Fourier space as

$\displaystyle\tilde{B}_{f}(\omega,\nu,\nu_{0})=\tilde{B}(\omega,\nu)\frac{e^{-\lambda^{2}(\nu_{0}-\nu^{\prime}_{0})^{2}/2}+e^{-\lambda^{2}(\nu_{0}+\nu^{\prime}_{0})^{2}/2}}{2e^{-\lambda^{2}\nu_{0}^{2}/2}}$

(22)

and becomes sharply peaked for a filter frequency $\nu_{0}$ around the frequency $\nu^{\prime}_{0}$ of the input function in the limit $\lambda\nu^{\prime}_{0}\gg 1$. This simple example shows that the Gaussian convolutional band-pass filter projects onto oscillatory contributions in time-series data and thus can be used analogously to ordinary Fourier transformation (which, obviously, does not correspond to a convolution). The Gaussian band-stop filter is the complement of the Gaussian band-pass filter and given by

$\displaystyle f_{\rm BS}(t,\nu_{0})$

$\displaystyle=\delta(t)-f_{\rm BP}(t)$

(23)

with the Fourier transform

$\displaystyle\tilde{f}_{\rm BS}(\nu,\nu_{0})$

$\displaystyle=1-\frac{e^{-\lambda^{2}(\nu-\nu_{0})^{2}/2}+e^{-\lambda^{2}(\nu+\nu_{0})^{2}/2}}{2e^{-\lambda^{2}\nu_{0}^{2}/2}}.$

(24)

It can be used to eliminate oscillatory dynamics with oscillation frequency $\nu_{0}$ in time-series data. Of course, one can apply an arbitrary combination of filter functions onto time-series data and the resulting function behaves as a regular phase-space observable, as explained in Sect. II.2.

Using the definition Eq. (9), we decompose the unfiltered Heisenberg observable $B(\omega,t)$ into the filtered Heisenberg observable $B_{f}(\omega,t)$ and its complement $B^{*}_{f}(\omega,t)$ according to

$\displaystyle B(\omega,t)$

$\displaystyle=B_{f}(\omega,t)+B^{*}_{f}(\omega,t).$

(25)

Note that this decomposition is exact and can be viewed as the definition of the complement. By adding many convolution-filtered Heisenberg observables according to $B_{f}(\omega,t)=\sum_{i}B^{i}_{f}(\omega,t)$, where the $i$ numbers different convolution filters, we can apply multiple low-pass, high-pass, band-pass and band-stop filters on the data. According to Eq. (25), the acceleration of the unfiltered Heisenberg observable can be decomposed as

$\displaystyle\ddot{B}(\omega,t)$

$\displaystyle=\ddot{B}_{f}(\omega,t)+\ddot{B}^{*}_{f}(\omega,t).$

(26)

As a matter of fact, one can analyze the two parts on the right side of Eq. (26) separately and using different types of theoretical description. For the example of a high-pass filter, one could use a stochastic description for the high-pass filtered part $B_{f}(\omega,t)$, which contains the fast fluctuations, while the low-pass complement $B^{*}_{f}(\omega,t)$, which contains the slow components, could be described by a deterministic model. In the following we will apply the projection formalism on $\ddot{B}_{f}(\omega,t)$ but stress that projection can be equally well applied on the complementary part $\ddot{B}_{f}^{*}(\omega,t)$.

Here we follow standard procedures [44]. We introduce a projection operator ${\cal P}$ that acts on phase space functions and its complementary operator ${\cal Q}$ via the relation $1={\cal Q}+{\cal P}$. Inserting this unit operator into the time propagator relation Eq. (II.2) and using Eq. (12), we obtain

		$\displaystyle\ddot{B}_{f}(\omega,t)=e^{(t-t_{P}){\cal L}}({\cal P}+{\cal Q}){\cal L}^{2}B_{f}(\omega,t_{P})$
		$\displaystyle=e^{(t-t_{P}){\cal L}}{\cal P}{\cal L}^{2}B_{f}(\omega,t_{P})+e^{(t-t_{P}){\cal L}}{\cal Q}{\cal L}^{2}B_{f}(\omega,t_{P}),$		(27)

where $t_{P}$ defines the time at which the projection is performed, which in principle can differ from the time $t_{0}$ at which the time propagation of the Heisenberg variable in Eq. (II.2) starts. The projection operators ${\cal Q}$ and ${\cal P}$ in general depend on the projection time $t_{P}$ (which is not explicitly written out) but not on the time $t$. This allows us to use the standard Dyson operator decomposition [46, 47, 48] for the propagator $e^{(t-t_{P}){\cal L}}$

$\displaystyle e^{(t-t_{P}){\cal L}}=e^{(t-t_{P})Q{\cal L}}+\int_{0}^{t-t_{P}}\mathrm{d}s\,e^{(t-t_{P}-s){\cal L}}{\cal P}{\cal L}e^{sQ{\cal L}}.$

(28)

Inserting this decomposition into the second term on the right hand side in Eq. (II.4), we obtain the GLE in general form

	$\displaystyle\ddot{B}_{f}(\omega,t)$	$\displaystyle=e^{(t-t_{P}){\cal L}}{\cal P}{\cal L}^{2}B_{f}(\omega,t_{P})+F(\omega,t)$
		$\displaystyle+\int_{0}^{t-t_{P}}\rm{d}s\,e^{(t-t_{P}-s){\cal L}}{\cal P} {\cal{\cal L}} F(\omega,s+t_{P}),$		(29)

where the complementary force is given by

$\displaystyle F(\omega,t)$

$\displaystyle\equiv e^{(t-t_{P}){\cal Q}{\cal L}}{\cal Q}{\cal L}^{2}B_{f}(\omega,t_{P}).$

(30)

The first term on the right-hand side in Eq. (II.4) will turn out to represent the conservative force from a potential, the third term represents non-Markovian friction effects and the force $F(\omega,t)$ represents all effects that are not included in the other two terms. $F(\omega,t)$ is a function of phase space and evolves in the complementary space, i.e. it satisfies ${\cal P}F(\omega,t)=0$ (as will be shown further below).

Clearly, the explicit form of Eq. (II.4) depends on the specific form of the projection operator $\cal P$. Here we choose the Mori projection, because it is straightforward to implement and for the derivation of the filtering effects based on two-point correlations functions is exact, as we will discuss below. We note that our filter-projection approach can also be used in conjunction with hybrid projection operators for which the resulting GLE explicitly contains a non-linear potential of mean force [49, 50, 51]. The Mori projection using $B_{f}(\omega,t_{P})$ as a projection function and applied on a general Heisenberg observable $A(\omega,t)$ is given by [31, 50]

		$\displaystyle{\cal P}A(\omega,t)=\langle A(\omega,t)\rangle+\frac{\langle A(\omega,t){\cal L} B_{f}(\omega,t_{P})\rangle}{\langle({\cal L}B_{f}(\omega,t_{P}))^{2}\rangle}{\cal L}B_{f}(\omega,t_{P})$
		$\displaystyle+\frac{\langle A(\omega,t)(B_{f}(\omega,t_{P})-\langle B_{f}\rangle)\rangle}{\langle(B_{f}(\omega,t_{P})-\langle B_{f}\rangle)^{2}\rangle}(B_{f}(\omega,t_{P})-\langle B_{f}\rangle).$		(31)

We define the expectation value of an arbitrary phase-space function $X(\omega)$ with respect to a time-independent projection distribution $\rho_{\rm P}(\omega)$ as

$\displaystyle\langle X(\omega)\rangle=\int{\rm d}\omega X(\omega)\rho_{\rm P}(\omega),$

(32)

which we here take to be the normalized equilibrium canonical distribution $\rho_{\rm P}(\omega)=e^{-H(\omega)/(k_{B}T)}/Z$, where $Z$ is the partition function. For this stationary projection distribution the average $\langle B_{f}\rangle=\langle B_{f}(\omega,t_{P})\rangle$ is independent of time. The Mori projection in Eq. (II.4) projects onto a constant, the filtered observable $B_{f}(\omega,t_{P})$ and its time derivative ${\cal L}B_{f}(\omega,t_{P})$ at time $t_{P}$, the projection time. Thus, the projection in Eq. (II.4) maps any observable $A(\omega,t)$ onto the subspace of all functions linear in the observables 1, $B_{f}(\omega,t_{P})$ and ${\cal L}B_{f}(\omega,t_{P})$, meaning that ${\cal P}1=1$, ${\cal P}B_{f}(\omega,t_{P})=B_{f}(\omega,t_{P})$ and ${\cal P}{\cal L}B_{f}(\omega,t_{P})={\cal L}B_{f}(\omega,t_{P})$. From this follows immediately that ${\cal Q}1={\cal Q}B_{f}(\omega,t_{P})={\cal Q}{\cal L}B_{f}(\omega,t_{P})=0$, which is a property that will become important later on.

The Mori projection is linear, i.e., for two arbitrary observables $A(\omega,t)$ and $C(\omega,t^{\prime})$ it satisfies ${\cal P} (c_{1}A(\omega,t)+c_{2}C(\omega,t^{\prime}))=c_{1}{\cal P}A(\omega,t)+c_{2}{\cal P}C(\omega,t^{\prime})$, it is idempotent, i.e., ${\cal P}^{2}={\cal P}$, and it is self-adjoint, i.e. it satisfies the relation

$\displaystyle\langle A(\omega,t){\cal P}C(\omega,t^{\prime})\rangle=\langle C(\omega,t^{\prime}){\cal P}A(\omega,t)\rangle.$

(33)

From these properties it follows that the complementary projection operator ${\cal Q}=1-{\cal P}$ is also linear, idempotent and self-adjoint. Thus, ${\cal P}$ and ${\cal Q}$ are orthogonal to each other, i.e. ${\cal P}{\cal Q}=0={\cal Q}{\cal P}$ [45].

Using all these properties, the GLE Eq. (II.4) takes the form [31, 44]

	$\displaystyle\ddot{B}_{f}(\omega,t)$	$\displaystyle=-K_{f}(B_{f}(\omega,t)-\langle B_{f}\rangle)+F(\omega,t)$
		$\displaystyle-\int_{0}^{t-t_{P}}{\rm d}s\,\Gamma_{f}(s)\dot{B}_{f}(\omega,t-s),$		(34)

details of the derivation are shown in Appendix A. The parameter $K_{f}$, which corresponds to the potential stiffness divided by the effective mass, is given by

$\displaystyle K_{f}$

$\displaystyle=\frac{\langle({\cal L}B_{f}(\omega,t_{P}))^{2}\rangle}{\langle(B_{f}(\omega,t_{P})-\langle B_{f}\rangle)^{2}\rangle}$

(35)

and the memory friction kernel is given by

$\displaystyle\Gamma_{f}(s)=\frac{\langle F(\omega,t_{P})F(\omega,s+t_{P})\rangle}{\langle({\cal L}B_{f}(\omega,t_{P}))^{2}\rangle}=\frac{\langle F(\omega,0)F(\omega,s)\rangle}{\langle({\cal L}B_{f}(\omega,t_{P}))^{2}\rangle}.$

(36)

Eq. (II.4) is an exact decomposition of the Liouville equation into three terms, the first term is a force due to a quadratic potential, the third term accounts for linear non-Markovian friction and depends on the memory kernel $\Gamma_{f}(s)$, which is related via Eq. (36) to the second moment of the force $F(\omega,t)$, defined in Eq. (30). In particular, eq. (II.4) is exact and time-reversible, as are the underlying Hamilton and Liouville equations. The explicit form of the memory function can be computed for simple model systems, as will be demonstrated later on. Note that the force $F(\omega,t)$ can be extracted from simulation or experimental data by different techniques [52, 53, 49]. Due to the specific form of the Mori projection in Eq. (II.4), several expectation values involving the force vanish, namely $\langle F(\omega,t)\rangle=\langle F(\omega,t)B_{f}(\omega,t_{P})\rangle=\langle F(\omega,t){\cal L}B_{f}(\omega,t_{P})\rangle=0$, which are important properties for extracting GLE parameters from time-series data.

For practical applications, one typically models the force $F(\omega,t)$ as a stochastic process with zero mean and a second moment given by Eq. (36), higher-order moments of $F(\omega,t)$ are typically neglected and the distribution of $F(\omega,t)$ is assumed to be Gaussian, which is valid for the calculation of two-point correlation functions. For non-linear systems and if one is interested in higher-order correlations, however, this assumption can not hold, since $F(\omega,t)$ is the only term in the GLE that accounts for non-linearities. Thus, imposing $F(\omega,t)$ to be a Gaussian variable can become a bad approximation for non-linear systems, which reflects a fundamental short-coming of the Mori projection scheme in conjunction with replacing $F(\omega,t)$ by a random Gaussian process. Alternative methods to derive GLEs with non-linear potential and friction terms have been recently proposed [49, 50, 54, 51]. We stay here with the Mori scheme because it simplifies analytical calculations and is exact for our calculations of two-point correlations. We note that high-pass filtering produces data that is Gaussian to a very good approximation.

We now interpret the GLE in Eq. (II.4) for the phase-space dependent filtered Heisenberg observable $B_{f}(\omega,t)$ as a stochastic equation. For a more stream-lined notation, we skip the phase-space dependence and define the filtered observable by subtracting its mean as

$\displaystyle x_{f}(t)=B_{f}(\omega,t)-\langle B_{f}\rangle.$

(37)

We also shift the projection time into the far past, $t_{P}\rightarrow-\infty$, define the memory kernel to be causal or single-sided, i.e. $\Gamma_{f}(t)=0$ for $t<0$, and define a stochastic random force as $F_{R}(t)=F(\omega,t)$. The GLE reads

$\displaystyle\ddot{x}_{f}(t)$

$\displaystyle=-K_{f}x_{f}(t)+F_{R}(t)-\int_{-\infty}^{\infty}{\rm d}s\,\Gamma_{f}(s)\dot{x}_{f}(t-s),$

(38)

where the potential parameter is, following Eq. (35), given by

$\displaystyle K_{f}$

$\displaystyle=\frac{\langle\dot{x}_{f}^{2}\rangle}{\langle x_{f}^{2}\rangle}$

(39)

and the memory friction kernel is, following Eq. (36), given by

$\displaystyle\Gamma_{f}(t)=\theta(t)\frac{\langle F_{R}(0)F_{R}(t)\rangle}{\langle\dot{x}_{f}^{2}\rangle}$

(40)

where $\theta(t)$ defines the Heavyside function and the mean-squared velocity $\langle\dot{x}^{2}\rangle$ would, for a standard equilibrium observable and according to the equipartition theorem, correspond to the thermal energy divided by the effective mass. The force $F_{R}(t)$ is characterized by its second moment according to Eq. (40), while its first moment vanishes, $\langle F_{R}(t)\rangle=0$, as explained before. This, however, does not mean that the random force is necessarily Gaussian. Note that from now on, all expectation values are defined as averages with respect to the random force distribution. This reinterpretation of the GLE in Eq. (II.4) as a stochastic integro-differential equation simplifies the notation significantly and is exact as long as care is taken to relegate all phase-space dependencies into the random force $F_{R}(t)$.

By Fourier transformation, Eq. (38) can be solved in terms of the linear-response relation as

$\displaystyle\tilde{x}_{f}(\nu)=\tilde{\chi}_{f}(\nu)\tilde{F}_{R}(\nu),$

(41)

where the response function is given as

$\displaystyle\tilde{\chi}_{f}(\nu)=\left[K_{f}-\nu^{2}+\imath\nu\tilde{\Gamma}_{f}(\nu)\right]^{-1}.$

(42)

We define the filtered two-point positional auto-correlation function as

$\displaystyle C_{xx}^{f}(t)=\langle x_{f}(t^{\prime})x_{f}(t+t^{\prime})\rangle,$

(43)

the Fourier transform of which reads

		$\displaystyle\tilde{C}_{xx}^{f}(\nu)=\int\frac{{\rm d}t^{\prime}}{2\pi} e^{\imath t^{\prime}(\nu+\nu^{\prime})}\langle\tilde{x}_{f}(\nu)\tilde{x}_{f}(\nu^{\prime})\rangle$
		$\displaystyle=\int\frac{{\rm d}t^{\prime}}{2\pi} e^{\imath t^{\prime}(\nu+\nu^{\prime})}\tilde{\chi}_{f}(\nu)\tilde{\chi}_{f}(\nu^{\prime})\langle\tilde{F}_{R}(\nu)\tilde{F}_{R}(\nu^{\prime})\rangle$		(44)

where in the last step we used Eq. (41). The random-force autocorrelation in Fourier space follows from Eq. (40) as

$\displaystyle\langle\tilde{F}_{R}(\nu)\tilde{F}_{R}(\nu^{\prime})\rangle=2\pi\delta(\nu+\nu^{\prime})\langle\dot{x}_{f}^{2}\rangle\left[\tilde{\Gamma}_{f}(\nu)+\tilde{\Gamma}_{f}(\nu^{\prime})\right].$

(45)

Combining Eqs. (III.1) and (45) we obtain

		$\displaystyle\tilde{C}_{xx}^{f}(\nu)=\langle\dot{x}_{f}^{2}\rangle\tilde{\chi}_{f}(\nu)\tilde{\chi}_{f}(-\nu)\left[\tilde{\Gamma}_{f}(\nu)+\tilde{\Gamma}_{f}(-\nu)\right]$
		$\displaystyle=\frac{\langle\dot{x}_{f}^{2}\rangle}{\imath\nu} \left[\tilde{\chi}_{f}(-\nu)-\tilde{\chi}_{f}(\nu)\right],$		(46)

where in the last step we used Eq. (42). Eq. (III.1) forms the starting point for a few further derivations and is exactly produced by the Mori GLE. First, we multiply Eq. (III.1) by $\imath\nu$ to obtain

$\displaystyle\tilde{C}_{x\dot{x}}^{f}(\nu)\equiv\imath\nu\tilde{C}_{xx}^{f}(\nu)=\langle\dot{x}_{f}^{2}\rangle \left[\tilde{\chi}_{f}(-\nu)-\tilde{\chi}_{f}(\nu)\right],$

(47)

where $\tilde{C}_{x\dot{x}}^{f}(\nu)$ is the Fourier transform of the derivative of the positional autocorrelation function $C_{x\dot{x}}^{f}(t)\equiv{\rm d}C_{xx}^{f}(t)/{\rm d}t=\langle x_{f}(0)\dot{x}_{f}(t)\rangle$. The right-hand side of Eq. (47) splits into a causal part, $\tilde{\chi}_{f}(\nu)$, and an anticausal part, $\tilde{\chi}_{f}(-\nu)$, therefore we can also split the left-hand side into causal and anticausal parts and finally obtain

$\displaystyle\tilde{C}_{x\dot{x}}^{+f}(\nu)=-\langle\dot{x}_{f}^{2}\rangle\tilde{\chi}_{f}(\nu),$

(48)

where $\tilde{C}_{x\dot{x}}^{+f}(\nu)$ is the Fourier transform of the single-sided correlation function, $C_{x\dot{x}}^{+f}(t)=\theta(t)C_{x\dot{x}}^{f}(t)$. Eq. (48) is the Fourier transform of the fluctuation dissipation theorem [44] and one can use it to extract all parameters of the filtered GLE from filtered time series data. To see this we combine Eqs. (48) and (42) to obtain

$\displaystyle-\frac{\langle\dot{x}_{f}^{2}\rangle }{\tilde{C}_{x\dot{x}}^{+f}(\nu)}=K_{f}-\nu^{2}-\nu\tilde{\Gamma}_{f}^{\prime\prime}(\nu)+\imath\nu\tilde{\Gamma}_{f}^{\prime}(\nu),$

(49)

where we split the memory function into its real and imaginary parts according to $\tilde{\Gamma}_{f}(\nu)=\tilde{\Gamma}_{f}^{\prime}(\nu)+\imath\tilde{\Gamma}_{f}^{\prime\prime}(\nu)$. It transpires that the real and imaginary parts of the memory function are determined by

	$\displaystyle\Gamma_{f}^{\prime}(\nu)=-\left(\frac{\langle\dot{x}_{f}^{2}\rangle }{\nu\tilde{C}_{x\dot{x}}^{+f}(\nu)}\right)^{\prime\prime},$		(50a)
	$\displaystyle\Gamma_{f}^{\prime\prime}(\nu)=\frac{K_{f}}{\nu}-\nu+\left(\frac{\langle\dot{x}_{f}^{2}\rangle }{\nu\tilde{C}_{x\dot{x}}^{+f}(\nu)}\right)^{\prime}.$		(50b)

The potential parameter $K_{f}$ follows from Eq. (39) or, alternatively, from Eq. (49) in the zero-frequency limit as

$\displaystyle K_{f}=-\frac{\langle\dot{x}_{f}^{2}\rangle }{\tilde{C}_{x\dot{x}}^{+f}(0)}=\frac{\langle\dot{x}_{f}^{2}\rangle }{\langle x_{f}^{2}\rangle },$

(51)

where we used that the Fourier transform of the memory function is finite in the zero frequency limit. In numerical applications, Eq. (51) can serve as a check on the accuracy of the numerical determination of $\tilde{C}_{x\dot{x}}^{+f}(\nu)$. Eqs. (50) and (51) can be used to extract all parameters of the filtered GLE in Eq. (38) from a filtered trajectory, since $\tilde{C}_{x\dot{x}}^{+f}(\nu)$ and the expectation values $\langle x_{f}^{2}\rangle$ and $\langle\dot{x}_{f}^{2}\rangle$ can be directly evaluated from time-series data.

Let us derive a second useful result from Eq. (III.1). Since $\chi_{f}(t)$ is a real function, it follows that $\tilde{\chi}_{f}^{\prime}(\nu)=\tilde{\chi}_{f}^{\prime}(-\nu)$ and $\tilde{\chi}_{f}^{\prime\prime}(\nu)=-\tilde{\chi}_{f}^{\prime\prime}(-\nu)$. Using this in Eq. (III.1), we obtain

$\displaystyle\tilde{C}_{xx}^{f}(\nu)=-\frac{2\langle\dot{x}_{f}^{2}\rangle\tilde{\chi}_{f}^{\prime\prime}(\nu)}{\nu},$

(52)

which is an alternative formulation of the fluctuation-dissipation theorem in Fourier space [44]. In the zero-frequency limit we obtain from Eq. (52)

$\displaystyle\tilde{C}_{xx}^{f}(0)=\frac{2\langle\dot{x}_{f}^{2}\rangle\tilde{\Gamma}_{f}(0)}{K_{f}^{2}}=\frac{2\langle x_{f}^{2}\rangle\tilde{\Gamma}_{f}(0)}{K_{f}},$

(53)

where we used Eqs. (42) and (39) and $\tilde{\Gamma}_{f}(0)$, the integral over the memory function, is the friction coefficient. For unconfined systems with $K_{f}=0$, $\tilde{\Gamma}_{f}(0)$ can be obtained from Eq. (52) using the velocity autocorrelation function $C_{\dot{x}\dot{x}}^{f}(t)\equiv\langle\dot{x}_{f}(0)\dot{x}_{f}(t)\rangle=-{\rm d^{2}}C_{xx}^{f}(t)/{\rm d}t^{2}$ as

$\displaystyle\tilde{C}_{\dot{x}\dot{x}}^{f}(0)=\frac{2\langle\dot{x}_{f}^{2}\rangle}{\tilde{\Gamma}_{f}(0)},$

(54)

which corresponds to the standard relation between the integral over the velocity autocorrelation function, $\tilde{C}_{\dot{x}\dot{x}}^{f}(0)$, and the diffusion constant [44]. Relations Eqs. (53) and (54) can be used to obtain the friction coefficient $\tilde{\Gamma}_{f}(0)$ from filtered trajectory data.

We now discuss the mapping between the filtered observable $x_{f}(t)$, described by the GLE in Eq. (38), and the unfiltered observable, denoted as $x(t)$ and which we describe by the same GLE but with different (i.e. unfiltered) parameters $K$, $\Gamma(t)$ and an unfiltered random force that is characterized by its second moment according to Eq. (40) where $\langle\dot{x}^{2}_{f}\rangle$ is replaced by $\langle\dot{x}^{2}\rangle$. Similarly to Eq. (13), the relation between $x_{f}(t)$ and $x(t)$ is given by

$\displaystyle\tilde{x}_{f}(\nu)=\tilde{f}(\nu)\tilde{x}(\nu).$

(55)

The question is, given the parameters of the GLE for the unfiltered observable, what are the parameters of the GLE for the filtered observable, or vice versa. As turns out, this is a rather non-trivial question.

First, using Eq. (55), the positional autocorrelation function for the filtered observable $x_{f}(t)$ in Eq. (III.1) can be written as

	$\displaystyle\tilde{C}_{xx}^{f}(\nu)$	$\displaystyle=\int\frac{{\rm d}t^{\prime}}{2\pi} e^{\imath t^{\prime}(\nu+\nu^{\prime})}\tilde{f}(\nu)\tilde{f}(\nu^{\prime})\langle\tilde{x}(\nu)\tilde{x}(\nu^{\prime})\rangle$
		$\displaystyle=\tilde{f}(\nu)\tilde{f}(-\nu)\tilde{C}_{xx}(\nu),$		(56)

where we followed the same steps leading to Eq. (III.1). This relation connects the autocorrelations for the filtered observable $\tilde{C}_{xx}^{f}(\nu)$ and the unfiltered observable $\tilde{C}_{xx}(\nu)$ and furnishes the central relation for understanding the effect of filtering on the system dynamics.

Combining Eqs. (53) and (III.2) leads to

$\displaystyle \frac{\langle\dot{x}_{f}^{2}\rangle\tilde{\Gamma}_{f}(0)}{K^{2}_{f}}= \frac{\tilde{f}^{2}(0)\langle\dot{x}^{2}\rangle\tilde{\Gamma}(0)}{K^{2}},$

(57)

which is a relation between the parameters of the GLEs describing the filtered and unfiltered observables. In the following sections we will disentangle the filtering effect on the GLE parameters.