text:
\section{Introduction}
\label{sec:Intro}
Brownian motion plays a key role in dynamics of microparticles and living microorganisms, and fundamental research of Brownian systems is of high importance for practical applications in physics, biology, and medicine.

A natural way to understand any physical system, including a Brownian one, is to study its response to external perturbations. A small perturbation, where the response is linear in perturbation forces, already allows to draw important conclusions~\cite{Altland2010}. As a better alternative to the direct measurement of the linear response, linear response theory can be used, which allows to predict the response via two-point correlation functions of the unperturbed system~\cite{Altland2010}. This direct correspondence between the response and unperturbed fluctuations is known as the fluctuation-dissipation-theorem (FDT)~\cite{Altland2010, Marconi2008}. Starting from the famous Sutherland-Einstein-Smoluchowski relation~\cite{Altland2010, Marconi2008, Sutherland1905, Einstein1905, Smoluchowski1906}, a big progress in linear response theory has been made, mostly regarding generalization of FDT to various systems and perturbations~\cite{Altland2010, Marconi2008, Callen1951, Weber1956, Green1954, Kubo1957, Kubo1966, Kubo1991, Hansen2009}. In recent years, the focus has been moving towards the response of nonequilibrium systems~\cite{Harada2005, Speck2006, Blickle2007, Chetrite2008, Baiesi2009_1, Baiesi2009_2, Baiesi2010, Prost2009, Kruger2009, Kruger2010, Seifert2010_1, Seifert2010_2, Seifert2012, Warren2012, Szamel2017, Caprini2018, Asheichyk2019_1, DalCengio2019, DalCengio2021, Asheichyk2020, Caprini2021_1, Caprini2021_2}.

Yet linear response theory for equilibrium Brownian systems is still far from being understood comprehensively. Although it is known that one can use different response formulas to compute a certain response for a particular scenario~\cite{Marconi2008, Speck2006, Seifert2010_1, Asheichyk2019_1, Asheichyk2020, Asheichyk2019_2}, the comparison of the formulas (e.g., in terms of their statistical efficiency) was only very recently performed~\cite{Asheichyk2019_1, Asheichyk2020, Asheichyk2019_2}. Such a comparison can help choosing the best formula for a particular case, in order to minimize computational resources or experimental measurement time. Another important aspect, which has not been studied in detail, concerns the role of system properties in response computations. For example, certain symmetries of the considered system allow to compute the response via alternative ways~\cite{Asheichyk2019_1, Asheichyk2020, Asheichyk2019_2}. One more fact is that underdamped Brownian particles have been investigated much less frequently than overdamped ones, because the latter model is more simple yet widely applicable in many practical cases. Even if the particle inertia is formally considered~\cite{Baiesi2010, Warren2012, Asheichyk2020, Asheichyk2019_2, Rzehak2003, Holzer2010, Lander2012, Kahlert2012}, typically,  little attention is paid to its effect on the system response. On the other hand, it was proven that inertial effects play an important role at short time scales~\cite{Lukic2005, Blum2006, Li2010, Huang2011, Kheifets2014, Pusey2011}. In that respect, the relaxation dynamics of a perturbed system towards the steady state is expected to be very sensitive to the particle mass.

In this work, we address the three aforementioned aspects in linear response theory of Brownian motion. As for the perturbation, we consider simple shear flow, an important paradigmatic force driving the system out of equilibrium~\cite{Kruger2009, Kruger2010, Seifert2010_1, Warren2012, Asheichyk2019_1, Asheichyk2020, Asheichyk2019_2, Rzehak2003, Holzer2010, Lander2012, Kahlert2012, Orihara2011, Derksen1990, Fuchs2015, Ortman2012, Ziehl2009, Fuchs2005, Amann2013, Uspal2015, tenHagen2011, Li2017, Sandoval2014, Sandoval2018}. Based on the approaches of Refs.~\cite{Asheichyk2019_1, Asheichyk2020, Asheichyk2019_2}, we demonstrate in detail how the response to shear can be equivalently replaced by the response to a potential, thanks to certain symmetries (Sec.~\ref{sec:Symmetries}). In Sec.~\ref{sec:Six_ways}, we collect six different routes for obtaining the linear response, including  a new Green-Kubo relation for underdamped Brownian particles. These six methods are demonstrated in a concrete example of underdamped interacting particles, where interesting relaxation effects due to the particle inertia are discussed and the four response relations are analyzed in detail in terms of their statistical efficiency (Sec.~\ref{sec:Investigation}). Finally, some aspects regarding the difference between the linear and nonlinear responses are investigated (Sec.~\ref{sec:Nonlinear}).

\section{System}
\label{sec:System}
\begin{figure}[!t]
\includegraphics[width=0.8\linewidth]{system.pdf}
	\caption{\label{fig:system}Schematic illustration of the considered system: Brownian particles (with a finite mass $ m $) perturbed by simple shear flow with the velocity profile $ \vct{V}(\vct{r}) = \dot{\gamma}(y, 0, 0)^T $ ($ \dot{\gamma} > 0 $ is the shear rate).}
\end{figure}

We consider a system of $ N $ interacting spherical Brownian particles, each with mass $ m $ and mobility $ \mu $, subject to external potential forces. Hydrodynamic effects are neglected. Notably, the chosen minimal model was found to predict the behavior of real Brownian systems both qualitatively and quantitatively~\cite{Lukic2005, Blum2006, Huang2011, Orihara2011, Henseler2010, Kreuter2012}. At time $ t = 0 $, the system is assumed to be in equilibrium. For time $ t > 0 $, it is perturbed by a simple shear flow with the velocity profile $ \vct{V}(\vct{r}) = \boldsymbol{\kappa}\cdot\vct{r} $, with the shear rate tensor $ \boldsymbol{\kappa} = \dot{\gamma}\hat{\vct{x}}\otimes \hat{\vct{y}}$ (where $ \dot{\gamma}$, $ \hat{\vct{x}} $, $ \hat{\vct{y}} $, and $ \otimes $ are the shear rate, the unit vectors, and the tensor product, respectively). Experimentally, such a flow can be realized in various ways, e.g., in a Couette device composed of two coaxial cylinders~\cite{Derksen1990, Fuchs2015}, in a sliding plate~\cite{Ortman2012} or cone-plate~\cite{Orihara2011} rheometer, or in a microfluidic device with two counter flows~\cite{Ziehl2009}. The dynamics of particle $ i $ is given by the Langevin equation for the velocity $ \vct{v}_i $ and the position $ \vct{r}_i $:
\begin{equation}
m\dot{\vct{v}}_i = \frac{1}{\mu}\boldsymbol{\kappa}\cdot\vct{r}_i -\frac{1}{\mu}\vct{v}_i + \vct{F}^{\textrm{int}}_i + \vct{F}^{\textrm{ext}}_i + \vct{f}_i, \ \ \ \ \dot{\vct{r}}_i = \vct{v}_i,
\label{eq:LE}
\end{equation}
where $ \vct{f}_i $ is a Gaussian white noise,
\begin{equation}
\langle \vct{f}_i(t) \rangle = 0, \ \ \ \ \langle \vct{f}_i(t) \otimes \vct{f}_j(t') \rangle = \frac{2k_{\textrm{B}}T}{\mu}\mathbb{I}\delta_{ij}\delta(t-t'),
\label{eq:RF}
\end{equation}
with $ k_{\textrm{B}} $, $ T $, and $ \mathbb{I} $ being the Boltzmann's constant, the temperature, and the identity matrix, respectively. $ \langle \cdots \rangle $ indicates average over the independent noise realizations. Interaction and external forces arise from the corresponding potentials: $ \vct{F}^{\textrm{int}}_i = -\boldsymbol{\nabla}_iU^{\textrm{int}} $ for internal (interaction) forces (where $ U^{\textrm{int}} $ is assumed to be radial),  $ \vct{F}^{\textrm{ext}}_i = -\boldsymbol{\nabla}_iU^{\textrm{ext}} $ for external forces. A schematic illustration of the system is given in Fig.~\ref{fig:system}.

\section{The role of symmetries: Response to shear via the response to a potential}
\label{sec:Symmetries}
The response of an observable $ A(t) $ is defined as $ \Delta A^{(\dot{\gamma})} \equiv {\left\langle A(t) \right\rangle}^{(\dot{\gamma})} - \left\langle A \right\rangle $ ($ t \geq 0 $), where the first and the second term on the right-hand side are the averages in the sheared and equilibrium system, respectively. Since we are interested in the \textit{linear} response, we consider only the term in $ \Delta A^{(\dot{\gamma})} $ which is linear in $ \dot{\gamma} $. In this section, we review the approach presented in Ref.~\cite{Asheichyk2019_2} to show that the linear response to shear is equivalent to the linear response to a potential if certain symmetry conditions are satisfied. The results of Ref.~\cite{Asheichyk2019_2} are complemented by new insights regarding these conditions: We illustratively demonstrate them based on the effect of system rotation and on the symmetry of equilibrium correlation functions.

\subsection{From the response to shear to the response to a potential}
\label{subsec:shear_pot}
Alongside with the shear perturbation force
\begin{equation}
\vct{F}_i^{\textrm{ptb}} = \frac{1}{\mu}\boldsymbol{\kappa}\cdot\vct{r}_i = \frac{\dot{\gamma}}{\mu}\left(y_i, 0, 0\right)^T,
\label{eq:F}
\end{equation}
consider a partner force~\cite{Asheichyk2020, Asheichyk2019_2, Dhont1996}
\begin{equation}
\vct{G}_i^{\textrm{ptb}} = \frac{1}{2\mu}\left(\boldsymbol{\kappa}^T - \boldsymbol{\kappa}\right)\cdot\vct{r}_i = \frac{\dot{\gamma}}{2\mu}\left(-y_i, x_i, 0\right)^T,
\label{eq:G}
\end{equation}
representing the half of the difference between shear flows in the $ y $ and $ x $ directions. Although both $ \vct{F}_i^{\textrm{ptb}} $ and $ \vct{G}_i^{\textrm{ptb}} $ are nonconservative forces, adding them together leads to a potential, $ \vct{F}_i^{\textrm{ptb}} + \vct{G}_i^{\textrm{ptb}} =  -\boldsymbol{\nabla}_iU^{\textrm{ptb}} $, where~\cite{Asheichyk2020, Asheichyk2019_2}
\begin{equation}
U^{\textrm{ptb}} = -\frac{\dot{\gamma}}{2\mu}\sum_{i=1}^Nx_iy_i.
\label{eq:Ushear}
\end{equation}
According to the superposition principle of the linear response, we get
\begin{equation}
\Delta A^{(\dot{\gamma})} + \Delta A\big|_{\vct{G}^{\textrm{ptb}}} = \Delta A\big|_{U^{\textrm{ptb}}},
\label{eq:SPP}
\end{equation}
where $ \Delta A^{(\dot{\gamma})} = \Delta A\big|_{\vct{F}^{\textrm{ptb}}} $, and $  \Delta A\big|_{U^{\textrm{ptb}}} $ is the linear response to $ U^{\textrm{ptb}} $ in Eq.~\eqref{eq:Ushear}.

The main idea of Ref.~\cite{Asheichyk2019_2} is the following: If certain symmetry conditions are satisfied, then $ \Delta A\big|_{\vct{G}^{\textrm{ptb}}} = 0 $, and hence,
\begin{equation}
\Delta A^{(\dot{\gamma})} = \Delta A\big|_{U^{\textrm{ptb}}},
\label{eq:ShearIsPotential}
\end{equation}
i.e., the response to shear is equivalent to the response to the potential in Eq.~\eqref{eq:Ushear}. This leads to the relation given in Eq.~\eqref{eq:LR_RK} below.

We note that the same idea can also be applied if the unperturbed system is out of equilibrium, as shown in Ref.~\cite{Asheichyk2019_1} for active Brownian particles.

\subsection{Symmetries}
\label{subsec:symmetries}
What are those symmetry conditions? This question can be answered using either of the following two facts. The first fact is that $ \vct{G}_i^{\textrm{ptb}} $ is orthogonal to the particle radius vector,
\begin{equation}
\vct{G}_i^{\textrm{ptb}} \cdot \vct{r}_i = 0,
\label{eq:Gr}
\end{equation}
which means that $ \vct{G}_i^{\textrm{ptb}} $ is a rotation force. Because of its construction, it lies in the plane perpendicular to the $ z $ axis~\cite{Dhont1996}. As a result, in the linear order, the overall effect of $ \vct{G}_i^{\textrm{ptb}} $ is rotation of the system around the $ z $ axis.

The second fact is that the linear response to $ \vct{G}_i^{\textrm{ptb}} $ is given by an equilibrium correlation function of an observable $ A $ with a quantity which is antisymmetric under the interchange $ x_i \leftrightarrow y_i, \ v_{ix} \leftrightarrow v_{iy} $ (this notation implies the interchange for all particles)~\cite{Asheichyk2019_2, Risken1996, Kurchan1998}:
\begin{align}
\notag & \Delta A\big|_{\vct{G}^{\textrm{ptb}}} = \frac{1}{k_{\textrm{B}}T}\int_0^tdt'\left\langle A(t)\sum_{i=1}^N\vct{G}_i^{\textrm{ptb}}(t') \cdot \vct{v}_i(t')\right\rangle\\
& = \frac{\dot{\gamma}}{2k_{\textrm{B}}T\mu}\int_0^tdt'\left\langle A(t)\sum_{i=1}^N\left[x_i(t')v_{iy}(t')-y_i(t')v_{ix}(t')\right]\right\rangle.
\label{eq:G_LR}
\end{align}
Derivation of this formula is given in Appendix~\ref{app:KKE}. Note that the first line in Eq.~\eqref{eq:G_LR} tells that the linear response is determined by how much $ A $ is coupled to the work $ \int_0^tdt'\sum_{i=1}^N\vct{G}_i^{\textrm{ptb}}(t') \cdot \vct{v}_i(t') $ done by $ \vct{G}_i^{\textrm{ptb}} $ on the system~\cite{Asheichyk2020, Asheichyk2019_2}.

Let us now introduce $ xy $ symmetry. Generally, we call any function of the phase space $ xy $ symmetric if this function remains unchanged under the interchange $ x_i \leftrightarrow y_i, \ v_{ix} \leftrightarrow v_{iy} $. A system is called $ xy $ symmetric if its distribution function is $ xy $ symmetric. For an equilibrium system, this is equivalent to saying that $ U^{\textrm{int}} $ and $ U^{\textrm{ext}} $ are $ xy $ symmetric.

Consider that the unperturbed system is $ xy $ symmetric. An example is Brownian spheres subject to gravity (acting along the $ z $ axis) and confined in a cubic box whose center lies on the $ z $ axis. Another example is a spherically symmetric system, i.e., $ U^{\textrm{int}} $ depends only on interparticle distances and $ U^{\textrm{ext}} $ is a function of $ |\vct{r}_i| $. All examples studied explicitly in this paper are spherically symmetric.

It is clear from physical grounds that a slow rotation of an $ xy $-symmetric system around the $ z $ axis does not change an $ xy $-symmetric observable if this observable depends only on particle positions. This argument is not so evident if $ A $ contains velocities, but it still holds thanks to the symmetry properties of equilibrium correlation functions. An equilibrium correlation function computed for an $ xy $-symmetric system is also $ xy $ symmetric (see Appendix~\ref{app:symm_eqcorr}). This means that
\begin{equation}
\left\langle A(t)x_i(t')v_{iy}(t')\right\rangle = \left\langle \widetilde{A}(t)y_i(t')v_{ix}(t')\right\rangle,
\label{eq:symmetry_correlator}
\end{equation}
where $ \widetilde{A} $ is obtained from $ A $ by the interchange $ x_i \leftrightarrow y_i, \ v_{ix} \leftrightarrow v_{iy} $. If $ A $ is $ xy $ symmetric, then $ \widetilde{A} = A $, and hence $ \Delta A\big|_{\vct{G}^{\textrm{ptb}}} = 0 $ according to Eqs.~\eqref{eq:G_LR} and~\eqref{eq:symmetry_correlator}.

The above discussions allow us to repeat: For $ xy $-symmetric systems and observables, $ \Delta A\big|_{\vct{G}^{\textrm{ptb}}} = 0 $, and Eqs.~\eqref{eq:ShearIsPotential} and \eqref{eq:LR_RK} follow. This statement generalizes a known consideration that an antisymmetric part of the strain tensor (from which $ \vct{G}_i^{\textrm{ptb}} $ stems) drops out in the linear response for isotropic systems and spatial observables~\cite{Nagele1998}.

We note that, if the system is $ xy $ symmetric, many of the observables which have a finite linear response are also $ xy $ symmetric: for example, such observables are $ \sum_{i=1}^Nx_iy_i $, $ \sum_{i=1}^Nv_{ix}v_{iy} $, or the $ xy $ component of the stress tensor defined in Eq.~\eqref{eq:sigma}. There are however observables, which show a linear response, but for which Eq.~\eqref{eq:ShearIsPotential} fails. These couple positions and velocities, e.g., $ \sum_{i=1}^Nx_iv_{iy} $ (see Fig.~\ref{fig:Transient}).

\section{Collection of (six) ways to compute the linear response}
\label{sec:Six_ways}
In this section, we show that the linear response of Brownian particles to shear flow can be computed in at least six different ways: perturbing the system directly by shear or alternatively by the shear potential, or using one of the four response relations [Eqs.~\eqref{eq:LR_TA},~\eqref{eq:LR_RF},~\eqref{eq:LR_GKU}, and~\eqref{eq:LR_RK}]. Notably, Eq.~\eqref{eq:LR_GKU} has not been given in the literature.

\subsection{Perturbing the system}
\label{subsec:Perturbing}
The direct way to compute $ \Delta A^{(\dot{\gamma})} $ is to apply shear. Alternatively, according to Eq.~\eqref{eq:ShearIsPotential}, one can perturb the system by the shear potential~\eqref{eq:Ushear} if the system and $ A $ are $ xy $ symmetric.

\subsection{Computing equilibrium correlation functions: Linear response formulas}
\label{subsec:LR}
Linear response theory gives the response in terms of equilibrium correlation functions. Perturbing the system by a time independent potential $U^{\textrm{ptb}}$ switched on at time $t=0$, one has, what we call in the following, the equilibrium FDT~\cite{Baiesi2009_2, Baiesi2010, Risken1996}
\begin{equation}
\Delta A = - \frac{1}{k_{\textrm{B}}T}\left[\langle AU^{\textrm{ptb}}\rangle - \langle A(t) U^{\textrm{ptb}}(0) \rangle\right].
\label{eq:FDT_pp}
\end{equation}
Formula~\eqref{eq:FDT_pp} follows immediately from Eq.~\eqref{eq:G_LR}, when replacing $ \vct{G}_i^{\textrm{ptb}} $ with $ -\boldsymbol{\nabla}_iU^{\textrm{ptb}} $. Since the shear perturbation is a nonconservative one (i.e., the force in Eq.~\eqref{eq:F} does not arise from a potential), the corresponding response relation cannot, in general, be given by the equilibrium FDT~\cite{Marconi2008, Kubo1966, Hansen2009, Asheichyk2019_2}.
Therefore, various forms of that relation are used in the literature.

One of those forms was already introduced in Eq.~\eqref{eq:G_LR}, stating that the linear response is related to the work done by the perturbation force on the system\add{~\cite{Asheichyk2019_2, Risken1996, Kurchan1998}}. In case of shear, we get [by replacing $ \vct{G}_i^{\textrm{ptb}} $ in Eq.~\eqref{eq:G_LR} with  $ \vct{F}_i^{\textrm{ptb}} $ given in Eq.~\eqref{eq:F}, see also Appendix~\ref{app:KKE}]
\begin{equation}
\Delta A^{(\dot{\gamma})} = \frac{\dot{\gamma}}{k_{\textrm{B}}T\mu}\int_0^tdt'\left\langle A(t)\sum_{i=1}^Ny_i(t')v_{ix}(t')\right\rangle.
\label{eq:LR_TA}
\end{equation}
Another response relation involves the random force $ f_{ix} $,
\begin{equation}
\Delta A^{(\dot{\gamma})} = \frac{\dot{\gamma}}{2k_{\textrm{B}}T}\int_0^tdt'\left\langle A(t)\sum_{i=1}^Ny_i(t')f_{ix}(t')\right\rangle.
\label{eq:LR_RF}
\end{equation}
Equation~\eqref{eq:LR_RF} can be derived using a path integral representation of the dynamics~\cite{Speck2006, Asheichyk2020}. It also follows from the response formula in terms of the Malliavin weight~\cite{Warren2012}.  Notably, the stochastic integrals in Eqs.~\eqref{eq:LR_TA} and~\eqref{eq:LR_RF} are independent of the convention for the stochastic calculus (i.e., Ito versus Stratonovich)~\cite{Cugliandolo2017, Gardiner2010, Wynants2010, Asheichyk2020}, because noise for different spatial directions is uncoupled.

The linear response can also be computed using the underdamped Green-Kubo relation (see derivations in Appendices~\ref{app:KKE} and~\ref{app:GKU_derivation})~\cite{Asheichyk2020},
\begin{align}
\notag & \Delta A^{(\dot{\gamma})} = \frac{\dot{\gamma}}{k_{\textrm{B}}T} \int_0^tdt'\left\langle A(t')\sigma_{xy}(0)\right\rangle\\
& + \frac{\dot{\gamma}m}{k_{\textrm{B}}T}\left[\left\langle A\sum_{i=1}^Ny_iv_{ix}\right\rangle - \left\langle A(t)\sum_{i=1}^Ny_i(0)v_{ix}(0)\right\rangle\right],
\label{eq:LR_GKU}
\end{align}
where \cite{Hansen2009}
\begin{equation}
\sigma_{xy} = -\sum_{i=1}^N\left[mv_{ix}v_{iy} +\left(F_{ix}^{\textrm{int}}+F_{ix}^{\textrm{ext}}\right)y_i\right]
\label{eq:sigma}
\end{equation}
is the $ xy $ component of the stress tensor (generalized to include external forces). Note that, in case of central forces, it is $ xy $ symmetric. To our knowledge, formula~\eqref{eq:LR_GKU} has not been previously derived, and only special cases were known~\cite{Hess1983}. A formula of the same structure but for a potential perturbation has been derived in Ref.~\cite{Baiesi2010}. The last two terms in Eq.~\eqref{eq:LR_GKU} (i.e., the terms without integral) are unexpected, but indeed turn out to be required in the numerical examples provided below. They are absent when considering the overdamped limit given by the famous Green-Kubo relation~\cite{Fuchs2005, Dhont1996, Heyes1994, Lin2017}:
\begin{equation}
\Delta A^{(\dot{\gamma})} = \frac{\dot{\gamma}}{k_{\textrm{B}}T} \int_0^tdt'\left\langle A(t')\sigma_{xy}(0)\right\rangle,
\label{eq:LR_GK}
\end{equation}
where $ \sigma_{xy} $ is given by Eq.~\eqref{eq:sigma} without the term containing velocities.

For formulas~\eqref{eq:LR_TA},~\eqref{eq:LR_RF}, and~\eqref{eq:LR_GKU}, we note that each of them directly follows from the two others, because the quantities they involve are related by the equation of motion~\eqref{eq:LE}.

Finally, if the system and $ A $ are $ xy $ symmetric, Eq.~\eqref{eq:ShearIsPotential} holds, and the response is given by the equilibrium FDT~\cite{Asheichyk2019_2}
\begin{equation}
\Delta A^{(\dot{\gamma})} = \frac{\dot\gamma}{2k_{\textrm{B}}T\mu}\left[\left\langle A\sum_{i=1}^Nx_iy_i \right\rangle - \left\langle A(t)\sum_{i=1}^Nx_i(0)y_i(0) \right\rangle\right],
\label{eq:LR_RK}
\end{equation}
obtained by inserting the shear potential~\eqref{eq:Ushear} into Eq.~\eqref{eq:FDT_pp}, and also derived in Appendix~\ref{app:KKE}. We refer to formula~\eqref{eq:LR_RK} as sFDT (shear FDT or FDT for shear).

Response relations~\eqref{eq:LR_TA} and~\eqref{eq:LR_RF} are general, in a sense that they are the same for the underdamped and overdamped cases [in contrast to Eqs.~\eqref{eq:LR_GKU} and~\eqref{eq:LR_GK}] and that they have no symmetry restrictions [in contrast to Eq.~\eqref{eq:LR_RK}]. Compared to Eqs.~\eqref{eq:LR_GKU},~\eqref{eq:LR_GK}, and~\eqref{eq:LR_RK}, they however require the instantaneous stochastic velocity $ v_{ix} $ [Eq.~\eqref{eq:LR_TA}] or the stochastic force $ f_{ix} $ [Eq.~\eqref{eq:LR_RF})] as an input. These quantities are typically hard to measure in experiments of Brownian systems.
For $ xy $ symmetric systems and observables, the response relation~\eqref{eq:LR_RK} is advantageous
because it contains no time integral and requires measurement of simple quantities. Further comparison of the response relations is given in Sec~\ref{sec:Investigation}, where they are demonstrated in a concrete example.

\section{Investigation of the linear response and comparison of the response relations}
\label{sec:Investigation}

\subsection{Setup and simulation parameters}
\label{subsec:Setup}
\begin{figure}[!t]
\includegraphics[width=0.8\linewidth]{systemHP.pdf}
	\caption{\label{fig:systemHP}Interacting Brownian particles (with a finite mass $ m $) confined in a harmonic potential and perturbed by simple shear flow in two space dimensions.}
\end{figure}

\begin{figure*}[!t]
\begin{tabular}{cc}
\includegraphics[width=0.49\linewidth]{XY.pdf}
&
\includegraphics[width=0.49\linewidth]{VXVY.pdf}\\
\includegraphics[width=0.49\linewidth]{XVY.pdf}
&
\includegraphics[width=0.49\linewidth]{YVX.pdf}
\end{tabular}
\caption{\label{fig:Transient}Main plots show the directly computed linear response as a function of time after start of shear for the system depicted in Fig.~\ref{fig:systemHP}. Four different observables are considered. For each observable, the results for three different masses are shown; for $ A = \sum_{i=1}^Nx_iy_i $, the overdamped ($ m = 0 $) response is also plotted. The inset plots compare the linear response for $ m = 0.4 $ computed via the six ways introduced in Sec.~\ref{sec:Six_ways}: (i) directly perturbing by shear (the same as in the main plots, labeled \enquote{Direct}); (ii) perturbing by the shear potential~\eqref{eq:Ushear} [labeled \enquote{Potential}]; (iii -- vi) using linear response formulas~\eqref{eq:LR_TA},~\eqref{eq:LR_RF},~\eqref{eq:LR_GKU}, and~\eqref{eq:LR_RK} [labeled \enquote{Work formula}, \enquote{RF (random force) formula}, \enquote{Green-Kubo}, and \enquote{sFDT}, respectively]. Parameters: $ N = 10 $, $ \dot{\gamma} = 0.01 $, and $ C = 10^8 $.}
\end{figure*}

As a specific example, we study the response to shear of a two-dimensional system of interacting Brownian particles confined in a harmonic potential (see Fig.~\ref{fig:systemHP}). A similar scenario was studied in Ref.~\cite{Asheichyk2019_2}; here, we extend that study by (i) considering underdamped  dynamics and (ii) computing the response using Eqs.~\eqref{eq:ShearIsPotential},~\eqref{eq:LR_TA}, and~\eqref{eq:LR_RF}.

The particles interact via a screened Coulomb potential
\begin{equation}
U^{\textrm{int}} = \frac{J}{2}\sum_{i=1}^N\sum_{j=1(j\neq i)}^N\frac{1}{r_{ij}} e^{-\frac{r_{ij}}{r^{\textrm{int}}}},
\label{eq:Uint}
\end{equation}
where $ r_{ij}\equiv|\vct{r}_i-\vct{r}_j|$, $J$, and $r^{\textrm{int}}$ are the interparticle distance, interaction strength, and interaction range, respectively. The external harmonic potential confining the particles reads
\begin{equation}
U^{\textrm{ext}} = \frac{k}{2}\sum_{i=1}^N|\vct{r}_i|^2.
\label{eq:Uext}
\end{equation}
For $N=1$, i.e.,  a single particle, we provide the analytical solution in Appendix~\ref{app:resp1},  complementing the results of Refs.~\cite{Rzehak2003, Holzer2010, Lander2012, Kahlert2012}  with the transient response. For $N>1$  particles, we evaluate the methods numerically, where we set $ k_{\textrm{B}}T = r^{\textrm{int}} = \mu = 1 $, $ J = 25 $, and $ k = 10 $; $ N $, $ m $, $ \dot{\gamma} $, and the number $ C $ of the independent noise realizations are varied between measurements. The dynamics is simulated using the Euler method with the time step $ \Delta t  = 5 \times 10^{-4} $ (note that in Figs.~\ref{fig:Var}(a),~\ref{fig:Var}(b), and~\ref{fig:respUptb} only every fourth time point is plotted).

\subsection{Transient response}
\label{subsec:Transient}
First, we study how the response of the system to shear depends on time. Interactions and a finite inertia give rise to interesting relaxation effects, as shown in Fig.~\ref{fig:Transient}. We consider observables $ A = \sum_{i=1}^Nx_iy_i $,  $ A = \sum_{i=1}^Nv_{ix}v_{iy} $, $ A = \sum_{i=1}^Nx_iv_{iy} $, and  $ A = \sum_{i=1}^Ny_iv_{ix} $, where the first two are $ xy $ symmetric, while the latter two are not. The shear rate $ \dot{\gamma} = 0.01 $ is chosen to be sufficiently small compared to other system parameters, such that the regime linear  in $ \dot{\gamma} $ applies.

The response for $ A = \sum_{i=1}^Nx_iy_i $ is given in Fig.~\ref{fig:Transient}(a). This quantity characterizes the morphology of the system: When shear is applied, the spatial distribution of the particles changes from circular to ellipsoidal (where principal axes of the ellipse are rotated with respect to the coordinate system), such that $ A $ becomes positive on average~\cite{Asheichyk2019_2, Rzehak2003, Holzer2010, Lander2012, Kahlert2012}. For overdamped particles, there is a simple monotonic relaxation towards the steady state, as was observed in Ref.~\cite{Asheichyk2019_2}. The situation is different if the particles have a finite mass, where oscillations appear. These oscillations become more pronounced as the mass increases: They have a larger amplitude, as well as a longer period and relaxation time. It is worth noticing that such an interacting system behaves qualitatively different compared to a single-particle case [see Fig.~\ref{fig:AE}(a)]. While the response of the former exceeds the steady-state result during oscillations, the response of the latter does not (here, the maxima of oscillations lie exactly on the steady-state result). The difference may be attributed to elastic energy stored in particle interactions, which leads to stronger oscillations.

Figure~\ref{fig:Transient}(b) shows the response for $ A = \sum_{i=1}^Nv_{ix}v_{iy} $. Here, the period of oscillations and the relaxation time increase with the mass, but the amplitude decreases. The response approaches zero for large times, as was also observed for a single particle (see Fig.~\ref{fig:AE}(b) and Refs.~\cite{Holzer2010, Kahlert2012}). However,  the transient response can be positive for a many-particle system, whereas it is always below or equals zero for a single particle [compare Figs.~\ref{fig:Transient}(b) and~\ref{fig:AE}(b)].

\begin{figure*}[!t]
\begin{tabular}{cc}
\includegraphics[width=0.5\linewidth]{var_tr_mean.pdf}
&
\includegraphics[width=0.5\linewidth]{var_tr_var.pdf}
\end{tabular}
\begin{tabular}{ccc}
\includegraphics[width=0.33\linewidth]{var_N_mean.pdf}
&
\includegraphics[width=0.33\linewidth]{var_N_var.pdf}
&
\includegraphics[width=0.33\linewidth]{var_N_vm.pdf}
\end{tabular}
\caption{\label{fig:Var}Panels~(a) and~(b) show the linear response (mean) and its standard deviation (SD) as functions of time  for the system depicted in Fig.~\ref{fig:systemHP} with $ N = 16 $ particles, computed using linear response formulas~\eqref{eq:LR_TA},~\eqref{eq:LR_RF},~\eqref{eq:LR_GKU}, and~\eqref{eq:LR_RK} [labeled \enquote{Work formula}, \enquote{RF (random force) formula}, \enquote{Green-Kubo}, and \enquote{sFDT}, respectively]. Panels~(c),~(d), and~(e) show the dependence of the stationary linear response (mean), its SD, and the relative SD (SD divided by the mean) on $ N $, also computed with the four formulas. Straight lines correspond to power-law fits. The considered observable is $ A = \sum_{i=1}^Nx_iy_i $. Parameters: $ m = 0.4 $, $ \dot{\gamma} = 0.01 $, and $ C = 10^6 $  (in addition, averaging over time in the steady state is performed; for details regarding the time average of SD, see the main text).}
\end{figure*}

The responses for $ A = \sum_{i=1}^Nx_iv_{iy} $ and $ A = \sum_{i=1}^Ny_iv_{ix} $ are shown in Figs.~\ref{fig:Transient}(c) and~\ref{fig:Transient}(d), respectively. The former is negative, while the latter is positive, and their steady-state values are related by a minus sign. However, this is not true for all times, as can be clearly seen from the peaks for $ m = 0.1 $ and $ m = 0.4 $, as well as from the behavior at $ t = 0 $ (the curves have zero derivatives for $ A = \sum_{i=1}^Nx_iv_{iy} $, but finite ones for $ A = \sum_{i=1}^Ny_iv_{ix} $). The reason for the difference originates in the different roles played by the velocity (viz.~$x$) and the gradient (viz.~$y$) direction. Also for these observables, one can identify qualitative differences compared to single-particle responses [see Figs.~\ref{fig:AE}(c) and~\ref{fig:AE}(d)]: For example, the curve for $ m = 0.1 $ in Fig.~\ref{fig:Transient}(c) goes below the steady-state result as the response develops in time, which is not observed for the curves in Fig.~\ref{fig:AE}(c).

It is interesting that the steady-state linear response does not depend on the mass, as can be seen in Fig.~\ref{fig:Transient}. In general, the mass dependence is expected even for spatial observables like $ A = \sum_{i=1}^Nx_iy_i $, because shear drives the system out of equilibrium leading to the coupling between spatial and velocity distributions. However, this effect appears only in a nonlinear order, as shown in Sec.~\ref{sec:Nonlinear}. We further note that, for $ A = \sum_{i=1}^Nx_iy_i $ and $ A = \sum_{i=1}^Nv_{ix}v_{iy} $, the mass independence is an exact and direct consequence from sFDT~\eqref{eq:LR_RK}: In the steady-state limit, the last term in Eq.~\eqref{eq:LR_RK} vanishes; the first one, being an equilibrium average, does not depend on the mass if $ A $ contains only particle positions, and vanishes if $ A = \sum_{i=1}^Nv_{ix}v_{iy} $ [confirming the zero result for large times in Fig.~\ref{fig:Transient}(b)]. This demonstrates another advantage of sFDT~\eqref{eq:LR_RK} compared to other response relations: Thanks to its simple structure, sFDT allows to make important statements without computations

The inset plots of Fig.~\ref{fig:Transient} show the response for $ m = 0.4 $ computed via the six ways introduced in Sec.~\ref{sec:Six_ways}. For $ A = \sum_{i=1}^Nx_iy_i $ and $ A = \sum_{i=1}^Nv_{ix}v_{iy} $ all the ways agree; this demonstrates their validity. For $ A = \sum_{i=1}^Nx_iv_{iy} $ and $ A = \sum_{i=1}^Ny_iv_{ix} $, the response to the shear potential~\eqref{eq:Ushear} and the response computed using sFDT~\eqref{eq:LR_RK} disagree from the other ways, as expected for not $ xy $-symmetric observables.

\subsection{Standard deviation}
\label{subsec:Variance}
Our next investigation concerns the standard deviation (SD) of linear response relations. According to the central limit theorem, an average has a statistical error $ \frac{\sigma}{\sqrt{C}} $, where $ \sigma $ (not to be confused with the stress tensor $ \sigma_{xy} $) is SD of the underlying distribution, and $ C $ is the number of independent measurements. A larger $ \sigma $ hence requires a larger $ C $ (and thus more computational resources) to obtain a given accuracy.

The response formulas introduced in Sec.~\ref{sec:Six_ways}, where the averaged quantity $ B $ is different, give the same average, or mean, value of $ B $. For example, in formula~\eqref{eq:LR_TA}, $ B =  \frac{\dot{\gamma}}{k_{\textrm{B}}T\mu}\int_0^tdt'A(t)\sum_{i=1}^Ny_i(t')v_{ix}(t') $, while in formula~\eqref{eq:LR_RK}, $ B = \frac{\dot{\gamma}}{2k_{\textrm{B}}T\mu}\left[A\sum_{i=1}^Nx_iy_i - A(t)\sum_{i=1}^Nx_i(0)y_i(0)\right] $. However, SD, defined  as $ \sigma = \sqrt{\left\langle B^2\right\rangle - {\left\langle B\right\rangle}^2} $, is, in general, different for different formulas, as we demonstrate below. Therefore, the practical usage of each formula is based not only on the quantity $ B $ which has to be measured, but also on SD of the formula. Since SD tells us about the statistical efficiency of the computed average, it is a crucial property of linear response formulas: The smaller $ \sigma $, the more efficient the formula.

In Ref.~\cite{Asheichyk2019_2}, we computed SD for formulas~\eqref{eq:LR_GK} and~\eqref{eq:LR_RK} using overdamped Brownian particles. Here, we compute SD using the underdamped system depicted in Fig.~\ref{fig:systemHP}, consider more response relations, and study the transient regime. The results are given in Fig.~\ref{fig:Var}, with $ A = \sum_{i=1}^Nx_iy_i $.

Figure~\ref{fig:Var}(a) shows the transient linear response to shear for $ N = 16 $ particles computed using formulas~\eqref{eq:LR_TA},~\eqref{eq:LR_RF},~\eqref{eq:LR_GKU}, and~\eqref{eq:LR_RK}. As expected, all four curves agree and relax to a definite stationary value. In contrast, the corresponding SD, plotted in Fig.~\ref{fig:Var}(b), is different for each formula. For sFDT, SD converges to a steady-state value, while it diverges with $t$ for the other formulas. For a single overdamped particle, we obtained a divergence with $ \sqrt{t} $ analytically (see Appendix~\ref{app:var}), and the numerical data in Fig.~\ref{fig:Var}(b) also suggest such law for the case of $N>1$. When $N>1$, this divergence is also analytically suggested, and it is  attributed to the fact that SD of these formulas involves double integration over time.

The absence of a genuine steady-state value for SD of formulas~\eqref{eq:LR_TA},~\eqref{eq:LR_RF},~\eqref{eq:LR_GKU} raises a question about a usefulness of a comparison. However, there is no fundamental problem, as, indeed, the statistical error of a numerically computed average accumulates with time due to time integration. A natural choice for the steady-state SD is then based on the steady-state mean. Here, the steady-state value for the mean is reached at $ t \approx 8 $ for all considered $ N $ [see Fig.~\ref{fig:Var}(a) for $ N = 16 $], and we estimate the steady-state value for SD at the same time, i.e.,  at $ t \approx 8 $. In order to perform time averaging in the steady state, we estimate the stationary mean and SD as an average over time in the interval $ t \in [8, 16] $.

Lower panels of Fig.~\ref{fig:Var} show the dependence of the stationary mean, SD, and the relative SD on the number of particles $ N $. For the mean, the four response relations agree and scale as $ N^{1.567} $ (for $ N \gtrapprox 4 $). However, SD is very different for each response formula [as can also be seen from the transient SD in Fig.~\ref{fig:Var}(b)], although the scaling behavior is similar. Formula~\eqref{eq:LR_RK}, sFDT, has the lowest SD, while formula~\eqref{eq:LR_RF} containing the random force -- the highest one. Estimated from SD and the central limit theorem, Eq.~\eqref{eq:LR_RK} thus needs roughly a factor of $ 10^4 $ smaller number of independent simulation runs than Eq.~\eqref{eq:LR_RF} for the same statistical error (for example, this is a minute of computational time versus 7 days). The efficiencies of the work formula~\eqref{eq:LR_TA} and the Green-Kubo relation~\eqref{eq:LR_GKU} are in between, respectively. Their SDs differ between each other roughly by a factor of $ 2 $, but they are separated from the curves for Eqs.~\eqref{eq:LR_RK} and~\eqref{eq:LR_RF} by an order of magnitude. Note that the relative SD, divided by the mean, is independent of $ N $ (for $ N \gtrapprox 4 $) for all response relations, as can be seen in Fig.~\ref{fig:Var}(e). This is different from the typical behavior of $1/\sqrt{N}$, as the system is not extensive with $N$.

We also would like to mention that, in contrast to the mean, SD computed for formulas~\eqref{eq:LR_TA} and~\eqref{eq:LR_RF} can depend on the discretization scheme used in numerical simulations, i.e., Ito or Stratonovich \cite{Asheichyk2020, Cugliandolo2017, Gardiner2010, Wynants2010}.

\section{Linear versus nonlinear}
\label{sec:Nonlinear}
Finally, we would like to demonstrate some differences between the linear and nonlinear responses to shear.

It was observed that the relaxation to the steady state is very different between the two cases~\cite{Asheichyk2020, Asheichyk2019_2, Amann2013}. What about the steady state itself? One important question is whether the steady-state response depends on the mass, and if it does,  for what observables. In the linear order, it does not, as can be seen in Fig.~\ref{fig:Transient} [see also discussions in Subsec.~\ref{subsec:Transient}].

The situation is different for the nonlinear case. Table~\ref{table:NL_mass} compares how the response for $ \dot{\gamma} = 0.01 $ and $ \dot{\gamma} = 20 $ depends on the mass, using the observable $ A = \sum_{i=1}^Nx_iy_i $, notably a function of position. When $ \dot{\gamma} = 0.01 $ (linear response), there is no dependence (up to statistical error), as we just mentioned. When $ \dot{\gamma} = 20 $, however, a clear dependence can be seen. This shows that the steady-state distribution couples inertia and potential parts. The dependence of the spatial distribution on the particle mass was very recently observed for a single Brownian gyrator~\cite{Bae2020}, which is also a nonequilibrium system.

\begin{table}[!t]
\caption{\label{table:NL_mass}The directly computed steady-state response to shear for different shear rates and masses. The system is the same as in Sec.~\ref{sec:Investigation}; $ A = \sum_{i=1}^Nx_iy_i $. Parameters: $ N = 10 $ and $ C = 10^6 $ (in addition, averaging over time in the steady state is performed).}
\begin{ruledtabular}
\begin{tabular}{c|ccc}
\diagbox{$\dot{\gamma}$}{$m$} & $0$ & $0.1$ & $0.4$\\
\hline
$0.01$ & $0.00881$ & $0.00877$ & $0.00923$\\
$20$ & $10.27$ & $14.71$ & $31.02$
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{figure}[!b]
\includegraphics[width=1.0\linewidth]{gammaDot_lin.pdf}
	\caption{\label{fig:NL_GD_lin}Overdamped steady-state responses to shear ($ \Delta A^{(\dot{\gamma})} $), the shear potential ($ \Delta A\big|_{U^{\textrm{ptb}}} $), and the rotation force ($ \Delta A\big|_{\vct{G}^{\textrm{ptb}}} $, the inset plot) as functions of the shear rate $ \dot{\gamma} $. The system is the same as in Sec.~\ref{sec:Investigation}; $ A = \sum_{i=1}^Nx_iy_i $. The black line represents the linear fit using the five smallest values of $ \Delta A^{(\dot{\gamma})} $. Parameters: $ N = 10 $ and $ C = 10^6 $ (in addition, averaging over time in the steady state is performed). Note that $ \Delta A\big|_{\vct{G}^{\textrm{ptb}}} =0$ within accuracy.}
\end{figure}

Another important question to investigate is whether our main observation, Eq.~\eqref{eq:ShearIsPotential}, remains valid beyond the linear order. Remember that Eq.~\eqref{eq:ShearIsPotential} follows from two facts: (i) the superposition principle [Eq.~\eqref{eq:SPP}] and (ii) the fact that the response to the rotation force $ \vct{G}_i^{\textrm{ptb}} $ is zero for $xy$-symmetric systems and observables.

Figure~\ref{fig:NL_GD_lin}, showing the dependence of $ \Delta A^{(\dot{\gamma})} $, $ \Delta A\big|_{U^{\textrm{ptb}}} $, and  $ \Delta A\big|_{\vct{G}^{\textrm{ptb}}} $ on $ \dot{\gamma} $, answers this question. The system and observable considered in the figure are $xy$-symmetric. One can see that $ \Delta A^{(\dot{\gamma})} = \Delta A\big|_{U^{\textrm{ptb}}}$ up to $ \dot{\gamma} \approx 1 $, where the response to shear is linear in $ \dot{\gamma} $. Once $ \Delta A^{(\dot{\gamma})} $ deviates from the linear behavior (it can be already seen for $ \dot{\gamma} \approx 2.5 $), Eq.~\eqref{eq:ShearIsPotential} fails. Since, for symmetry reasons, $ \Delta A\big|_{\vct{G}^{\textrm{ptb}}} $ remains zero even when this deviation occurs, Eq.~\eqref{eq:ShearIsPotential} fails only because the superposition principle~\eqref{eq:SPP} fails. We note, however, that for other $xy$-symmetric observables, e.g. $ A = \sum_{i=1}^N(x_i^2+y_i^2) $, the nonlinear response to $ \vct{G}_i^{\textrm{ptb}} $ is not necessarily zero, because the effect of $ \vct{G}_i^{\textrm{ptb}} $ for large $ \dot{\gamma} $ may be different from a pure rotation.

It is also worth noticing that the system perturbed by $ U^{\textrm{ptb}} $ has a transition at $ \dot{\gamma} = 2\mu k $ (see Appendix~\ref{app:respUptb}). For  $ \dot{\gamma} \geq 2\mu k $ (which corresponds to $ \dot{\gamma} \geq 20 $ for the chosen parameters), the system is unstable, because the sum of $ U^{\textrm{ptb}} $ and $ U^{\textrm{ext}} $ form a saddle potential, see Fig.~\ref{fig:respUptb}. The response thus diverges with time for $ \dot{\gamma} \geq 2\mu k $, and the steady-state response diverges when approaching $ \dot{\gamma} = 2\mu k $ from below. The points for $ \Delta A\big|_{U^{\textrm{ptb}}}$ and $ \dot{\gamma} \geq 2\mu k $ are absent in Fig.~\ref{fig:NL_GD_lin}.

\section{Conclusion}
\label{sec:Conclusion}
In this paper, we studied the linear response of interacting Brownian particles with a finite mass to simple shear flow. The superposition principle of the linear response [Eq.~\eqref{eq:SPP}], and the symmetry of the system and the observable with respect to the interchange of the $ x $ and $ y $ axes (shear axes)  make the responses to shear and to the shear potential [Eq.~\eqref{eq:Ushear}] equivalent. This fact leads to two efficient ways for the linear response computation: perturbation by the shear potential in Eq.~\eqref{eq:Ushear}, and FDT for shear [sFDT, Eq.~\eqref{eq:LR_RK}].

Using six different computational methods (direct perturbation by shear, perturbation by the shear potential, and applying four linear response relations), we computed the transient response to shear for four different observables. It was found that interparticle interactions and a finite inertia lead to nontrivial time dependence of the response, which is very different compared to a single underdamped particle or interacting overdamped particles. The four response relations were compared in terms of their statistical efficiency, where sFDT was found to be the most efficient relation, requiring about 100 times less computational resources than the next efficient one. This is to be expected due to the absence of a time integral, which also simplifies the computation in other regards. Equation~\eqref{eq:LR_RK}, for the steady state, can even be evaluated in a simple Monte Carlo simulation, in contrast to other formulas, where the dynamics needs to be resolved for the time integration.

Going beyond the linear order, we observed that, in contrast to the linear response, the steady-state nonlinear response depends on the particle mass. Also, for large shear rates, the response to shear cannot be computed via the response to the shear potential, because the superposition principle fails.

Future work may study the response to shear of active Brownian particles with a finite mass~\cite{Lowen2020}. Application of the discussed here computational methods to the viscosity of Brownian suspensions is also a promising avenue to explore. Another interesting point is the connection between the response relations discussed here and those following from Newtonian dynamics (where DOLLS and SLLOD equations are used to model shear)~\cite{Evans2008, Chong2009, Suzuki2013}.

\begin{acknowledgments}
K.A. was supported by the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes) and by the Physics Department of the University of Stuttgart. M.F. and M.K. acknowledge funding by the DFG within SFB 1432, through projects C06 and C05, respectively. K.A. also acknowledges support by S.~\mbox{Dietrich} and Alexander N. Furs.
\end{acknowledgments}

\begin{appendix}
\section{Linear response in Klein-Kramers equation}
\label{app:KKE}
The Langevin equation~\eqref{eq:LE} corresponds to the Klein-Kramers equation $\partial_t \Psi = \Omega \Psi$ for the probability distribution function  (pdf) $\Psi$ given by the Fokker-Planck operator~\cite{Hess1983}
\begin{eqnarray}\label{eq1}
\notag  \Omega &=& \Omega_0+\delta\Omega= - \sum_{i=1}^N\left(\vct{F}_i \cdot \frac{\partial}{  \partial m \vct{v}_i}+ \vct{v}_i  \cdot \frac{\partial}{\partial \vct{r}_i}\right)   \\
                &+&
                 \frac{1}{\mu } \sum_{i=1}^N   \frac{\partial}{\partial m \vct{v}_i} \cdot\left( k_{\textrm{B}} T \frac{\partial}{\partial m\vct{v}_i}+ \vct{v}_i- \boldsymbol{\kappa}\cdot \vct{r}_i \right),
\end{eqnarray}
where $\vct{F}_i$ denotes all potential forces acting on particle $i$ and the flow rate tensor $\boldsymbol{\kappa}$ has nonvanishing element $\kappa_{xy}=\dot\gamma\Theta(t)$, where $\Theta(t)$ is the Heaviside step function. Note in passing that the perturbation $\delta \Omega=- \frac{1}{\mu}\sum_{i=1}^N \frac{\partial}{\partial m\vct{v}_i}\cdot  \boldsymbol{\kappa}\cdot \vct{r}_i $ arises in the diffusion term of the Fokker-Planck operator and not in its drift term where a coupling to an external flow field would enter in Liouvillian response~\cite{Kubo1966}.

To linear order, the change of the pdf obeys the differential equation
\begin{equation}\label{eq2}
\left(\partial_t - \Omega_0 \right) \; \delta \Psi = \delta \Omega\;  \Psi_{\rm eq} = \frac{1}{k_{\textrm{B}}T} \; \Pi \; \Psi_{\rm eq},
\end{equation}
 where $\Psi_{\rm eq}$ is the canonical equilibrium pdf and the inhomogeneity is produced by the power $\Pi$~\cite{Kurchan1998}
\begin{equation}\label{eq3}
\Pi= \frac{1}{\mu}\; \sum_{i=1}^N\;  \vct{v}_i \cdot \boldsymbol{\kappa}\cdot \vct{r}_i.
\end{equation}
For the specified constant perturbation starting at $t=0$, the change of the pdf becomes
\begin{equation}\label{eq4}
 \delta \Psi(t) = \frac{1}{k_{\textrm{B}}T}  \int_0^t dt'\;  e^{ \Omega_0(t-t')}\;  \Pi\; \Psi_{\rm eq}.
\end{equation}
This leads to Eq.~\eqref{eq:LR_TA}. For an arbitrary perturbation $ \vct{K}^{\textrm{ptb}}_i $, the power in Eq.~\eqref{eq3} is replaced by $ \Pi= \sum_{i=1}^N\;  \vct{v}_i \cdot \vct{K}^{\textrm{ptb}}_i $, leading to Eq.~\eqref{eq:G_LR} [with $ \vct{K}^{\textrm{ptb}}_i =  \vct{G}^{\textrm{ptb}}_i] $.

In the general case, the condition of detailed balance, $\Omega_0(\{\vct{r}_i, \vct{v}_i\})\,A\, \Psi_{\rm eq} = \Psi_{\rm eq}\, \Omega^\dagger_0(\{\vct{r}_i, -\vct{v}_i\})\,A$ with $\Omega^\dagger_0$ the adjoint and $A$ an arbitrary function~\cite{Risken1996}, can be used to show that
\begin{equation}\label{eq5}
  \Pi\; \Psi_{\rm eq}  =  \boldsymbol{\sigma} : \boldsymbol{\kappa}\;  \Psi_{\rm eq}  - \mu m\;\Omega_0\;  \Pi\; \Psi_{\rm eq},
\end{equation}
where $\boldsymbol{\sigma}$ is the Irving-Kirkwood stress tensor~\cite{Hansen2009} whose $xy$ element is given in Eq.~\eqref{eq:sigma}.
Integrating Eq.~\eqref{eq4} for $\delta\Psi$ gives two terms in this case:
\begin{align}
\notag \delta \Psi(t) & = \frac{1}{k_\textrm{B}T} \int_0^tdt'\; e^{ \Omega_0 t'} \;\boldsymbol{\sigma} : \boldsymbol{\kappa}\;  \Psi_{\rm eq}\\
& + \frac{\mu m}{k_{\textrm{B}}T} \left(1- e^{ \Omega_0 t}\right) \; \Pi\; \Psi_{\rm eq},
\label{eq6}
\end{align}
which directly leads to Eq.~\eqref{eq:LR_GKU}.

In the case of $xy$ symmetry, the flow can be described by the potential, $ \frac{1}{2\mu}\left(\boldsymbol{\kappa}+\boldsymbol{\kappa}^T\right)\cdot\vct{r}_i = -\boldsymbol{\nabla}_iU^{\textrm{ptb}} $, as discussed in Sec.~\ref{sec:Symmetries}. This allows to rewrite $\Pi\Psi_{\rm eq} = \Omega_0  U^{\rm ptb} \Psi_{\rm eq}$ [where $ \Pi $ is given by Eq.~\eqref{eq3}, with $ \boldsymbol{\kappa} $ replaced by $ \frac{1}{2}(\boldsymbol{\kappa}+\boldsymbol{\kappa}^T)$, and $ U^{\textrm{ptb}} $ is given by Eq.~\eqref{eq:Ushear}], and to perform the time integration in Eq.~\eqref{eq4} leading to $ \delta \Psi(t) = \frac{-1}{k_\textrm{B}T} (1- e^{ \Omega_0 t}) \; U^{\rm ptb} \Psi_{\rm eq}$. Performing an average with this $\delta\Psi$ leads to Eq.~\eqref{eq:LR_RK}.

\section{The symmetry of an equilibrium time correlation function}
\label{app:symm_eqcorr}
An equilibrium time dependent correlation function of two variables $ A $ and $ B $ can be written as (we consider $ t \geq t' $)~\cite{Risken1996}
\begin{align}
\notag \left \langle A(t)B(t') \right\rangle = & \left \langle A(t-t')B(0) \right\rangle =\\
 & \int d\Gamma A(\Gamma)e^{\Omega_0(\Gamma)(t-t')}B(\Gamma)\Psi_{\textrm{eq}}(\Gamma),
\label{eq:EqCorr}
\end{align}
	where $ \Gamma $ is the phase space, and $ \Omega_0 $ and $ \Psi_{\textrm{eq}} $ are the equilibrium Fokker-Planck operator (see Eq.~\eqref{eq1} and Refs.~\cite{Risken1996, Hess1983}) and distribution function, respectively. For an $ xy $-symmetric system, $ \Omega_0 $ and $ \Psi_{\textrm{eq}} $ are $ xy $ symmetric. Because of this, making the interchange $ x_i \leftrightarrow y_i, \ v_{ix} \leftrightarrow v_{iy} $ for $ A $ and $ B $ in Eq.~\eqref{eq:EqCorr} is equivalent to simply renaming the integration variables in the same way. The latter action does not affect the value of the integral. Therefore, an equilibrium correlation function computed for an $ xy $-symmetric system is also $ xy $ symmetric.

\section{Derivation of the underdamped Green-Kubo relation}
\label{app:GKU_derivation}
We demonstrate here the derivation of Eq.~\eqref{eq:LR_GKU} \cite{Asheichyk2020}. It is based on a path integral formalism~\cite{Altland2010, Asheichyk2019_1, Asheichyk2020}.

The path weight $ \mathcal{W}^{(\dot{\gamma})} $, representing the probability that within a time interval $ [0, t] $ the system described by Eq.~\eqref{eq:LE} follows a certain trajectory, reads~\cite{Altland2010, Asheichyk2019_1, Asheichyk2020, Cugliandolo2017, Onsanger1953, Machlup1953, Martin1973, Janssen1976, DeDominicis1978}
\begin{equation}
\mathcal{W}^{(\dot{\gamma})} \propto e^{-\mathcal{A}^{(\dot{\gamma})}}, \ \ \ \ \mathcal{A}^{(\dot{\gamma})} = \frac{\mu}{4k_{\textrm{B}}T}\textrm{Ito}\int_0^tdt'\sum_{i=1}^N\boldsymbol{\mathcal{X}}_i^2(t', \dot{\gamma}),
\label{eq:PW}
\end{equation}
where $ \mathcal{A}^{(\dot{\gamma})} $ is the system action and $ \boldsymbol{\mathcal{X}}_i^2 $ measures the deviation from the noise-free path,
\begin{equation}
\boldsymbol{\mathcal{X}}_i(\dot{\gamma}) = m\dot{\vct{v}}_i - \frac{1}{\mu}\boldsymbol{\kappa}\cdot\vct{r}_i +\frac{1}{\mu}\vct{v}_i -  \vct{F}^{\textrm{int}}_i - \vct{F}^{\textrm{ext}}_i .
\label{eq:X}
\end{equation}
Since Eq.~\eqref{eq:PW} is derived using the Ito time-discretization scheme~\cite{Asheichyk2020, Cugliandolo2017, Gardiner2010, Wynants2010}, the integral in Eq.~\eqref{eq:PW} is understood as a stochastic Ito integral~\cite{Asheichyk2020, Cugliandolo2017, Gardiner2010, Wynants2010}, as indicated by the notation \enquote{Ito}.

Expanding the path weight~\eqref{eq:PW} in powers of $ \dot{\gamma} $, we get
\begin{equation}
\mathcal{W}^{(\dot{\gamma})} - \mathcal{W} = \mathcal{W}\frac{\dot{\gamma}}{2k_{\textrm{B}}T}\textrm{Ito}\int_0^tdt'\sum_{i=1}^N\mathcal{X}_{ix}(t')y_i(t') + \mathcal{O}(\dot{\gamma}^2),
\label{eq:PW_expansion_Ito}
\end{equation}
where $ \mathcal{X}_{ix} $ and $ \mathcal{W} $ correspond to the unperturbed system. One can show that the stochastic Ito integral $ \textrm{Ito}\int_0^tdt' \cdots $ in Eq.~\eqref{eq:PW_expansion_Ito} is equivalent to the stochastic Stratonovich integral $ \int_0^tdt' \cdots $  obeying conventional integration rules~\cite{Asheichyk2020, Cugliandolo2017, Gardiner2010, Wynants2010}. We can hence write
\begin{equation}
\mathcal{W}^{(\dot{\gamma})} - \mathcal{W} = \mathcal{W}\frac{\dot{\gamma}}{2k_{\textrm{B}}T}\int_0^tdt'\sum_{i=1}^N\mathcal{X}_{ix}(t')y_i(t') + \mathcal{O}(\dot{\gamma}^2).
\label{eq:PW_expansion_Str}
\end{equation}
From Eq.~\eqref{eq:PW_expansion_Str}, the linear response  relation follows immediately:
\begin{equation}
\Delta A^{(\dot{\gamma})} = \frac{\dot{\gamma}}{2k_{\textrm{B}}T}\int_0^tdt'\left\langle A(t)\sum_{i=1}^N\mathcal{X}_{ix}(t')y_i(t')\right\rangle.
\label{eq:GKU_der1}
\end{equation}

Next, we use the time reversal symmetry of the equilibrium linear response: the term containing $ \sum_{i=1}^Nv_{ix}y_i $ in Eq.~\eqref{eq:GKU_der1} is antisymmetric with respect to the time reversal, and it equals the other term which is time symmetric~\cite{Baiesi2009_1, Baiesi2009_2, Baiesi2010, Basu2015}. Therefore, the response can be given by two times either of the terms. For the Green-Kubo relation, we need the time-symmetric one:
\begin{align}
\notag &\Delta A^{(\dot{\gamma})} = \frac{\dot{\gamma}}{k_{\textrm{B}}T}\int_0^tdt'\\
&\times \left\langle A(t)\sum_{i=1}^N\left[m\dot{v}_{ix}(t')- F^{\textrm{int}}_{ix}(t') - F^{\textrm{ext}}_{ix}(t')\right]y_i(t')\right\rangle.
\label{eq:GKU_der2}
\end{align}

Finally, we use partial integration for the first term in Eq.~\eqref{eq:GKU_der2}:
\begin{align}
\notag & \int_0^tdt'\sum_{i=1}^N\dot{v}_{ix}(t')y_i(t') = -\int_0^tdt'\sum_{i=1}^Nv_{ix}(t')v_{iy}(t')\\
& +\sum_{i=1}^Ny_i(t)v_{ix}(t) - \sum_{i=1}^Ny_i(0)v_{ix}(0).
\label{eq:GKU_der3}
\end{align}
Substituting Eq.~\eqref{eq:GKU_der3} into Eq.~\eqref{eq:GKU_der2} and identifying the stress tensor~\eqref{eq:sigma}, we obtain formula~\eqref{eq:LR_GKU}.

\onecolumngrid
\section{The linear response of a single underdamped particle confined in a harmonic potential}
\label{app:resp1}
The response of a single underdamped particle can be evaluated analytically~\cite{Rzehak2003, Holzer2010, Lander2012, Kahlert2012}. This appendix complements the results of
Refs.~\cite{Rzehak2003, Holzer2010, Lander2012, Kahlert2012} with the transient response. A derivation of the expressions given here as well as a check of formulas~\eqref{eq:LR_GKU}
and~\eqref{eq:LR_RK} can be found in Ref.~\cite{Asheichyk2020}.

\begin{figure*}[!t]
\begin{tabular}{cc}
\includegraphics[width=0.49\linewidth]{AE_xy.pdf}
&
\includegraphics[width=0.49\linewidth]{AE_vxvy.pdf}\\
\includegraphics[width=0.49\linewidth]{AE_xvy.pdf}
&
\includegraphics[width=0.49\linewidth]{AE_yvx.pdf}
\end{tabular}
\caption{\label{fig:AE}Rescaled response to shear flow for $ A = xy $ [(a), Eq.~\eqref{eq:AE_xy_direct}], $ A = v_xv_y $ [(b), Eq.~\eqref{eq:AE_vxvy_direct}], $ A = xv_y $
[(c), Eq.~\eqref{eq:AE_xvy_direct}], and $ A = yv_x $ [(d), Eq.~\eqref{eq:AE_yvx_direct}] of a single underdamped particle confined in a harmonic trap as a function of
rescaled time $ \tilde{t} \equiv \frac{t}{\tau_k} = \mu kt $ after the flow is applied. The results are shown for different values of a characteristic ratio
$ \tau \equiv \frac{\tau_m}{\tau_k} $, which is the ratio between the inertia time $ \tau_m = \mu m $ and the trap relaxation time $ \tau_k = \frac{1}{\mu k} $. Different values of $ \tau $
correspond to different values of $ m $, while $ \mu $ and $ k $ are assumed to be fixed and finite; $ \tau \to 0 $ is the overdamped limit.
$ \textrm{Pe} \equiv \dot{\gamma}\tau_k = \frac{\dot{\gamma}}{\mu k} $, $ l^2 \equiv k_{\textrm{B}}T\mu\tau_k = \frac{ k_{\textrm{B}}T}{k} $, and
$ u^2 \equiv \frac{l^2}{\tau_k^2} = k_{\textrm{B}}T\mu^2k $ are the Peclet number, the unit of squared length, and the unit of squared velocity, respectively.}
\end{figure*}

The responses for $ A = xy $, $ A = v_xv_y $, $ A = xv_y $, and $ A = yv_x $ read (we note that $ \left\langle A \right\rangle = 0 $ for all considered $ A $, i.e., $ \Delta A^{(\dot{\gamma})} = {\left\langle A(t) \right\rangle}^{(\dot{\gamma})}$)
\begin{align}
\notag {\left\langle x(t)y(t) \right\rangle}^{(\dot{\gamma})} = & \ \frac{2\dot{\gamma}k_{\textrm{B}}T\mu}{1-4\mu^2km}\Bigg\{\left(\frac{\mu m}{1-\sqrt{1-4\mu^2km}}\right)^2\left(1-e^{-\frac{1}{\mu m}\left(1-\sqrt{1-4\mu^2km}\right)t}\right)\\
& +\left(\frac{\mu m}{1+\sqrt{1-4\mu^2km}}\right)^2\left(1-e^{-\frac{1}{\mu m}\left(1+\sqrt{1-4\mu^2km}\right)t}\right) - \frac{m}{2k}\left(1-e^{-\frac{t}{\mu m}}\right)\Bigg\},
\label{eq:AE_xy_direct}
\end{align}
\begin{equation}
{\left\langle v_x(t)v_y(t) \right\rangle}^{(\dot{\gamma})} = \frac{\dot{\gamma}k_{\textrm{B}}T\mu}{2\left(1-4\mu^2km\right)}\Bigg\{\left(1-e^{-\frac{1}{\mu m}\left(1-\sqrt{1-4\mu^2km}\right)t}\right) +\left(1-e^{-\frac{1}{\mu m}\left(1+\sqrt{1-4\mu^2km}\right)t}\right) - 2\left(1-e^{-\frac{t}{\mu m}}\right)\Bigg\},
\label{eq:AE_vxvy_direct}
\end{equation}
\begin{align}
\notag {\left\langle x(t)v_y(t) \right\rangle}^{(\dot{\gamma})} = & \ -\frac{\dot{\gamma}k_{\textrm{B}}T\mu}{1-4\mu^2km}\Bigg\{\frac{\mu m}{1-\sqrt{1-4\mu^2km}}\left(1-e^{-\frac{1}{\mu m}\left(1-\sqrt{1-4\mu^2km}\right)t}\right)\\
& +\frac{\mu m}{1+\sqrt{1-4\mu^2km}}\left(1-e^{-\frac{1}{\mu m}\left(1+\sqrt{1-4\mu^2km}\right)t}\right) - 2\mu m\left(1-e^{-\frac{t}{\mu m}}\right)\Bigg\},
\label{eq:AE_xvy_direct}
\end{align}
\begin{align}
\notag {\left\langle y(t)v_x(t) \right\rangle}^{(\dot{\gamma})} = & \ -\frac{\dot{\gamma}k_{\textrm{B}}T\mu}{1-4\mu^2km}\Bigg\{\frac{\mu m}{1-\sqrt{1-4\mu^2km}}\left(1-e^{-\frac{1}{\mu m}\left(1-\sqrt{1-4\mu^2km}\right)t}\right)\\
& +\frac{\mu m}{1+\sqrt{1-4\mu^2km}}\left(1-e^{-\frac{1}{\mu m}\left(1+\sqrt{1-4\mu^2km}\right)t}\right) - \frac{1-2\mu^2km}{\mu k}\left(1-e^{-\frac{t}{\mu m}}\right)\Bigg\}.
\label{eq:AE_yvx_direct}
\end{align}
These results are valid for any $ \dot{\gamma} $, i.e., the linear response for a single particle and the considered observables equals the total response, in contrast to a many-particle
system (see Fig.~\ref{fig:NL_GD_lin} and Ref.~\cite{Asheichyk2019_2}). In order to visualize Eqs.~\eqref{eq:AE_xy_direct},~\eqref{eq:AE_vxvy_direct},~\eqref{eq:AE_xvy_direct}, and~\eqref{eq:AE_yvx_direct}, we make the
equations dimensionless~\cite{Asheichyk2019_2} and plot them in Fig.~\ref{fig:AE}; definitions of the parameters are given in the caption.

\twocolumngrid
\section{Standard deviation for a single overdamped particle}
\label{app:var}
This appendix gives analytical results for the standard deviation (SD) of the response relations in case of a single overdamped particle. We consider the same system as in Sec.~\ref{sec:Investigation}, and choose $ A = xy $. As it is shown in Subsec.~\ref{subsec:Variance}, SD is different for different response relations. It is defined as
\begin{equation}
\sigma = \sqrt{\left\langle B^2\right\rangle - {\left\langle B\right\rangle}^2},
\label{eq:var_def}
\end{equation}
where, for the chosen system and observable,
\begin{subequations}
\begin{alignat}{4}
& B_{\textrm{Work}} = \frac{\dot{\gamma}}{k_{\textrm{B}}T\mu}\int_0^tdt'x(t)y(t)y(t')\dot{x}(t'),\label{eq:B_TA}\\
& B_{\textrm{RF}} = \frac{\dot{\gamma}}{2k_{\textrm{B}}T}\int_0^tdt'x(t)y(t)y(t')f_x(t'),\label{eq:B_RF}\\
& B_{\textrm{GK}} = \frac{\dot{\gamma}k}{k_{\textrm{B}}T} \int_0^tdt'x(t')y(t')x(0)y(0),\label{eq:B_GK}\\
& B_{\textrm{sFDT}} = \frac{\dot{\gamma}}{2k_{\textrm{B}}T\mu}\left[x^2y^2 - x(t)y(t)x(0)y(0)\right],\label{eq:B_RK}
\end{alignat}
\end{subequations}
for formulas~\eqref{eq:LR_TA},~\eqref{eq:LR_RF},~\eqref{eq:LR_GK}, and~\eqref{eq:LR_RK}, respectively. The difference in SDs originates from the term $ \left\langle B^2\right\rangle $ in Eq.~\eqref{eq:var_def}, while the result for $ {\left\langle B\right\rangle}^2 $ is the same for all the aforementioned $ B $ and equals~\cite{Asheichyk2019_1, Asheichyk2020}
\begin{equation}
{\left\langle B\right\rangle}^2 = \frac{\dot{\gamma}^2(k_{\textrm{B}}T)^2}{4\mu^2k^4}\left(1-e^{-2\mu kt}\right)^2.
\label{eq:Bav2}
\end{equation}

In order to evaluate $ \sigma $ for $ B $ in Eq.~\eqref{eq:B_TA}, we use the Langevin equation to replace $ \dot{x} $ with $ \dot{x} = -\mu kx + \mu f_x $.  The double integrals containing $ f_x $ are evaluated using standard (Stratonovich) integration rules. The corresponding SDs computed in numerical simulations can hence differ from the analytical results provided below [Eqs.~\eqref{eq:varTA} and~\eqref{eq:varRF}] if the discretization scheme different from the Stratonovich one is used; however, the scaling with $ \sqrt{t} $ for large $ t $ should not be affected by this issue.

Here are our results:
\begin{subequations}
\begin{alignat}{4}
&\sigma_{\textrm{Work}} = \frac{\dot{\gamma}k_{\textrm{B}}T}{2\mu k^2}\Big(4\mu kt + 13 -16e^{-2\mu kt} + 3e^{-4 \mu kt}\Big)^{\frac{1}{2}},\label{eq:varTA}\\
\notag & \sigma_{\textrm{RF}} = \frac{\dot{\gamma}k_{\textrm{B}}T}{2\mu k^2}\Big(2\mu kt + 9 - 8\mu kte^{-2\mu kt} - 12e^{-2\mu kt}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 3e^{-4 \mu kt}\Big)^{\frac{1}{2}},\label{eq:varRF}\\
\notag & \sigma_{\textrm{GK}} = \frac{\dot{\gamma}k_{\textrm{B}}T}{2\mu k^2}\Big(4\mu kt + 9 - 16\mu kte^{-2\mu kt} - 12e^{-2\mu kt}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 3e^{-4 \mu kt}\Big)^{\frac{1}{2}},\label{eq:varGK}\\
& \sigma_{\textrm{sFDT}} = \frac{\dot{\gamma}k_{\textrm{B}}T}{2\mu k^2}\Big(9 -12e^{-2\mu kt} + 3e^{-4 \mu kt}\Big)^{\frac{1}{2}},\label{eq:varRK}
\end{alignat}
\end{subequations}
corresponding to response relations~\eqref{eq:LR_TA},~\eqref{eq:LR_RF},~\eqref{eq:LR_GK}, and~\eqref{eq:LR_RK}, respectively. In the limit of large time, $ \mu kt \gg 1 $, SD scales as $ \sqrt{t} $ for the work formula, random force formula, and Green-Kubo relation, whereas it approaches a constant for sFDT. Considering that the steady state is reached at $ \mu kt \approx 5 $ [see Eq.~\eqref{eq:Bav2}], we conclude from Eqs.~\eqref{eq:varTA},~\eqref{eq:varRF},~\eqref{eq:varGK}, and~\eqref{eq:varRK} that, in the steady state, $ \sigma_{\textrm{sFDT}} < \sigma_{\textrm{RF}} < \sigma_{\textrm{GK}} < \sigma_{\textrm{Work}} $. Note that this inequality differs from what we found numerically (using the Euler discretization) in Subsec.~\ref{subsec:Variance} for a many-particle underdamped system, because of the aforementioned dependence of $ \sigma_{\textrm{Work}} $ and $ \sigma_{\textrm{RF}} $ on the discretization procedure.

\section{Response to the shear potential}
\label{app:respUptb}
When the system of particles confined in a harmonic trap $ U^{\textrm{ext}} = \frac{k}{2}\sum_{i=1}^N(x_i^2+y_i^2) $ is perturbed by the shear potential~\eqref{eq:Ushear}, the resulting external potential becomes
\begin{equation}
U^{\textrm{ext}} = \frac{k}{2}\sum_{i=1}^N\left(x_i^2+y_i^2\right) - \frac{\dot{\gamma}}{2\mu}\sum_{i=1}^Nx_iy_i.
\label{eq:U}
\end{equation}
Each particle hence feels the external potential
\begin{equation}
u = \frac{k}{4}\left[(1-\delta)(x+y)^2 + (1+\delta)(x-y)^2\right],
\label{eq:u}
\end{equation}
with $ \delta = \frac{\dot{\gamma}}{2\mu k} $.

\begin{figure}[!b]
\includegraphics[width=1.0\linewidth]{gammaDotTime.pdf}
\caption{\label{fig:respUptb}Response to the shear potential~\eqref{eq:Ushear} as a function of time for different shear rates. The insets illustrate the sum of external potential and perturbation potential. The system, observable, and parameters are the same as in Fig.~\ref{fig:NL_GD_lin}.}
\end{figure}

If $ \delta < 1 $ (i.e., $ \dot{\gamma} < 2\mu k $), the point $ (0, 0) $ is a minimum of $ u $ [see the insets of Figs.~\ref{fig:respUptb}(a) and~\ref{fig:respUptb}(b)], i.e., there is a well-defined equilibrium. Therefore, $ \Delta A\big|_{U^{\textrm{ptb}}} $ converges to a steady-state value [see Figs.~\ref{fig:respUptb}(a) and~\ref{fig:respUptb}(b)].

If $ \delta = 1 $ (i.e., $ \dot{\gamma} = 2\mu k $), $ x = y $ is a \enquote{minimum line} [see the inset of Fig.~\ref{fig:respUptb}(c)], i.e., the particles tend to go along the diagonal $ x = y $ with no confinement on that way. Therefore, $ \Delta A\big|_{U^{\textrm{ptb}}} $ diverges [see Fig.~\ref{fig:respUptb}(c)].

Finally, if $ \delta > 1 $ (i.e., $ \dot{\gamma} > 2\mu k $), $ (0, 0) $ is a saddle point, with the line $ x = y $ being unstable yet preferable to go along [see the inset of Fig.~\ref{fig:respUptb}(d)]. Therefore, $ \Delta A\big|_{U^{\textrm{ptb}}} $ diverges exponentially  [see Fig.~\ref{fig:respUptb}(d)].

\end{appendix}

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10 ways to get more out of Lego Mindstorms »
What if you could use a single programming environment to learn to code at an early age, then use that same environment to explore programming for robots, IoT, Augmented Reality, and then take those projects into a full-blown programming environment? Apple's Swift Playgrounds is a free app that aspires to do all of these things. I spent some time working with their new features for controlling educational robots, and here is what I found.
The Swift programming language was introduced in 2014, along with a feature called Playgrounds in Xcode, Apple's main developer tool. The Swift Playgrounds app for iPad was introduced in 2016 as a "learn to code" environment aimed at children, but it is very much real programming. This spring (2017) it received substantial updates, including the ability to program several educational robots,
I spent some time using Swift Playgrounds with these robots:
Wonder Workshop Dash
Sphero (R2-D2 and BB-8)
Other robots are supported, including Parrot drones and Meebot, but I didn't have a chance to test them.
What I found is a VERY promising beginning. The core Swift Playgrounds tutorials (Learn to Code) are really solid, and provide for a seamless transition into robotics. The robotics support is good, even though the level of finish and polish is variable. The ability to bring projects into XCode (Apple's IDE) is an important bonus, and opens the path to more sophisticated projects.
Lego Mindstorms is very polished, with a useful Live View, and very complete support for the API. Documentation is extremely thorough. You can do everything you can do with the Lego software. It feels like an advanced step for students used to the Lego apps.
Wonder Workshops Dash is also well designed, but it seems like the API support is limited. It feels like you can only use the robot to a fraction of its capabilities, it may benefit from further api testing to maximize the capabilities of the API and robots communication. The tutorials are very easy, but without much of a growth path. It feels like an extension to basic Swift tutorials, instead of using this tool as a way to explore more advanced concepts.
The Sphero playgrounds are very solid, with complete API support, Live Views, and thorough documentation. Well structured Star Wars themed tutorials for the R2-D2 robot, and more general tutorials for the entire Sphero product line. This also feels like an advanced step past the native apps.
Keep reading for more detail on my experience.
In the Swift Playgrounds app, new Playgrounds can be added from scratch, or from a library. The robot programming setups are all in the Accessory tab. They often include a more structured tutorial as well as a blank template.
Authoring these tutorials is straightforward, including the addition of "dashboard" type elements for live control, images and video. Apple has published a full specification for creating your Playgrounds. (Interactive features are themselves coded in Swift, which is neat.)
In order to save typing, Playgrounds will autocomplete most of the robot API commands. You can get through the tutorials by pointing and clicking. When typing, Playgrounds has an interesting alternative keyboard layout that lets you add common coding constructs by swiping down on keys.
I tried Swift Playgrounds with some of the robots I have around the house, connecting and writing simple programs.
Swift Playgrounds for Mindstorms includes both a structured tutorial and a blank template.
The Lego Mindstorms tutorial is well constructed, with a narrative about creating animals and helping them survive. The set of lessons ramps up the difficulty level slowly, starting with tips on connecting the EV3 Brick and testing with a limited set of motors, and progressing to the construction of a moderately complex "turtle" bot.
A very attractive feature is a Live View that is active when a robot is connected, which displays real-time status of any connected motors and sensors. A Graph View page keeps track of values over the time. These can be logged and saved.
The implementation of the EV3 API (application programming interface) is extremely thorough. Every function available in Lego's own software seems to be supported, including movement functions for robots with two motor drivetrains. The documentation is completely populated, with both the Glossary and pop-help features providing extremely detailed information, including assembly instructions for a sample robot that works with the tutorials.
The blank template includes the complete API, the Live View and the Graph View, but none of the structured tutorials.
I found the Lego Mindstorms EV3 support in Swift Playground very thorough and easy to use, with real potential to build interesting apps. If you are already using Mindstorms, this will substantially expand your possibilities at no extra cost.
The one downside is that, unlike the native Lego apps, programs are not compiled and uploaded to the robot. The programs you write run on the iPad, and control the robot remotely. This means you need to keep an iPad connected while using the robot.
Wonder Workshops seems to have taken a completely different tack.
The Dash robot is aimed at younger children, and the Swift Playgrounds support reflects that. The graphics and lessons are consistent with the Wonder Workshop apps. I'm not sure this is entirely a good idea: why not use the native apps to serve the younger audience, and use Swift Playgrounds to enable more sophisticated projects? The API support is very limited, with MoveForward() and TurnLeft() and TurnRight() functions like those in the beginning Swift tutorials (move a step, turn left or right 90 degrees), whereas the robot is capable of much more. The native apps have much better control of movement speed, turning on the various LEDs, etc.
The built-in documentation features in Swift Playgrounds are not used at all, which is frustrating. You have to rely on the AutoComplete features to figure out what commands are available.
In general, a child that has completed the first few Swift tutorials will easily understand this playground, but might get bored rather quickly. Given that this system requires much more reading and writing than the native apps, there seems a bit of a mismatch here.
I was surprised that a robot positioned for "learning to code" didn't make more of the Swift Playgrounds opportunity – perhaps they are trying to steer users to their own apps. If you already have a Wonder Workshops Dash robot, Swift Playgrounds does not add much (if anything) beyond the native apps.
Like the others, there is both a structured tutorial and a blank template for programming.
Unlike Mindstorms, the Dash robots already rely on a tablet or phone for their programming, and that is the same with Swift.
Sphero R2-D2 and BB-8
Sphero seems to have really jumped on the opportunity to do something interesting with their robot offering. The Playground templates include one specific to the newly released R2-D2 toy, with a series of detailed Star Wars themed challenges. Another template is called Sphero Arcade, and is more focused on Sphero's original spherical robots (which are compatible with their BB-8 toy). Finally there is a blank template.
All the materials look good, with thorough documentation, full support of the API for R2-D2 and BB-8, a nice sensor view, and a dashboard built in Swift that allows manual driving of the robots.
The Sphero Playgrounds suffered from connection issues early on, but those appear to have been fixed with a recent IOS update. I was able to detect and connect to both R2-D2 and BB-8.
If you already have a Sphero robot, Swift Playgrounds is a good way to increase their possibilities. In particular, since some of the Spheros are marketed as toys, this is a way to gain some more educational value from them. Compared to the native apps (both the toy control and the Sphero EDU apps), you are able to write more complex programs.
Taking it further – connecting with XCode
One of the more intriguing features is that Swift Playgrounds are themselves built in Swift, and can be created and extended using XCode. Apple has provided complete instructions to the Playground format.
Playgrounds are saved in iCloud Drive, and can be directly opened in XCode on a Mac. Since everything is built in Swift, including the code to connect to robots, it is possible to take a project started on the iPad and convert it to a full-blown IOS app (with some work). I found it really interesting to browse the code for connecting to the different robots with Bluetooth, and to provide the Live View functionality. You can also build your own Playgrounds templates following Apple's instructions and add support for new robots this way.
This is a very interesting development. Someone can learn to code from the most basic level, and progress on to controlling various different robots. It is possible to take code from a Playground and eventually use it to build a self-contained app for MacOS or IOS.
Swift Playgrounds is worth a look by anyone involved in educational robotics. It provides a seamless path from the very beginnings of programming all the way to a variety of robot platforms. For some of the supported platforms, Playgrounds is at least as good as the native apps supplied with the robots, with the benefit of being a real, generally applicable programming language. Many robot programming solutions are very specific to the world of robots, and transferring lessons learned to and from other systems is not easy. Swift makes this seamless.
The ability to examine the Playgrounds themselves, and borrow code from them to make real apps, is also interesting.
October 14th, 2017 | Tags: apple, lego, mindstorms, programming, sphero, swift, wonder workshop | Category: Education, Programming, Robotics
2 comments to Hands-on with Swift Playgrounds for educational robots
where are the playgrounds associated with sphero and Mindstorms?
vill30
When you click "New Playground", at the bottom of the screen you see the options Learn to Code, Challenges, Accessories and Starting Points. The robot playgrounds are all in Accessories.
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Like the sparrows, finches can be very hard to differentiate, unless you know some facts. Being a compulsive bird addict, making time for Ornithology or for the truest scientific study of birds isn't always my favorite part about watching birds. I'd rather watch birds, as opposed to read about birds.
However, because it's a cold day, snowing outside, which is very rare for our region and because it's time for wintering goldfinches to invade the backyard, they seemed like a good bird to evaluate for a post. To help out...I've found and inserted an image from Birding is Fun.com.
First, let's look at the Lesser Goldfinch in Adult male "Western form". They are a small and compact songbird with a black crown, short pointy gray bill, lemon-yellow underparts that contrast with black and white patterned wings and an olive back. "Texas" males show solid black back and bright yellow underparts. While male birds in the southern and eastern parts of the lesser goldfinch's range have a black back and western and northern birds have an olive green back. Both share the bright yellow underparts.
While you can't see this bird's tail, back or underpants, its short pointy gray bill and dark cap are giveaways.
So with all that information we should be able to tell them apart. Sounds easy right? Then how about this chick? Is it a Lessor or an American? My guess, it's a Lesser. Mainly because her parents were here and I saw their black backs! I cheated.
Last year a female and male tended to a nest in our olive tree, which is nestled near the house's front roof. The female Lesser Goldfinch selected the nest site, choosing a spot in a fork of branches. They like spots concealed by clusters of leaves, or shaded by lichens or grapevines. The babies are hatched naked, blind, and totally dependent on parents for food.
While it doesn't relate to I.D. some fun facts about Lessers. They are monogamous birds that mate after a courtship ritual of the male ceremoniously feeding the female, just as he will provide food for her while she incubates their eggs and while both parents feed the chicks.
Now, let's look at American Goldfinches. These birds are slightly larger than the Lesser Goldfinch. They're very bright yellow above and below, with boldly patterned black and white wings (not backs) and have a white undertail, not yellow like the Lesser.
Here's where it gets tricky. Winter birds are drab, unstreaked brown, with blackish wings and two pale wingbars.
Head: The head is a warm tan color that may show a yellow or gray wash, though the extent of the color varies widely.
Bill: The bill remains sharply pointed in winter but turns a dull dark gray or black.
Throat: Winter American goldfinches have varying amounts of yellow on their throat, though the color is usually dull and can be hard to see. A brighter finch may have more yellow extending onto the upper breast or to the cheeks.
Wings: In winter, the black wings contrast less with the dull body but still show one wide white wing bar, no matter which gender the bird may be.
Plumage: The overall coloration of winter American goldfinches is a dull tan with a warm brown or yellow wash. The underparts are slightly paler than the upperparts.
Legs and Feet: Even in winter, the legs and feet of these goldfinches remain pale.
The goldfinch's main natural habitats are weedy fields and floodplains, where plants such as thistles and asters are common. They're also found in cultivated areas, roadsides, orchards, and backyards. Goldfinches can be found at feeders any time of year, but most abundantly during winter, hence my posting. I've been seeing more of these little guys in recent weeks, as they really love the tall pine trees in our backyard.
Well, I hope this helps you...half as much as it has helped me. Happy Birding.
Thanks for the info - very helpful. I loved the typo (?), just above the second photo: "underpants"! That's exactly what they should be called!


I've came across the brand Talika sometime back during work, and when I asked some colleagues about it they raved about it, especially about their best-selling eyelash serum. Had the pleasure to be invited by dear Mag from Modgam to know more about the brand, as well as to check out the launch of their latest product: Talika Light Duo+; a revolutionary iontherapy home treatment appliance.
Held at the scenic Forlino, I took a closer look at the oval-shaped gadget. I found it interesting and rather cool that this was inspired by NASA light technology. Since the 1970s, Talika has been constantly researching the benefits of light wavelengths on skin.
A breakthrough in 2008 saw its first Light Therapy product, and since then it currently has 3 different light therapy products: LIGHT 590 (a collagen synthesis stimulator), LIGHT 525 (skin lightener), LIGHT DUO (A combination of LIGHT 590 & 525).
Mr Alexis Broode himself gave a personal anecdote while introducing the Duo+. During a rather eventful surfing trip, he had gotten a huge gash on his forehead that left him with numerous gashes. But rather than let it get to him, he tested out the Duo+ on his wound, and was amazed to see how it closed in just days.
Miraculous? Maybe. But he credits Duo+'s light therapy, as traumatised or agressed skin caused but exteral factors (toxins, sunburn, stress, surgery, smoking, hormonal changes) loses its natural 'intellegence' to heal, hence recovery period stretches longer. But by restimulating the cells with this light therapy, it helps to regains its ability to reboot the natural healing process.
I thought that the item was quite thoughtfully crafted; the size was perfect and curved to fit the curve of a palm. Its smart-technology has a sensor such that if no skin or surface is detected under its light rays, it will switch off automatically. When switched on, there is a gentle vibrating motion, and can be used to soothe and help penetrate any existing beauty regimes by simply placing gently or holding it close to the face.
Basically, the Talika Duo+ helps to convey messages to the skin cell, which help to reactivate its original "intelligence". As our skin is both comples and multifactorial, skin problems can start with the simplest of things; like deepening of wrinkle lines, which is then usually followed after with inflammations and redness.
Hence, the Duo+ comes with 3 different light settings (each with different frequencies) to tackle three general skin problem areas: Green for lightening dark spots, Yellow for reversing the signs of wrinkles and fine lines, and Red for effective reduction of redness.
And of course, was a pleasure to meet new and old friends at beautiful setting like this! So many beautiful blogger ladies in one single picture, though sadly poor Jo from Missjology is missing here due to overexposure which caused her face to appear as a super white blank. I feel really bad to be laughing about it but it was really quite hilarious, especially when I told her she doesn't need to buy anymore whitening products.
My thoughts: I think The Talika Duo+ could be the answer to a number of chronic and long-lasting skin problems, albeit a rather costly pricetag of $500. Although, if one measures it cost ratio, it might actually be an economic choice as compared to paying a few thousands of dollars for repeated facial treatments. I'm intrigued and I'd love to have a chance to review this to check out its rather amazing results.
Thank You Talika for the invite!


June 2002 – With the help of TAM, Jimmy Rodriguez earned his first B since 8th grade, and enrolled in John Muir Charter School to earn his high school diploma.
In June 2004, 13-year-old Aviation Explorer, Breean Farfan from Bellflower, completed a cross-country flight with an FAA Certified Flight Instructor Devin Reed. Breean flew from Compton, CA to Oshkosh, WI to attend a four-day aviation camp. Breean's roundtrip covered 3,671 nautical miles, and included stops in Arizona, Colorado, and Nebraska.
In September 2004, 11-year-old Jimmy Haywood and 14-year-old Kenny Roy, Aviation Explorers from Compton, set world records. Jimmy became the youngest African-American male to fly an airplane on an International round-trip and Kenny became the youngest to solo an airplane. Kenny, supervised by a Certified Flight Instructor, flew a Cessna airplane to Canada where he earned his license to fly solo. In Canada, the age requirement to fly solo is 14, as opposed to 16 in the U.S.A. Press and family welcomed the two young pilots from their return flight to the Compton Air Fair.January 2005 – Partnered with Compton Unified School District Regional Occupation Program. TAM students participated in Avionics CAD/CAM class, and one graduate, Adam Ifill, was placed at a Boeing internship.
On September 22nd, 2005, TAM participated in the Congressional Black Caucus Convention in Washington DC. Three young Aviation Explorers, Diamond Hooper, Richard Olmos, and Kenny Roy, competed in the first "youth versus military" precision flying competition demonstration at College Park Airport, where airplane inventors Orville and Wilbur Wright first began giving flying lessons.
June 2006, 14-year-old Aviation Explorer from Inglewood, Jonathan Strickland, made aviation history by becoming the youngest to solo an airplane and a helicopter on the same day. To accomplish this goal, he had to travel to Canada. The trip earned Jonathan three other world records: the youngest African-American to solo a helicopter; the youngest African-American to fly a helicopter internationally; and the youngest African-American to fly a helicopter on an International round-trip.
In 2008, on her 16th birthday, Kelly Anyadike made her way into the Guinness Book of world Records by being the youngest African-American female to solo in four different fixed-wing aircraft on the same day. In July 2009, Kelly's younger sister Kimberly Anyadike made aviation history by becoming the youngest African-American female to pilot an airplane transcontinental when she flew with a safety pilot and 86 year-old Tuskegee Airman Levi Thornhill "from sea-to-shining-sea" from Compton, CA to Newport News, VA and back.
In September 2009, Kimberly and Kelly Anyadike were invited by California Governor Schwarzenegger to the State Capitol to be honored for their achievements on both the Senate and Assembly floors. Kelly is a Junior at Biola University. Kimberly was named top 25 Most Influential People by ESSENCE MAGAZINE, a SEVENTEEN MAGAZINE Pretty Amazing Finalist and a Freshman at the University of California Los Angeles.
My name is MacArthur Dandridge. I'm sixteen years old. I started coming to TAM when I was six years old. I'm a helicopter and airplane pilot. When I first came to TAM I was scared to fly. One day Robin bet me $50.00 to get 5 feet off the ground. Now, to this day, I am not scared to fly. I come to TAM to stay off the streets. TAM is the best place to be.
Hi! My name is Danyale Denise Dotson and I am a member at Tomorrow's Aeronautical Museum, also known as TAM. My first time coming to TAM was when I was in the 7th grade and I had come with my friend Kyra who was currently enrolled in the program. My mother and I live across the street from TAM on Grandee Avenue, we moved there October 3, 2011.
During the summer of the year 2011, I decided to join TAM's flying program, known as Flight Class and my flight teacher was Isaac. My reason for coming to TAM again after 2 tears was because my boyfriend had kept on asking me to go with him but I kept telling him no because I was shy since I hadn't been there in so long.
My purpose in joining TAM's flying program was to learn about how to fly, I wanted to learn about flying because it is a beautiful talent and it is also a great way to travel.
I love being at TAM because I love meeting new people, eating snacks during my Youth Leadership Council meeting (I am the Secretary) and going on field trips.
My name is Daniel Kinzy and I started coming to TAM 10 years ago when I graduated from elementary school. All of my friends started coming here so I said, "Hey that sounds fun, I want to go." And I did.
Now, when I first started coming to TAM, I was getting into a lot of trouble as a child and I wasn't that much into flying, but as I started coming I figured, why get into trouble when I could learn to fly and this can be a place to have fun after school.
TAM works out for me.
TAM has helped me a lot through my struggles at home and at school. If I didn't want to be at home, I could come to TAM as a safe haven. As for school, if I was having problems with homework or any subject, there would be someone there to help me. As I've gotten older, the people at TAM have become like family to me.
In 2003, I began my journey toward the career of my dreams. The venue is Tomorrow's Aeronautical Museum. It's a facility that is dedicated to helping the youth achieve greatness in the aviation industry. That youth includes myself. My name is Brian Monette and I am a commercial pilot.
I was 18 years young when I began my flight training. It began with me trying to just have a little fun but little did I know that I was to endure upon a thrilling journey of a lifetime. The Aviation Explorer program was the vehicle that took me to my goals in aviation. I worked very weekday afterschool towards my Private Pilot license.
paid no "out of pocket" expense towards my license which is a huge accomplishment.
In 2007 I obtained my Commercial Pilot license and was immediately hired to be a sightseeing tour pilot. Till this day I am a pilot for TAM/Celebrity Helicopters and will always be around to give my support. My goal is to become a corporate pilot or an airline captain. That goal is extremely close to me, I can almost taste it. Just this past January I received my multi-engine license that will allow me to fly airplanes with more than one engine.
Through TAM I have met many pilots and entrepreneurs. Many of them I remain in contact with to this day. I am currently job searching for anything available as far as flying. I'm also pursuing my flight instructor's license so that I can pass on my knowledge to the next young aspiring pilots. From living and growing up in the streets of Compton to flying over the entire city of Los Angeles and I'm not done yet. All of this was possible because of TAM and the founder Robin Petgrave who has been a great mentor since I started working there years back.


In this digital age, your wedding may be the only time you ever need to ask your friends and family for their mailing addresses. It can be a somewhat tedious task, especially if you have a large guest list. A new website called Postable aims to make that job much easier and is free of charge. Signing up takes barely a minute and this creates a personal url that you can copy and email to your wedding guests. Using your personal link, your guests enter their mailing addresses and an address book is populated for you. When you log back into your account, you can export the addresses to Excel, or as a CSV file for Gmail, your Apple address book, or your Microsoft Outlook address book. The nice format in Excel also means you can pass this on to your wedding stationery designer with little additional work. Postable even gives you an option to print the address labels yourself. What a great idea!


Guild provides a modern, luminous space with nautical touches for executive dinners, intimate celebrations, corporate gatherings, and more. A full buy-out can accommodate up to 160 seated guests or 250 standing guests. Our private dining room equipped with AV setup and display accommodates 40 seated guests or 60 standing guests. There are also multiple patios surrounding the restaurant for semi-private gatherings large or small.
We have multiple rooms and patios for different sized events from 20 seated guests in our private dining room to 160 seated guests in the entire restaurant.
Food & Beverage minimums vary based on date. Email events@chameleongp.com for pricing.