Inhomogeneous entropy production in active crystals with point imperfections

The stochastic thermodynamics [41, 42, 79, 80] is a powerful tool to measure how far from thermodynamic equilibrium is a system, i.e. its degree of irreversibility. Such information is contained in the so-called entropy production rate (EPR), $\dot{s}$, which can be determined by considering the probabilities of the trajectories connecting two different states of the system. The EPR is expressed in terms of path-probability by resorting to path-integral techniques [81, 50, 82, 83] as

$$\dot{s}=\lim_{t\to\infty}\frac{1}{t}\left\langle\log\left(\frac{\mathcal{P}}{\mathcal{P}_{r}}\right)\right\rangle\,,$$

(7)

where the symbol $\langle\cdot\rangle$ represents the steady-state average performed over the realizations of the noise and $\mathcal{P}$ and $\mathcal{P}_{r}$ are the path probabilities of the forward and backward trajectories of the system, respectively. These probabilities depend on the whole time history of the dynamical variables of the system $(\mathbf{x}_{p},\mathbf{v}_{p},\mathbf{f}^{a}_{p})$ conditioned to their initial values $(\mathbf{x}_{p}(0),\mathbf{v}_{p}(0),\mathbf{f}^{a}_{p}(0))$. In the case of equilibrium systems, in virtue of the detailed balance condition, which is tantamount to the probabilistic time-reversal symmetry, this ratio is one and the EPR vanishes. To estimate $\mathcal{P}$ and $\mathcal{P}_{r}$, let us remark that the probability of the trajectory of a stochastic system is uniquely determined by the probability of observing a path-trajectory of the stochastic noises, that in our case have a Gaussian distribution. Therefore, one performs a transformation from the noise variables to the dynamical variables by using the equation of motion (1) together with Eq. (3). In doing so, we neglect the determinant of the transformation because, in the present case of additive noise, this term does not contribute to the EPR. Applying this procedure, the probability of forward and backward trajectories are expressed as $\mathcal{P}\sim e^{\mathcal{A}}$ and $\mathcal{P}_{r}\sim e^{\mathcal{A}_{r}}$, respectively, where $\mathcal{A}$ and $\mathcal{A}_{r}$ are actions associated to backward and reverse dynamics. The action $\mathcal{A}$ is obtained by expressing the Gaussian distribution of the noise variables, $\boldsymbol{\xi}_{p}$, in terms of the state variables $(\mathbf{x}_{p},\mathbf{v}_{p},\mathbf{f}^{a}_{p})$ using the relation between the two sets of variables given by Eq. (1) with the result:

$$\mathcal{A}=-\sum_{p}\frac{m_{p}^{2}}{4T\gamma}\int dt\left[\dot{\mathbf{v}}_{p}-\frac{\mathbf{F}_{p}}{m_{p}}-\frac{\mathbf{f}^{a}_{p}}{m_{p}}+\frac{\gamma}{m_{p}}\mathbf{v}_{p}\right]^{2}$$

(8)

Here, we have not included in the action the contribution associated with the rotational noise, $\eta_{p}$, since it is known that the simple Brownian process of Eq. (3) does not generate entropy. The action of the backward trajectory $\mathcal{A}_{r}$ can be obtained by applying the time-reversal transformation to the dynamics (1) and considering the parity of the dynamical variables, $(\mathbf{x}_{p},\mathbf{v}_{p},\mathbf{f}^{a}_{p})$, under this transformation. By denoting with the subscript $r$ the time-reversed variables, we assume:

	$\displaystyle\mathbf{x}_{r}\to\mathbf{x}$		(9a)
	$\displaystyle\mathbf{v}_{r}\to-\mathbf{v}$		(9b)
	$\displaystyle\mathbf{f}^{a}_{r}\to\mathbf{f}^{a}$		(9c)

for each particle (above, the particle index has been suppressed for notational convenience). In this way, the backward action, $\mathcal{A}_{r}$, reads

$$\mathcal{A}_{r}=-\sum_{p}\frac{m_{p}^{2}}{4T\gamma}\int dt\left[\dot{\mathbf{v}}_{p}-\frac{\mathbf{F}_{p}}{m_{p}}-\frac{\mathbf{f}^{a}_{p}}{m_{p}}-\frac{\gamma}{m_{p}}\mathbf{v}\right]^{2}\,,$$

(10)

where we have again neglected the irrelevant contribution of the angular dynamics, for the same reasons given above.

Performing algebraic calculations, the expression for $\dot{s}$ can be analytically derived and reads:

$$\dot{s}=\sum_{p}\dot{s}_{p}$$

(11)

where $\dot{s}_{p}$ is the entropy production generated by a single particle given by

$$\dot{s}_{p}=\frac{1}{T}\langle\mathbf{f}^{a}_{p}\cdot\mathbf{v}_{p}\rangle+{\text{b.t.}}\,.$$

(12)

The expression b.t. means boundary terms, i.e. the additional terms that do not contribute to the steady-state entropy production and vanish in the long-time limit. Such a result holds, in general, for underdamped active particles subject to a persistent active force and in contact with a thermal bath, independently of their density and of the dimension of the system.

By introducing the spatial Fourier transform of the dynamical variables denoted by hat-symbols, and by neglecting boundary terms, $\dot{s}_{p}$ can be calculated in the Fourier space, in particular, in the frequency domain, obtaining

$\displaystyle\dot{s}_{p}$

$\displaystyle=\lim_{t\to\infty}\frac{1}{t}\int\frac{d\omega}{2\pi}\frac{\langle\hat{\mathbf{f}}^{a}_{p}(-\omega)\cdot\hat{\mathbf{v}}_{p}(\omega)\rangle}{T}+c.c.$

(13)

where $\hat{\mathbf{f}}^{a}_{p}(\omega)$ and $\hat{\mathbf{v}}_{p}(\omega)$ are the Fourier transforms in the frequency domain of the active force and velocity of the $p$ particle, respectively, defined in appendix A and the symbol ”$c.c$” stands for complex conjugate. The EPR, $\dot{s}$, is proportional to the frequency integral of the real part of the cross-correlation between the Fourier components of active force and velocity. We also introduce the spectral entropy $\sigma_{p}(\omega)$ for each particle as the integrand in Eq. (13)

$$\sigma_{p}(\omega)=\lim_{t\to\infty}\frac{1}{t}\frac{\langle\hat{\mathbf{v}}_{p}(\omega)\cdot\hat{\mathbf{f}}^{a}_{p}(-\omega)\rangle}{T}+c.c.$$

(14)

Note that this term corresponds to the spectral dissipation of the particle $p$ due to the active force, divided by the temperature of the bath.

To proceed analytically, we choose $m_{1}=m+\delta m$ with $|\delta m|\ll m$ and apply a perturbative method by expanding the solution in powers of the small parameter $\delta m/m\ll 1$. Explicitly the modified equation of motion for the imperfect lattice reads

$$(m+\delta_{p0}\delta m)\dot{\mathbf{v}}_{p}=-\gamma\mathbf{v}_{p}+\mathbf{F}_{p}+\sqrt{2T\gamma}\boldsymbol{\xi}_{p}+\mathbf{f}^{a}_{p}$$

(15)

where the Kronecker delta function $\delta_{p0}$ selects the particle $p=0$ corresponding to the imperfection. The force $\mathbf{F}_{p}$ is approximated within the harmonic approximation of Eq. (5).

By introducing the continuous Fourier transforms in the frequency domain $\omega$ of the displacement $\hat{\mathbf{u}}_{p}=\hat{\mathbf{u}}_{p}(\omega)$, the velocity $\hat{\mathbf{v}}_{p}=i\omega\hat{\mathbf{u}}_{p}(\omega)$, the active force $\hat{\mathbf{f}}^{a}_{p}(\omega)$ and the white noise $\hat{\boldsymbol{\xi}}_{p}(\omega)$ Eq. (15) can be rewritten as

$$L_{pk}(\omega)\hat{\mathbf{u}}_{k}-\delta m\omega^{2}\hat{\mathbf{u}}_{0}\delta_{p0}=\hat{\mathbf{f}}^{a}_{p}+\sqrt{2\gamma T}\hat{\boldsymbol{\xi}}_{p}$$

(16)

where the time-Fourier transform of a generic observable is denoted by the hat symbol and the Einstein summation convention used. The matrix elements $L_{pk}(\omega)$ have the following form

$$L_{pk}(\omega)=(-m\omega^{2}+i\omega\gamma)\delta_{pk}-m\omega^{2}_{E}\sum_{j}^{*}\delta_{p,k+j}$$

(17)

where the sum over the index $j$ runs over the nearest neighbors $(k+j)$ of the particle $p$. When $\delta m=0$, Eq. (16) corresponds to the dynamics of a perfect lattice and can be rewritten with the help of the lattice Green’s function, $G_{pn}(\omega)$, in the $\omega$-representation as:

$$\hat{\mathbf{u}}_{p}(\omega)=G_{pn}(\omega)(\hat{\mathbf{f}}^{a}_{n}+\sqrt{2\gamma T}\hat{\boldsymbol{\xi}}_{n})\,.$$

(18)

where $G_{pn}(\omega)$ is expressed with the help of the discrete spatial-Fourier transform:

$$G_{pn}(\omega)=L^{-1}_{pn}(\omega)=\frac{1}{N}\sum_{\mathbf{q}}\frac{e^{-i\mathbf{q}\cdot(\mathbf{r}^{0}_{\mathbf{n}}-\mathbf{r}^{0}_{\mathbf{p}}})}{-m\omega^{2}+i\omega\gamma+m\omega^{2}(\mathbf{q})}\,,$$

(19)

where the sum runs over the dimensionless reciprocal lattice wave vectors $\mathbf{q}$ (see appendix C). The quantity $\omega(\mathbf{q})$ represents the dispersion relation of the vibrational modes of the lattice and depends on the Einstein frequency, $\omega_{E}$ and on the lattice structure. It is obtained by solving the secular equation associated with the lattice harmonic oscillations [84]. Since the perturbative method discussed below is independent of the specific lattice structure, being the relative information contained in $\omega(\mathbf{q})$, we postpone (see Eq. (34)) the presentation of the explicit expression of $\omega(\mathbf{q})$ for the case of the triangular lattice in two dimensions and nearest neighbor interactions. By solving Eq. (16) when $\delta m\neq 0$, we obtain

$$\hat{\mathbf{u}}_{p}(\omega)=G_{pn}(\hat{\mathbf{f}}^{a}_{n}+\sqrt{2\gamma T}\hat{\boldsymbol{\xi}}_{n})+\delta m\omega^{2}G_{p0}\,\hat{\mathbf{u}}_{0}\,.$$

(20)

The entropy production $\dot{s}$ can now be calculated by using Eq. (13), which requires the knowledge of the cross-dynamical correlation between the active force and velocity. We multiply Eq. (20) by $\mathbf{f}^{a}_{n}(-\omega)$, take the average over the realizations of the noise, and obtain

	$\displaystyle\langle\hat{\mathbf{u}}_{p}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle=$	$\displaystyle G_{pn}(\omega)\langle\hat{\mathbf{f}}_{n}^{a}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle$		(21)
		$\displaystyle+\delta m\,\omega^{2}G_{pn}\langle\hat{\mathbf{u}}_{n}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle\delta_{n0}\,.$

Multiplying both sides of Eq. (21) by $(i\omega)$ we find a relation for the average $\langle\hat{\mathbf{v}}_{p}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle$. To proceed further, we need the explicit expression of the correlation $\langle\hat{\mathbf{f}}^{a}_{p}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle$, which is estimated by approximating the ABP dynamics with the AOUP model [85, 86, 87, 88, 89]. This strategy has been often adopted with success in the literature to get analytical predictions [90, 91, 43, 92]. We obtain

$$\lim_{t\to\infty}\frac{1}{t}\langle\hat{\mathbf{f}}^{a}_{n}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle=2v_{0}^{2}\gamma^{2}\frac{\tau}{1+\omega^{2}\tau^{2}}\delta_{np}\,,$$

(22)

and

$$\langle\hat{\mathbf{f}}^{a}_{p}(\omega)\rangle=0$$

(23)

as discussed in Appendix D. By replacing the expression (22) in Eq. (21), we finally get the equation describing the dynamical cross correlation between active force and velocity

	$\displaystyle\lim_{t\to\infty}\frac{1}{t}$	$\displaystyle\langle\hat{\mathbf{v}}_{p}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle=2i\omega G_{np}(\omega)v_{0}^{2}\gamma^{2}\frac{\tau}{1+\omega^{2}\tau^{2}}\delta_{np}$		(24)
		$\displaystyle+\lim_{t\to\infty}\frac{i}{t}\delta m\,\omega^{3}G_{p0}(\omega)\langle\hat{\mathbf{u}}_{0}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle\,.$

By dividing by the temperature $T$ and taking the real part of Eq. (24), we obtain the equation to determine $\sigma_{p}(\omega)$, that reads

	$\displaystyle T$	$\displaystyle\sigma_{p}(\omega)=-2\omega v_{0}^{2}\gamma^{2}\frac{\tau}{1+\omega^{2}\tau^{2}}\text{Im}[G_{np}(\omega)]\delta_{np}$		(25)
		$\displaystyle-\lim_{t\to\infty}\frac{1}{t}\delta m\omega^{3}\text{Im}[G_{p0}(\omega)\langle\hat{\mathbf{u}}_{0}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle]\,.$

Equation (25) is not closed and cannot be used to determine the entropy production rate because contains the dependence on the unknown correlation $\langle\hat{\mathbf{u}}_{0}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle$. To proceed we employ a perturbative method and expand the solution in powers of $\lambda=\delta m/m\ll 1$:

$$\hat{\mathbf{u}}_{p}(\omega)=\hat{\mathbf{u}}_{p}^{(0)}(\omega)+\lambda\hat{\mathbf{u}}_{p}^{(1)}(\omega)+\lambda^{2}\hat{\mathbf{u}}_{p}^{(2)}(\omega)+...\,.$$

(26)

where the superscript ⁽ⁿ⁾ denotes the order of the perturbative solution that is consistent with a perturbative solution of the spectral entropy production:

$${\sigma}_{p}(\omega)={\sigma}_{p}^{(0)}(\omega)+\lambda{\sigma}_{p}^{(1)}(\omega)+\lambda^{2}{\sigma}_{p}^{(2)}(\omega)+...\,.$$

(27)

and of its integral over $\omega$, i.e. $\dot{s}$, so that

$$\dot{s}_{p}=\dot{s}_{p}^{(0)}+\lambda\dot{s}_{p}^{(1)}+\lambda^{2}\dot{s}_{p}^{(2)}+...\,.$$

(28)

Note that the zeroth-order entropy production is independent of the particle index $p$, which can be dropped at this order. Instead, the EPR is spatially dependent starting from the first correction in $\delta m/m$.

In our discrete formalism, this implies that $\dot{s}_{p}^{(0)}=\dot{s}^{(0)}$ and $\sigma_{p}^{(0)}(\omega)=\sigma^{(0)}(\omega)$ for every $p$, while, in general, we expect a dependence on $p$, or in other words a spatial dependence, only starting from the first correction in $\delta m/m$.

The zero-order solution of Eq. (25) (obtained for $\delta m=0$) corresponds to the spectral entropy, ${\sigma}^{(0)}(\omega)$, of the homogeneous solid, in the absence of the impurity.

Setting $\delta m=0$ in Eq. (25), we obtain the zeroth-order value of the spectral entropy production per particle

$\displaystyle T\sigma^{(0)}(\omega)=$

$\displaystyle-2\omega v_{0}^{2}\gamma^{2}\frac{\tau}{1+\omega^{2}\tau^{2}}\text{Im}[G_{00}(\omega)]\,,$

(29)

which is independent of the position. By integrating over $\omega$, we get the zeroth-order expression for the entropy production rate

	$\displaystyle T\dot{s}^{(0)}$	$\displaystyle=\frac{2}{N}\sum_{q}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{\tau\gamma}{1+\omega^{2}\tau^{2}}\frac{\omega^{2}v_{0}^{2}\gamma^{2}}{m^{2}(\omega^{2}(\mathbf{q})-\omega^{2})^{2}+\omega^{2}\gamma^{2}}$		(30)
		$\displaystyle=\frac{v_{0}^{2}\tau\gamma}{\tau+\tau_{I}}\mathcal{G}_{00}.$

The second equality introduces the propagator, $\mathcal{G}_{00}$, whose expression is:

$$\mathcal{G}_{00}=\frac{1}{N}\sum_{q}\frac{1}{1+\frac{\tau^{2}\tau_{I}}{\tau+\tau_{I}}\omega^{2}(\mathbf{q})}\,.$$

(31)

The $\omega$-integration in Eq.(76) leading to Eq.(31) is reported in appendix E (see Eq.(76)). In practice, we convert the sum over wave-vectors into an integral over the Brillouin zone of volume $\Omega$ by replacing $\frac{1}{N}\sum_{q}\to\int_{\Omega}\frac{d{\mathbf{q}}}{\Omega}$ and $\mathcal{G}_{00}\to\mathcal{G}(0)$. The prefactor in Eq.(30) is identified with the EPR of the non-interacting system, $\dot{s}_{\text{free}}$, according to the relation:

$$\dot{s}_{\text{free}}=\frac{v_{0}^{2}\tau\gamma}{T}\frac{1}{\tau_{I}+\tau}\,,$$

(32)

and rewrite:

$$\dot{s}^{(0)}=\dot{s}_{\text{free}}\mathcal{G}(0)\,.$$

(33)

The quantity $\dot{s}_{\text{free}}$ is a function of the ratio between the active temperature, $v_{0}^{2}\gamma\tau$, and the thermal temperature. At fixed active temperature, it decreases as the persistence time, $\tau$, and inertial time, $\tau_{I}$ increase . Instead, the term $\mathcal{G}_{00}$ accounts for the interactions among the particles in the solid, is $1$ in the non-interacting limit, and $\mathcal{G}_{00}\leq 1$ for the interacting case since $\omega^{2}(\mathbf{q})\geq 0$. In other words, the interaction decreases the value of the EPR with respect to the free-particles case because the interactions characterizing the solid hinder the particle’s ability to move with the same speed as free particles so that $|\mathbf{v}|\ll v_{0}$. As a consequence, the entropy production of a solid formed by $N$ active particles is always smaller than the entropy production of $N$ potential-free active particles, $\dot{s}\leq\dot{s}_{\text{free}}$.

Imperfections always break the discrete spatial translational symmetry of the crystal. Hence, some observables, including the local entropy production rate, are expected to become spatially dependent on the distance from the defect and take a constant value away from it, the one characterizing the perfect crystal.

To analytically predict the local EPR, we employ the first-order perturbative expression, $\sigma_{p}^{(1)}(\omega)$, given by Eq. (25):

$\displaystyle T\sigma_{p}^{(1)}(\omega)=$

$\displaystyle-m\omega^{3}\text{Im}[G_{p0}(\omega)\langle\hat{\mathbf{u}}^{(0)}_{0}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle]\,,$

(37)

where $\hat{\mathbf{u}}^{(0)}_{0}(\omega)$ is the unperturbed displacement at the location of the impurity. Eq. (20) gives the expression $\hat{\mathbf{u}}^{(0)}_{0}(\omega)=G_{0n}(\hat{\mathbf{f}}^{a}_{n}+\sqrt{2\gamma T}\hat{\boldsymbol{\xi}}_{n})$ that substituted in Eq. (37) yields:

	$\displaystyle T\sigma_{p}^{(1)}(\omega)=$	$\displaystyle-m\omega^{3}\text{Im}[G_{p0}(\omega)G_{0n}(\omega)\lim_{t\to\infty}\frac{1}{t}\langle\hat{\mathbf{f}}^{a}_{n}(\omega)\hat{\mathbf{f}}_{p}^{a}(-\omega)\rangle]$		(38)
	$\displaystyle=$	$\displaystyle-m\omega^{3}\text{Im}[G_{p0}(\omega)G_{0p}(\omega)]\frac{2v_{0}^{2}\gamma^{2}\tau}{1+\omega^{2}\tau^{2}}\,.$

where we employed the correlator of the active force, Eqs.(22), to obtain the last equality.

After replacing the expression for $G_{0p}(\omega)$ and $G_{p0}(\omega)$ using Eq.(19) and integrating over the frequency $\omega$, we obtain the first-order perturbative correction to the EPR. Since the calculations are rather lengthy, here, we show only the final results while details of the derivation are reported in Appendix C:

$$\dot{s}^{(1)}_{p}=-\frac{v_{0}^{2}\tau\gamma\tau_{I}}{T(\tau+\tau_{I})^{2}}\mathcal{G}_{0p}\mathcal{G}_{p0}=-\frac{\dot{s}_{\text{free}}\tau_{I}}{\tau+\tau_{I}}\mathcal{G}_{0p}\mathcal{G}_{p0}\,,$$

(39)

where

$$\mathcal{G}_{0p}=\frac{1}{N}\sum_{q}\frac{e^{-i\mathbf{q}\cdot\mathbf{r}_{p}^{0}}}{1+\frac{\tau^{2}\tau_{I}}{\tau+\tau_{I}}\omega^{2}(\mathbf{q})}$$

(40)

and $\mathcal{G}_{p0}=\mathcal{G}^{*}_{0p}$. By approximating the sum $\sum_{q}$ by an integral,

$\displaystyle\mathcal{G}_{0p}\to\mathcal{G}(r)=$

$\displaystyle\int_{\Omega}\frac{d{\mathbf{q}}}{\Omega}\frac{e^{-i\mathbf{q}\cdot\mathbf{r}_{p}^{0}}}{1+\frac{\tau^{2}\tau_{I}}{\tau+\tau_{I}}\omega^{2}(\mathbf{q})}\,.$

(41)

The above integral depends on the distance from the impurity, $r$ and converges to the value $\mathcal{G}(0)$ as $r\to 0$. Thus, the first-order correction to the entropy production, at distance $r$ from the impurity, becomes:

$$\dot{s}^{(1)}_{p}\to\dot{s}^{(1)}(r)=-\dot{s}_{\text{free}}\mathcal{G}(r)^{2}\frac{\tau_{I}}{\tau_{I}+\tau}\,.$$

(42)

.

According to Eq. (42) the first order perturbative correction to the EPR at the position of the impurity, $r=0$, is

$$\dot{s}^{(1)}(0)=-\frac{\dot{s}_{\text{free}}\tau_{I}}{\tau+\tau_{I}}[\mathcal{G}(0)]^{2}=-\dot{s}^{(0)}\mathcal{G}(0)\frac{\tau_{I}}{\tau+\tau_{I}}\,.$$

(43)

We estimated numerically $\dot{s}^{(1)}(0)$ from the difference $(\dot{s}(0)-\dot{s}^{(0)})$, i.e. by subtracting the zeroth-order EPR from the total EPR obtained from the simulations. The resulting value of $\dot{s}^{(1)}(0)$ together with the zeroth-order theoretical prediction $\dot{s}^{(0)}$, shown for comparison, is plotted in Fig. 3 as a function of the reduced inertial time $\tau_{I}/\tau$. Both $\dot{s}^{(0)}$ and $\dot{s}^{(1)}(0)$ are bell-shaped curves but the two maxima do not coincide. Indeed, $\dot{s}^{(1)}(0)$ tends to zero as equilibrium is approached, i.e. when $\tau_{I}/\tau\gg 1$. By decreasing the ratio $\tau_{I}/\tau$ the system departs from equilibrium and $\dot{s}^{(1)}(0)$ grows, as $\dot{s}^{(0)}$ also does, and reaches a peak for $\tau_{I}/\tau\approx 10$. Such an increase when $\tau_{I}/\tau$ decreases is mainly due to the prefactor $\dot{s}_{\text{free}}$ in Eq. (43).

In the large persistence regime, $\tau_{I}/\tau\leq 10$, the first-order correction $\dot{s}^{(1)}(0)$ displays a decrease similar to $\dot{s}^{(0)}$. From Eq. (43) we see that $\left|\frac{\dot{s}^{(1)}(0)}{\dot{s}^{(0)}}\right|=\mathcal{G}(0)\frac{\tau_{I}}{\tau+\tau_{I}}\leq 1\,$. The inequality holds because both $\mathcal{G}(0)$ and $\tau_{I}/(\tau+\tau_{I})$ are less or equal to 1. Moreover, $\frac{\dot{s}^{(1)}(0)}{\dot{s}^{(0)}}$ is a decreasing function of $\tau$ and an increasing function of $\tau_{I}$. In the limit of infinite $\tau$, the system approaches an arrested state (with decreasing speed, $|\mathbf{v}|$), and, as a consequence, not only the entropy production of the bulk $\dot{s}^{(0)}$ decreases but also the EPR due to the imperfection at $r=0$ does.