At this point, an important ingredient of the bath force (its noise) has disappeared and the basic recipe of stochastic thermodynamics cannot be applied straightforwardly. Still, it is useful to find a measure of “distance from equilibrium” and relate it to parameters and observable quantities. Considering that

${f}_{a}$ is random, one is tempted to consider

$-\gamma \dot{x}+{f}_{a}(t)$ as an effective bath and define a heat as

$\dot{x}\circ [-\gamma \dot{x}+{f}_{a}(t)]$. However, the random force

${f}_{a}(t)$ is non-Markovian and therefore the standard recipe of stochastic thermodynamics brings in complications [

34,

35,

36].

The simplest way to get rid of the non-Markovian character of the noise is to time-derive Equation (16), obtaining

where we have introduced the effective mass

$\mu =\gamma \tau $ and the space-dependent viscosity correction

$\Gamma (x)=1+\frac{\tau}{\gamma}{\varphi}^{\u2033}(x)$.

As highlighted by the two versions in Equations (17b)–(17c), the evolution of the effective velocity u is affected by the conservative force $-{\varphi}^{\prime}(x)$ and by an additional force that can be interpreted in two different ways: (1) an equilibrium bath at temperature ${T}_{a}$ plus a non conservative force ${f}_{nc}=-\tau {\varphi}^{\u2033}(x)u$ which is odd under time-reversal; or (2) a non-equilibrium bath with space-dependent viscosity modulated according to the function $\Gamma (x)$. In the next two subsections, we see the consequences of such different interpretations, which change both the definition of entropy production as well as of heat.

#### 3.2.1. Equilibrium Bath with a Non-Conservative Force: Conjugated Entropy Production

This interpretation is considered in [

23]. The authors propose to define heat as the energy injected by the force

$-\gamma udt+\gamma \sqrt{2{D}_{a}}dW$, as if it were an equilibrium bath

To derive the entropy production, the authors consider the formula (6) with the probability of the time-reversed path (which appears in the denominator) computed according to a dynamics where the force

${f}_{nc}(t)$ is replaced by

$-{f}_{nc}(t)$, as discussed at the end of

Section 2.2. This idea is justified by the authors by showing that such a change of sign is necessary in order to make invariant under time-reversal the dynamics without the bath. However such an argument is not really compelling. The terms

$-\gamma udt+\gamma \sqrt{2\gamma {T}_{a}}dW$ do not correspond to any well-defined part of the physical system which could be identified as an equilibrium bath: the first term is the viscous damping due to the solvent, the second term is the fluctuating part of the derivative of the self-propulsion. It is a mathematical coincidence that together they form a Ornstein-Uhlenbeck process of the same form of equilibrium bath forces. In our opinion, it is quite arbitrary detaching them from Equation (17) and there is no reason why the rest of the equation (once those terms are removed) should satisfy the invariance under time-reversal.

According to the “conjugated” prescription, one gets for the case of a single particle considered here (see

Appendix A)

where

${(du)}^{2}\approx 2{T}_{a}dt/(\gamma {\tau}^{2})$. The average can be written as

with

$\delta {Q}_{1}=\langle \delta {q}_{1}\rangle $.

A peculiarity of this recipe is that it gives a non-zero average entropy production also for the harmonic case $\varphi (x)\sim {x}^{2}$. Since in the harmonic case Equation (17) satisfies detailed balance, it is unclear if such a peculiarity is an advantage or not. Moreover, as already discussed, entropy production computed with the conjugated reversed dynamics is not accessible in experiments. Most importantly, in our opinion Equation (20) hardly deserves the name “Clausius relation”, as it does not give the same important information about the sign of the average heat.

#### 3.2.2. Equilibrium Bath with a Non-Conservative Force: Standard Entropy Production

In [

22] the authors consider Formula (6) (without conjugation for the reversed dynamics) applied to the dynamics in Equation (17c). However all terms giving exact deterministic differentials are thrown away, leading to an approximate formula (see

Appendix A)

In the steady state the neglected terms have zero average, and indeed only the average formula is reported in [

22]. Fluctuations and large deviations functions, however, may keep the memory of those terms [

37,

38,

39].

Another difficulty of formula (21) is its connection with heat. In the end of their paper, the authors manage to show that Equation (16) can be mapped exactly into a generalized Langevin equation with memory. This equation can be broken into a viscoelastic bath at equilibrium at temperature

T plus a non-conservative force. Within such a description, the average of the entropy production rate in Equation (21) can be written as

$\mathcal{J}/T$, where

$\mathcal{J}$ is the heat flux dissipated into the bath. A simple formula for such a “viscoelastic” heat or its—local or global—average is not given in [

22]. Most importantly, some terms of fluctuations of entropy production are neglected which could be relevant for large deviation functions and the validity of the Fluctuation Relation [

37,

38,

39].

#### 3.2.3. Non-Equilibrium Bath

If the standard recipe of stochastic thermodynamics, Equation (

6), is used without neglecting any term, one gets (see

Appendix A)

with the “active bath heat” defined as

the “active bath force” as

and the “local active temperature”

$\theta (x)={T}_{a}/\Gamma (x)$. It is clear that—as in the general formulation, Equation (9)—

$\delta {q}_{2}$ corresponds to the variation of the total energy

$e=\frac{\mu {u}^{2}}{2}+\varphi (x)$ due to the bath force. The interpretation of

$\theta (x)$ as a local active temperature is supported by the observation that a local Maxwellian with temperature

$\theta (x)$ is an approximate solution for the local velocity distribution, with “small” violations of detailed balance, see [

21] for details. We underline that

$\theta (x)$ is immediately accessible in experiments: indeed the external potential

$\varphi (x)$ is directly controlled by the experimentalist (for instance by means of optical fields). The parameters

$\gamma $,

$\tau $ and

${D}_{a}$ can be measured by independent measurements with single particles in the fluid without potential.

Averaging Equation (22), one gets

which in the steady state (

$dS=0$) is identical to the Clausius relation [

21]. Interestingly, the local average

$\dot{\tilde{q}}(x)$ of the dissipated heat flux reads

where

$n(x)=\int dvp(x,u)$. This is an additional argument in favour of the simplicity and consistency of the picture discussed in the present section: the “active heat” is exactly proportional to the difference between the local active temperature

$\theta (x)$ and the empirical temperature

${\langle {u}^{2}\rangle}_{x}$. The empirical temperature is “attracted” by the local active temperature but the non-uniformity of such a temperature prevents full relaxation: the mismatch is a source of flowing heat. Equation (27) shows a straightforward way to measure such “active heat”. Indeed such a measurement only amounts to measure

$n(x)$ and

${\langle {u}^{2}\rangle}_{x}$ (for instance by means of a fast camera attached to a microscope), while all other variables are parameters of the experimental setup, controlled by the experimentalist. Once one has measured the local heat

$\dot{\tilde{q}}(x)$ an experimental verification of the Clausius relation, Equation (26), is immediately accessible, since

We note that when the potential does not depend upon time, as in all our equations up to this point, the active heat

$\delta {q}_{2}$ has zero average. Nevertheless, the average entropy production

$\delta \Sigma $ has non-zero average, apart from the harmonic case

$\varphi (x)\sim {x}^{2}$ which is a special case where

$\theta (x)$ is uniform [

21].

When more particles are involved, a (local and time-dependent) diagonalisation procedure can always set back the problem in the case of a single particle. The multi-particles and multi-dimensional version of Equation (17) reads

with

${\Gamma}_{ij}={\delta}_{ij}+\frac{\tau}{\gamma}{\partial}_{j}{\partial}_{i}\varphi $ and indexes running over all particles and all Cartesian components and the Einstein summation convention is assumed. The potential

$\varphi $ includes both external and internal forces. Since the matrix

${\Gamma}_{ij}(\mathbf{r})$ is symmetric, an orthogonal matrix

${P}_{ij}(\mathbf{r})$ always exists such that

$P\Gamma {P}^{T}=D$ with

${D}_{ij}(\mathbf{r})={\lambda}_{i}(\mathbf{r}){\delta}_{ij}$. By defining the rotated coordinates

$\mathbf{R}=P\mathbf{r}$ and velocities

$\mathbf{U}=P\mathbf{u}$, and recalling that the gradient rotates as a vector and the rotation of the vector of independent white noises gives again a vector of independent white noises, it is straightforward to get the formula:

Computation of the entropy production leads, therefore, to

with

${\theta}_{i}(\mathbf{R})={T}_{a}/{\lambda}_{i}(\mathbf{R})$ the

i-th component of the local active temperature and

Notice that Equation (31) generalizes the Clausius relation to a system with different temperatures ${\theta}_{i}$.

As an example, in the case of an active particle moving in a plane a subject to a central potential $\varphi (r)=\varphi (\mathbf{r})$, we have the following Cartesian representation of the matrix ${D}_{ij}(r)={D}_{r}(r){\widehat{r}}_{i}{\widehat{r}}_{j}+{D}_{t}(r)({\delta}_{1j}-{\widehat{r}}_{i}{\widehat{r}}_{j})$ with ${D}_{r}(r)=1+\frac{\tau}{\gamma}{\varphi}^{\u2033}(r)$ and ${D}_{t}(r)=1+\frac{\tau}{\gamma}{\varphi}^{\prime}(r)/r$. The two temperatures are ${\theta}_{r}(r)=1/{D}_{r}(r)$ and ${\theta}_{t}(r)=1/{D}_{t}(r)$.