Linear response theory and Green-Kubo relations for active matter

TL;DR

This paper develops a theoretical framework for understanding how active matter systems, specifically Active Brownian Particles, respond to external forces, revealing deviations from equilibrium relations due to activity and interactions.

Contribution

It extends fluctuation-dissipation theorem and Green-Kubo relations to active matter, providing analytical expressions and simulation validation for transport properties.

Findings

Derived Green-Kubo formulas for active particle diffusivity and mobility.

Quantified deviations from Stokes-Einstein relation in active systems.

Validated theoretical predictions with unperturbed simulation data.

Abstract

We address the question of how interacting active systems in a non-equilibrium steady-state respond to an external perturbation. We establish an extended fluctuation-dissipation theorem for Active Brownian Particles (ABP) which highlights the role played by the local violation of detailed balance due to activity. By making use of a Markovian approximation we derive closed Green-Kubo expressions for the diffusivity and mobility of ABP and quantify the deviations from the Stokes-Einstein relation. We compute the linear response function to an external force using unperturbed simulations of ABP and compare the results with the analytical predictions of the transport coefficients. Our results show the importance of the interplay between activity and interactions in the departure from equilibrium linear response.

Equations

$$\frac{\partial \Psi }{\partial t}=\Omega \Psi =({\Omega }_0+{\Omega }_{\text{ext}})\Psi .$$

$$⟨A{⟩}_t-⟨A{⟩}_0={\int }_0^{t}ds⟨\frac{{\Omega }_{\text{ext}}{\Psi }_0}{{\Psi }_0}A(s){⟩}_0.$$

$${\Omega }_{\text{ext}}{\Psi }_0=-\beta {\mu }_0\mathbf{f}\cdot {\mathbf{F}}_n{\Psi }_0.$$

$$⟨A{⟩}_t-⟨A{⟩}_0=-\beta {\mu }_0\mathbf{f}\cdot {\int }_0^t\text{mbox}ds⟨{\mathbf{F}}_n{e}^{{\Omega }^{†}s}A{⟩}_0$$

$${\dot{\mathbf{r}}}_i(t)={\mu }_0{\mathbf{F}}_i+v_0{\mathbf{n}}_i(t)+{\mathbf{\xi }}_i(t),{\dot{\theta }}_i(t)={\zeta }_i(t).$$

$${\Omega }_0=\sum _i\left[{\partial }_i\cdot (D_0{\partial }_i-{\mu }_0{\mathbf{F}}_i-v_0{\mathbf{n}}_i)+D_{\theta }\frac{{\partial }^2}{\partial {\theta }_i^2}\right]$$

$${\Psi }_0(\Gamma )\sim e^{-\beta \left[U(\{{\mathbf{r}}_i\})+v_0\tau \Xi (\{{\mathbf{r}}_i,{\theta }_i\})\right]}$$

$$v_0{\partial }_i\cdot (-{\mu }_0\tau {\partial }_i\Xi -{\mathbf{n}}_i){\Psi }_0+{\tau }^{-1}{\partial }_{{\theta }_i}^2{\Psi }_0=-{\nabla }_i\cdot {\mathbf{j}}_i$$

$${\mathbf{\nu }}_i\equiv {\mathbf{j}}_i/{\Psi }_0=v_0\left[({\mathbf{n}}_i+{\mu }_0\tau {\partial }_i\Xi ),\beta {\partial }_{{\theta }_i}\Xi \right].$$

$${\Omega }_0^{†}-{\overline{\Omega }}_0^{†}=2\mathcal{V}\cdot \nabla$$

$$⟨A{⟩}_t-⟨A{⟩}_0=$$

$$\beta \left({\int }_0^{t}ds⟨\dot{V}(0)A(s){⟩}_0-{\int }_0^{t}ds⟨\mathcal{V}\cdot \nabla V(0)A(s){⟩}_0\right)$$

$${\dot{\mathbf{r}}}_i(t)={\mu }_0{\mathbf{F}}_i+{\mathbf{\eta }}_i(t)$$

$${\Omega }_0^M=\sum _{i=1}^N{\partial }_i\cdot {\mathcal{D}}_i(\Gamma )\left[{\partial }_i-\beta {\mathbf{F}}_i^{eff}(\Gamma )\right]$$

$${\mathcal{D}}_i(\Gamma )=D_0+\frac{v_0^2\tau }{2}\left(1+\frac{\tau {\mu }_0{\partial }_i\cdot {\mathbf{F}}_i}{1-\tau {\mu }_0{\partial }_i\cdot {\mathbf{F}}_i}\right)$$

$${\mathbf{F}}_i^{eff}(\Gamma )=(D_0{\mathbf{F}}_i-{\beta }^{-1}{\partial }_i{\mathcal{D}}_i)/{\mathcal{D}}_i$$

$${\Omega }^M{\Psi }_0=({\Omega }_0^M+{\Omega }_{\text{ext}}){\Psi }_0=-\beta {\mu }_0\mathbf{f}\cdot {\mathbf{F}}_n^{eff}{\Psi }_0,$$

$$⟨A{⟩}_t-⟨A{⟩}_0=-{\mu }_0\beta \mathbf{f}\cdot {\int }_0^t\text{mbox}ds⟨{\mathbf{F}}_n^{eff}{e}^{({\Omega }^M{)}^{†}s}A{⟩}_0$$

$$\mu ={\mu }_0\left(1-{\mu }_0\beta {\int }_0^{\infty }ds⟨F_n^{eff,x}(0)F_n^x(s){⟩}_0\right)$$

$$\frac{\mu }{{\mu }_0}=$$

$$+\frac{v_0^2{\tau }^2{\mu }_0^3}{2D_a^2}{\int }_0^{\infty }ds⟨F_n^x(0){\partial }_n\cdot {\mathbf{F}}_n(0)F_n^x(s){⟩}_0$$

$$+\frac{v_0^2{\tau }^2{\mu }_0^2}{2D_a}{\int }_0^{\infty }ds⟨{\partial }_n^x{\partial }_n\cdot {\mathbf{F}}_n(0)F_n^x(s){⟩}_0$$

$${\mu }_0^2⟨F_n^x(0)\cdot F_n^x(t){⟩}_0=⟨{\mathcal{D}}_n{⟩}_0\delta (t)-⟨{\dot{x}}_n(0){\dot{x}}_n(t){⟩}_0$$

$$D=D_a+\frac{{\mu }_0{v}_0^2{\tau }^2}{2}⟨{\partial }_n\cdot {\mathbf{F}}_n{⟩}_0-{\mu }_0^2{\int }_0^{\infty }ds⟨F_n^x(0)F_n^x(s){⟩}_0$$

$$\frac{\mu }{{\mu }_0}=\frac{D}{D_a}-\frac{{\mu }_0{v}_0^2{\tau }^2}{2D_a}⟨{\partial }_n\cdot {\mathbf{F}}_n{⟩}_0+\text{h.o.t.}$$

$$\chi (t)=N^{-1}\sum _i⟨x_i(t)q_i(t)⟩$$

Full text

Linear response theory describes how a (small) external perturbation affects the macroscopic properties of a system MarconiReview . If the unperturbed system is initially in equilibrium, its response to small perturbations is generically related to equilibrium fluctuations through the fluctuation-dissipation theorem (FDT). In this situation, one can derive exact expressions for the transport coefficients in terms of equilibrium correlations, the so-called Green-Kubo relations, which are among the very rare general results in non-equilibrium statistical mechanics KuboBook . However, intrinsically non-equilibrium systems, such as active matter, lie beyond the scope of this framework. Indeed, active matter stands for systems made of components which typically convert energy from their environment into motion in a way that breaks detailed balance ReviewMarchetti ; ReviewCates ; ReviewFodor ; battle2016broken . Therefore, equilibrium states cannot be considered as the reference unperturbed states in that case.

Extensions of the FDT to non-equilibrium states have been recently derived BaiesiMaesPRL ; BaiesiMaesNJP ; ChetriteFDT ; ProstJoanny ; SeifertSpeck ; SpeckSeifert . However, these approaches have not been applied to locally driven active systems and general Green-Kubo-like expressions, relating their transport coefficients with non-equilibrium steady-state correlations, have not been established yet. Establishing the nature of these relationships has deep consequences because of the insight they provide in the nature of non-equilibrium response in active matter. For example, experiments on microswimmer suspensions have revealed interesting rheological behaviour BacteriaClement ; RafaiPRL2010 and rectification phenomena in the presence of asymmetric boundaries diLeonardoWheel ; KoumakisDiLeonardo ; SamuelWheel , opening up the possibility to exploit the non-equilibrium character of active matter to control transport at the microscale and extract energy from it. Harvesting the potential of these systems thus needs the development of an extended response theory: This is the overall aim of the present work.

We start by considering a generic time evolution of the probability $\Psi(\Gamma,t)$ to find the system in a state $\Gamma$:

[TABLE]

The generator of the dynamics $\Omega$ can be split into an unperturbed, $\Omega_{0}$, and perturbed, $\Omega_{\text{ext}}$, part. Using the operator identity $e^{\Omega t}=1+\int_{0}^{t}dt^{\prime}e^{\Omega t}\Omega$, we can derive the following expression of the average of an observable $A$ at any time $t\geq 0$, $\langle A\rangle_{t}=\int d\Gamma\Psi(\Gamma,t)A(\Gamma)$, as

[TABLE]

Here $\langle*\rangle_{0}$ denotes averages over $\Psi_{0}$, the steady-state initial distribution, verifying $\Omega_{0}\Psi_{0}=0$, while $A(s)=e^{\Omega^{\dagger}\,s}A(0)$ has been evolved with the full evolution operator (in the Heisenberg representation of the ensemble average). Note that, while the former expression is completely general and does not rely on any perturbation expansion (therefore also valid in the non-linear regime), its application demands some knowledge of the distribution $\Psi_{0}$, or at least, the action of $\Omega_{\text{ext}}$ on it.

For the sake of clarity, let us focus first on an equilibrium system of $(N+1)$ overdamped Brownian particles interacting by means of potential forces $\bm{F}_{i}=-\partial_{i}U$ ($\partial_{i}$ denotes the spatial gradient $\partial/\partial\bm{r}_{i}$) with diffusion coefficient $D_{0}$, mobility $\mu_{0}$ and inverse temperature $\beta=\mu_{0}/D_{0}$. In that case, Eq. (1) corresponds to the Smoluchovski equation describing the time evolution of the probability density of a point in configuration space $\Gamma\equiv(\bm{r}_{1},...,\bm{r}_{N},\bm{r}_{n})$, with $\Omega_{0}\equiv\Omega_{eq}=\sum_{i}\left[{\partial_{i}}\cdot(D_{0}{\partial_{i}}-\mu_{0}\bm{F}_{i})\right]$. We perturb the system by applying a constant force $\bm{f}=f\bm{u}_{x}$ to a tracer particle described by $\bm{r}_{n}$, thus $\Omega_{\text{ext}}=-\mu_{0}\bm{f}\cdot\partial_{n}$ and

[TABLE]

From Eq. (2) we find

[TABLE]

where $\Omega^{\dagger}$ is the adjoint (backwards) Smoluchovski operator Risken . Eq. (4) constitutes a generalized nonlinear Green-Kubo expression, relating a non-equilibrium average at time $t$ with an equilibrium time-correlation. Such approach, originally introduced in the context of glassy rheology, is usually referred to as integration-through-transients (ITT) FuchsCatesPRL ; fuchs2005 ; fuchs2009 ; ThomasITT ; GazuzFuchs ; SharmaBrader2016 ; ThomasABPMCT . If we now choose $A\equiv F^{x}_{n}=\bm{F}_{n}\cdot\bm{u}_{x}$ in Eq. (4), and define the tracer mobility $\mu\equiv\lim_{t\to\infty}{\langle\dot{\bm{r}}_{n}\rangle}/{\bm{f}}$, we find, in the linear regime $\bm{f}\to 0$, the standard Green-Kubo relation: $\mu=\beta\int_{0}^{t}dt^{\prime}\langle\dot{x}_{n}(t^{\prime})\dot{x}_{n}(0)\rangle_{0}=\beta D$ (see SM for a proof).

We consider now Active Brownian Particles (ABP) LutzABP , self-propelled with a constant velocity $v_{0}$ in the 2D plane along their orientation $\bm{n}_{i}=(\cos{\theta_{i}},\,\sin{\theta_{i}})$. The dynamic equations read

[TABLE]

where $\bm{F}_{i}$ accounts for all inter-particle potential forces (typically short-range repulsions) and $\mu_{0}$ is the mobility. The noise terms, $\bm{\xi}_{i}$ and $\zeta_{i}$ are Gaussian and white, with zero mean and variance $2D_{0}$ and $2D_{\theta}$, with $D_{0}=\mu_{0}/\beta$ the thermal Brownian diffusivity and $D_{\theta}$ the rotational diffusion coefficient, introducing the persistence time $\tau=1/D_{\theta}$. Equilibrium is recovered both in the limit of $v_{0}\to 0$ or $\tau\to 0$. The ABP system above is one of the reference models for active matter, and it has been studied extensively. ABP has been characterized in terms of thermodynamic quantities such as pressure and chemical potential Brady2014 ; solonPRL ; WinklerPressure ; JoanSoft ; SolonThermo ; paliwalChemical and its phase behaviour analysed in great detail FilyMarchetti ; RednerPRL ; BialkeEPL ; FarageBrader ; Siebert2017 ; LinoPRL . Several recent works have studied its linear response (and of similar active particles modelled in terms of an Ornstein-Uhlenbeck process) from different viewpoints: (i) introducing an effective temperature characterizing FDT violations LoiTeff ; LevisTeff ; szamelTeff ; LeticiaFNL ; (ii) taking equilibrium as the reference state and considering activity as a perturbation of it SharmaBrader2016 ; ThomasABPMCT ; FodorPRL ; (iii) deriving expressions of linear response functions in terms of weighted averages over the unperturbed dynamics szamelMalliavin ; SolonResponse ; LeticiaFNL , in the same spirit as the Malliavin weights sampling WarrenAllenPRL . In the present letter, we first characterize the violations of the FDT in ABP, showing how the non-equilibrium character of activity comes into play in an extended FDT that we establish, and then derive Green-Kubo expressions for its transport coefficients in terms of its non-equilibrium fluctuations.

With the aim of characterizing transport in active matter, we first analyze how ABP (in a non-equilibrium steady state) respond to an external perturbation. Our starting point is the Smoluchowski operator corresponding to Eq. (5)

[TABLE]

with a perturbation due to an external force $\bm{f}=-\partial_{n}V$, where $V=-fx_{n}$, applied to a tagged particle located in $\bm{r}_{n}=(x_{n},y_{n})$. A salient feature of ABP is the absence of zero-flux steady-state solutions of the Smoluchovski equation (which can be seen from the impossibility to simultaneously satisfy the zero-current conditions for the angular and positional degrees of freedom). In the absence of odd variables under time-reversal, as it is the case for this overdamped dynamics, the absence of zero-flux solution is a necessary and sufficient condition for the violation of detailed balance GardinerBook . [In contrast to Active Ornstein-Uhlenbeck Particles (AOUP) which fulfil detailed balance in the small drive limit FodorPRL ; Bonilla ].

Because the probability density must be positive we may write

[TABLE]

in terms of a generalized potential $v_{0}\tau\,\Xi(\{\bm{r}_{j},\theta_{j}\})$ - related to the so-called information potential in stochastic thermodynamics (see e.g. BaiesiMaesNJP ) - encoding deviations from Boltzmann statistics. The steady state condition $\Omega_{0}\Psi_{0}=0$ leads to

[TABLE]

where $\nabla_{i}=(\partial_{i},\partial_{\theta_{i}})$, introducing the steady-state local velocity

[TABLE]

In order to make the connection between the steady-state current and the local violation of detailed balance explicit, we consider the time-reversed adjoint operator $\overline{\Omega}_{0}^{\dagger}$ (defined as $A(-t)=e^{\overline{\Omega}_{0}^{\dagger}t}A(0)$) and find SM

[TABLE]

where $\bm{\mathcal{V}}=\{\bm{\nu}_{i}\}$ and $\bm{\nabla}=\{\partial_{i}\,,\partial_{\theta_{i}}\}$ . Eq. (10) shows that violations of detailed balance and non-zero steady current are two faces of the same coin. In the passive limit $v_{0}\to 0$ detailed balance is recovered and $\bm{\nu}_{i}\to 0$.

Inserting Eq. (7) into Eq. (2) yields

[TABLE]

In the linear regime, the latter expression constitutes an extension of the FDT far-from-equilibrium BaiesiMaesPRL ; BaiesiMaesNJP . When $\bm{\mathcal{V}}=0$ one easily recovers the standard Kubo expression $\beta^{-1}R_{A}(t)=\frac{\partial}{\partial t}\langle x_{n}(0)A(t)\rangle_{eq}$ where $R_{A}(t)=\delta\langle A\rangle/\delta f$. Activity is responsible for the second term $\propto\bm{\mathcal{V}}$ in Eq. (Linear response theory and Green-Kubo relations for active matter), which quantifies the local dissipation of energy required to maintain the nonequilibrium steady state, often referred to as housekeeping heat PhysRevLett.86.3463 .

As opposed to the equilibrium FDT, the response of an active system is not completely determined by its fluctuations, but depends on the specific form of its steady-state distribution Risken ; SpeckSeifert ; FodorPRL . Our aim being to derive explicit Green-Kubo relations for ABP (which do not depend on an unknown generalized potential), we forbid ourselves to make any assumption about steady-state properties but only rely on the dynamics. Our starting point should thus be Eq. (5), although for this dynamics we cannot derive an analog of Eq. (3). Thus, the ITT construction that leads us to Eq. 4 cannot be readily followed. To overcome this difficulty we integrate out the angular variables and work with the following reduced dynamics FilyMarchetti ; FarageBrader

[TABLE]

where the noise $\bm{\eta}_{i}$ is approximately gaussian with zero mean and variance $\langle\bm{\eta}_{i}(t)\bm{\eta}_{j}(s)\rangle=(2D_{0}\delta(t-s)+{v_{0}^{2}}e^{-|t-s|/\tau}/2)\delta_{ij}\mathbf{1}$. As usual, by integrating away some degrees of freedom one generates memory, here with a time-correlation $\tau\equiv 1/D_{\theta}$. Even if in the non-interacting limit particles described by Eq. (12) diffuse at long times with a diffusivity $D_{a}=D_{0}+v_{0}^{2}\tau/2$, the difficulty resides on the non-Markovianity of the evolution, which cannot be formulated in terms of a Smoluchovski operator; a long standing problem in statistical mechanics HanggiRev ; Adelman ; MaxiSancho ; Fox . At small correlation times, Fox developed a small$-\tau$ expansion that leads to an effective Smoluchowski equation and which has proven useful in the context of ABP and AOUP FarageBrader ; Wittmann2017effective . To lower order in this expansion, Eq. (12) reduces to FarageBrader

[TABLE]

where we have introduced an effective diffusivity and interaction force

[TABLE]

[Note that $\tau\mu_{0}\partial_{i}\cdot\bm{F}_{i}<1$ must be ensured.] The dynamics encoded in Eq. (13) fulfils detailed balance. The non-equilibrium character of the problem is now encoded in the effective diffusivity (which now depends on the relative positions of all the particles) and forces (which do not derive from a potential). Although the steady-state distribution is non-Boltzmann, it has now zero net current, thus lifting the difficulties associated with the operator Eq. (6) and enabling us to proceed with the ITT construction. Indeed, if we consider the constant force perturbation we find

[TABLE]

which allows us to derive an analogue of Eq. (4)

[TABLE]

Eq. 17 allows us to derive Green-Kubo expressions of the diffusivity and mobility which do not rely of the Stokes-Einstein relation, but only on the time-evolution of the system under the Markovian approximation Eq. (14). By choosing $A\equiv F_{n}^{x}$ in Eq. (17) we get the following Green-Kubo relation for the mobility

[TABLE]

which, to first order in ${\tau\mu_{0}\partial_{i}\cdot\bm{F}_{i}}$, reads

[TABLE]

In equilibrium, only the terms in the first line survive, capturing how interactions affect the ideal gas mobility. Here, activity plays a role in the statistics of collisions, thus the force self-correlation function, and in the value of the pre-factor via the single particle diffusivity. The remaining two terms $\propto(v_{0}\tau)^{2}$ correspond to subdominant higher order correlations involving many-body interactions.

To further characterize the departure from equilibrium linear response and, in that case, the Stokes-Einstein relation, we make use of the expression (see SM for a proof)

[TABLE]

which relates the force and the velocity autocorrelation functions, and whose functional form depends only on the properties of the evolution operator (the special case for equilibrium dynamics was derived in Klein ). Once we identify the diffusivity with the velocity self-correlation function we get the Green-Kubo expression

[TABLE]

which allows us to express Eq. (Linear response theory and Green-Kubo relations for active matter) as

[TABLE]

where h.o.t. refers to higher order terms (see Eq. (Linear response theory and Green-Kubo relations for active matter))and which reduces to the usual Stokes-Einstein relation in the passive limit. If inter-particle forces are divergence-free, a modified Stokes-Einstein relation holds, $\mu=\beta_{eff}D$, but with an effective temperature $k_{B}T_{eff}=D_{a}/\mu_{0}$ (see Eq. (22)). For instance, in the dilute limit, ABP behave as an equilibrium ideal gas at a higher $T_{eff}$ PalacciPRL ; LevisTeff ; szamelTeff ; LeticiaFNL ]. Indeed, genuine non-equilibrium behaviour (with no effective equilibrium description) results from the combined effect of interactions and activity, as observed in active colloidal suspensions Ginot and proven for AOUP FodorPRL ; StefanoPRX ; Bonilla (among other examples). Incidentally, AOUP behave as an effective equilibrium system when the potential has zero third derivatives, while in our case equilibrium-like response is recovered if the potential has zero second derivatives.

To illustrate our results and put them into test, we run particle based simulations of ABP Eq. (5) with periodic boundary conditions. We consider the pair potential $U(r)=(\sigma/r)^{12}$ and the following set of parameters: $\tau=0.02$, $\mu_{0}=1$ and $\beta=1$. We vary the Peclet number $\text{Pe}=v_{0}\tau/\sigma$ and the mean density $\phi=\frac{\pi\sigma^{2}N}{4L^{2}}$ in a range for which the system remains in its homogeneous phase ($\text{Pe}\in(0\,:\,10)$ and $\phi\in(0.01\,:\,0.2)$).

We analyze the integrated response $\chi(t)=N^{-1}\lim_{f\to 0}\sum_{i}^{N}(\langle x_{i}(t)-x_{i}(0)\rangle)/f$ of the particles’ positions due to a constant force applied to all them $\bm{f}_{i}=\epsilon_{i}f\bm{u}_{x}$, where $\epsilon_{i}=\pm 1$ with equal probability LevisTeff . We compute $\chi$ using two different strategies: (i) we explicitly apply a small force and measure the particle displacements it induces; (ii) we track the appropriate stochastic variables needed to compute the response function of interest using simulations of the unperturbed dynamics. The first ’direct method’ involves computing displacements generated by a small perturbation that guarantees the linear regime. The second ’Malliavin weight (MW) method’ overcomes the considerable numerical uncertainties (and cost) related to the control of a small perturbation parameter in Brownian dynamics simulations: we thus make an extensive use of it in the following. This method, originally introduced in the context of Monte Carlo simulations of spin systems Chatelain , and then generalized to Brownian dynamics WarrenAllenPRL ; WarrenAllen , was recently extended to active particle systems szamelMalliavin ; LeticiaFNL .

We compute the diffusion coefficient $D=N^{-1}\lim_{t\to\infty}\sum_{i}\langle(\bm{r}_{i}(t)-\bm{r}_{i}(0))^{2}\rangle/(4t)$ from the long-time behaviour of the mean-square displacement (MSD). According to the MW method, the response function of interest can be expressed as WarrenAllenPRL ; LeticiaFNL

[TABLE]

where $q_{i}$ is a Malliavin weight that evolves accordingly to $\dot{q}_{i}(t)=\sqrt{{\beta}/{2}}\sum_{i}\xi^{x}_{i}(t)$, and averages are taken over independent realizations of the unperturbed dynamics. We thus compute $\chi$ using Eq. (23) and extract $\mu=\lim_{t\to\infty}\chi(t)/t$ (after checking consistency with the ’direct method’, see SM ). We also compute $D$ and $\mu$ using our Green-Kubo expressions. To be concrete, we compute from simulations of ABP, the different terms involving correlations and gradients of the potential that appear in Eq. (21) and Eq. (Linear response theory and Green-Kubo relations for active matter). The results obtained are shown in Fig. (1). The diffusion coefficient follows the same growth $\sim\text{Pe}^{2}$ as the ideal gas in the parameter range explored, but decreases with $\phi$. Eq. (21) underestimates the value of $D$ obtained from the MSD but retains its functional dependence. Mobility is not affected by activity in the dilute regime and its value decreases as the density increases, as expected from Eq. (Linear response theory and Green-Kubo relations for active matter) and previous works LevisTeff ; LeticiaFNL . At finite density and Pe, $\mu$ decreases with density but remains roughly constant for $\text{Pe}\gtrsim 2$. Such behaviour is reproduced by Eq. (Linear response theory and Green-Kubo relations for active matter), although it overestimates the numerical value. Our Green-Kubo expressions predict the qualitative behavior of $D$ and $\mu$, despite we cannot reach a precise quantitative agreement at high Pe and $\phi$. Although such mismatch, expected from the basic assumptions behind the approximations made to reach Eq (21) and Eq. (Linear response theory and Green-Kubo relations for active matter)(constrained to small values of $\tau$ and $\tau\mu_{0}\partial_{i}\cdot\bm{F}_{i}$; neither too active nor too dense), prevents quantitative agreement, the derived Green-Kubo expression provides a general understanding on how the interplay between activity and interactions affects the transport properties of active particle systems.

The response of non-equilibrium systems is typically sensitive to details of the unperturbed initial state, hence the difficulty to establish a general theory. Such lack of universality is encoded in the presence of the generalized potential in extended fluctuation-dissipation relations, arising from the breakdown of detailed balance at the microscopic level. In this letter we set the stage for a systematic response theory of active systems obtained on pure dynamical grounds. Via a Markovian approximation scheme we overcome the aforementioned difficulty and we put forward a closed Green-Kubo expression for the mobility and diffusivity. This allows us to quantify the breakdown of the Stokes-Einstein relation due to the interplay between activity and inter-particle interactions. In presence of the latter, active systems have proven to substantially depart from the response of inanimate matter, contrary to the non-interacting limit for which an exact equilibrium mapping exist. Extending the present approach to other transport coefficients will then provide a theoretical framework to gain insights on the rheological properties of active matter BacteriaClement ; RafaiPRL2010 ; diLeonardoWheel ; KoumakisDiLeonardo ; SamuelWheel .