Clausius relation for active particles: what can we learn from fluctuations?

TL;DR

This paper compares different theoretical definitions of heat and entropy production in active particles, emphasizing an 'active heat' that satisfies the Clausius relation and can be experimentally tested.

Contribution

It analyzes and compares three recent proposals for defining heat and entropy production in active particle models, highlighting an experimentally verifiable active heat.

Findings

Active heat satisfies the Clausius relation.

Comparison of three definitions clarifies their physical meaning.

Active heat can be experimentally measured.

Abstract

Many kinds of active particles, such as bacteria or active colloids, move in a thermostatted fluid by means of self-propulsion. Energy injected by such a non-equilibrium force is eventually dissipated as heat in the thermostat. Since thermal fluctuations are much faster and weaker than self-propulsion forces, they are often neglected, blurring the identification of dissipated heat in theoretical models. For the same reason, some freedom - or arbitrariness - appears when defining entropy production. Recently three different recipes to define heat and entropy production have been proposed for the same model where the role of self-propulsion is played by a Gaussian coloured noise. Here we compare and discuss the relation between such proposals and their physical meaning. One of these proposals takes into account the heat exchanged with a non-equilibrium active bath: such an "active heat" satisfies the original Clausius relation and can be experimentally verified.

Equations

$$\delta \Sigma =dS-\frac{\delta Q}{T}\ge 0.$$

$$dS\ge \frac{\delta Q}{T}$$

$$TdS\ge dE-\delta W\to \delta W\ge dE-TdS=dF.$$

$$\oint \frac{\delta Q(t)}{T(t)}\le 0,$$

$$\delta \Sigma =dS-\frac{\delta Q}{T}+{\Sigma }_{an}\ge 0,$$

$${\int }_0^t\delta \sigma (t^{\prime })$$

$$={\int }_0^{t}ds+{\int }_0^t\delta s_m,$$

$$dx(t)$$

$$mdu(t)$$

$$\delta q=de-\delta w=u\circ [-\gamma udt+\sqrt{2\gamma T}dW(t)]$$

$$\delta \sigma =ds-\frac{u\circ [-\gamma udt+\sqrt{2\gamma T}dW]}{T}=ds-\frac{\delta q}{T},$$

$$dx(t)$$

$$mdu(t)$$

$$d{f}_a(t)=-\frac{f_a(t)}{\tau }dt+\frac{\gamma \sqrt{2D_a}}{\tau }d{W}_2(t),$$

$$dx(t)=\frac{\sqrt{2\gamma T_b}dW(t)+f_a(t)dt-{\varphi }^{\prime }[x(t)]dt}{\gamma }.$$

$$\delta q_b=u\circ [-\gamma udt+\sqrt{2\gamma T_b}dW(t)],$$

$$\delta Q_b=⟨\delta q_b⟩\le 0.$$

$$\dot{x}=\frac{f_a(t)-{\varphi }^{\prime }(x)}{\gamma }.$$

$$dx(t)$$

$$\mu du(t)$$

$$=-\gamma \Gamma (x)u(t)dt+\gamma \sqrt{2D_a}dW(t)-{\varphi }^{\prime }[x(t)]dt$$

$$\delta q_1=u\circ (-\gamma udt+\gamma \sqrt{2\gamma T_a}dW).$$

$$\delta \Sigma =ds-\frac{\delta q_1}{T_a}+\frac{{\tau }^2}{2T_a}(du{)}^2{\varphi }^{\prime \prime },$$

$$\delta \Sigma =dS-\frac{\delta Q_1}{T_a}+\frac{dt}{\gamma }⟨{\varphi }^{\prime \prime }⟩\ge 0,$$

$$\delta \Sigma \approx \frac{{\tau }^{2}u{\varphi }^{\prime \prime }\circ du}{T_a}.$$

$$\delta \sigma =ds-\frac{\delta q_2}{\theta (x)},$$

$$\delta q_2=u\circ d{f}_{ab},$$

$$d{f}_{ab}(t)$$

$$=-\gamma \Gamma [x(t)]u(t)dt+\sqrt{2\gamma \Gamma [x(t)]\theta (x)}dW(t)$$

$$\delta \Sigma =dS-\frac{⟨\delta q_2(x)⟩}{\theta (x)}\ge 0,$$

$$\dot{\tilde{q}}(x)=\gamma \Gamma (x)\left[\frac{\theta (x)}{\mu }n(x)-\int dv{u}^{2}p(x,u)\right],$$

$$\mu d{u}_i=-\gamma {\Gamma }_{ij}(\mathbf{r})u_{j}dt-{\partial }_i\varphi (\mathbf{r})dt+\gamma \sqrt{2D_a}d{W}_i,$$

$$\mu d{U}_i=-\gamma {\lambda }_i(\mathbf{R})U_{i}dt-{\partial }_{R_i}\varphi +\gamma \sqrt{2D_a}d{W}_i.$$

Full text

I Introduction

Active particle systems have recently attracted the increasing interest of scientists of different disciplines since they sit at the intersection of biology, chemistry and physics Ramaswamy (2010); Marchetti et al. (2013). A central feature of these materials is that its elementary constituents convert energy from the environment via metabolic or chemical reactions into direct motion but also dissipate energy producing heat by friction in order to move inside a surrounding solvent Bechinger et al. (2016). Therefore, the complex behavior of active particles can only be described by the tools of non equilibrium statistical physics, such as kinetic theory, statistical mechanics of non-equilibrium processes and stochastic thermodynamics Cates (2012). A fascinating question, which naturally comes to our mind, is how thermodynamics shapes biological functions in living organisms England (2013) such as motility and self-propulsion and in particular which is the entropy production associated with their non equilibrium steady states de Groot and Mazur (1984). This question requires a notion of heat generated by self-propulsion and dissipated in the thermostatted solvent.

Within careful calorimetric experiments, one is able to measure the heat dissipated into the solvent by a microbial colony Gustafsson (1991). Such total heat depends upon many biological functions which are not included in active models, but one could devise smart experiments (e.g. by varying motility without changing other functions) in order to assess the fraction of heat strictly generated by self-propulsion. Observing the associated fluctuations is perhaps a much harder task, if not impossible. However, before encountering experimental limitations, one finds limits in the theory.

At our mesoscopic level, the definition of heat has to be framed within stochastic thermodynamics Lebowitz and Spohn (1999); Seifert (2005); Marconi et al. (2008); Sekimoto (2010). A problem, however, arises when thermal fluctuations are discarded: such an approximation is adopted in many models of active particles, since temperature is negligible with respect to the energy associated with both self-propulsion and external forces. In active models, some noise is retained to describe the non-deterministic nature of the self-propulsion force, but it usually acts on time-scales and energy scales much larger than molecular agitation of the solvent. This “athermal” nature of active particle models is similar to that in granular models Puglisi et al. (2005) or in models of macroscopic friction Baule et al. (2011). While it is very useful - sometimes even inevitable - for the purpose of analytic calculations or numerical simulations, it leads to a mismatch between entropy production and heat Cerino and Puglisi (2015). Basically, the relation between total entropy production and dissipated heat loses its similitude with the original Clausius form and involves additional terms. For this reason, it does not provide a clear constraint on heat divided by temperature, as it occurs for the Clausius relation in macroscopic thermodynamics. Such a problem has already been noticed in some models of active particles Ganguly and Chaudhuri (2013), and in systems with feedback Kim and Qian (2004); Munakata and Rosinberg (2012, 2014).

Recently it has been shown that the above fallacy is bypassed in a model of active particles where Gaussian colored noise plays the role of self-propulsion Maggi et al. (2015); Marconi et al. (2017). Even if thermal fluctuations are neglected, a notion of “coarse-grained heat” can be introduced, together with a spatial-dependent effective temperature, such that the original Clausius relation is fully recovered. The crucial point here is that both such a “coarse-grained heat” and the effective temperature can be measured in experiments and therefore a test of this active Clausius relation can be attempted. In the last year other two proposals have appeared in the literature Fodor et al. (2016); Mandal et al. (2017), devoted to define entropy production and heat in the same identical model. The purpose of the present paper is to discuss the connection between those different Clausius relations. Apart from this comparison, two novelties are present here with respect to Marconi et al. (2017): 1) a single trajectory level of description is adopted, while in Marconi et al. (2017) an ensemble average had been considered; 2) a generalization of Marconi et al. (2017) to more than one dimension and interacting particles is presented. In order to simplify the discussion, the main discussion is focused on the 1d case with a time-independent potential, which is sufficient to show the main difference between the definitions of heat and entropy production in the three works considered Fodor et al. (2016); Marconi et al. (2017); Mandal et al. (2017). The generalisation of the proposal in Marconi et al. (2017) to multi-dimensional cases with a time-dependent potential is also discussed at the end.

In Section 2 we review the basic facts of Clausius relation in macroscopic thermodynamics and in stochastic (or mesoscopic) thermodynamics. In Section 3, we introduce the active model with Gaussian colored noise with thermal fluctuations, where a “microscopic Clausius relation” is trivially satisfied, and then show what happens at the coarse-grained level, when inertia and thermal fluctuations are neglected. The three recipes appeared in Fodor et al. (2016); Marconi et al. (2017); Mandal et al. (2017) to connect entropy production, dissipated heat and temperature are reviewed and compared.

II Heat and entropy production: from macroscopic to stochastic thermodynamics

Here we revise a few elementary facts of thermodynamics, at the macroscopic level and at the mesoscopic one. The macroscopic level is the one presented in thermodynamic textbooks, where there are no fluctuations: we denote quantities at this level with capital letters. The mesoscopic level is the topic of an intense research exploded roughly in the last two decades, and is dominated by fluctuations: we denote quantities at this level with small letters. Averaging out the fluctuations of the mesoscopic level (an operation which - in general - is automatically obtained in the limit of a very large number of constituents) brings back the results of the macroscopic one. Through the whole paper, we set the Boltzmann constant $k_{B}=1$.

II.1 Macroscopic level

For a large system one may measure heat, for instance with a calorimeter, and call it $\delta Q$: we consider it positive when going from the bath to the system. The second principle of thermodynamics guarantees that in a transformation where entropy $S$ changes by a quantity $dS$ there is a non-negative entropy production 111We use a shorthand notation where - for instance $\delta\Sigma$ and $\delta Q$ are non-exact differentials, while $dS$ is an exact differential.

[TABLE]

In a “quasi-static” transformation the equal sign holds, i.e. $\delta\Sigma=0$.

In the absence of a direct way of measuring the entropy of a system (e.g. if quasi-static transformations are not available), other relations derived from Eq. (1) and involving measurable observables are useful. For instance from Eq. (1) it follows

[TABLE]

and therefore the existence of a “minimum work” that can be extracted, also called difference of free energy $dF$:

[TABLE]

Another consequence that is derived way from Eq. (1) is to consider a cyclical transformation ($dS=0$), where it implies

[TABLE]

which is the celebrated Clausius relation. This can be tested in experiments and is the founding principle of the theory of heat engines, efficiency, etc.

It is important to underline that if - hypothetically - the total entropy production was something different from $dS-\delta Q/T$, i.e. if

[TABLE]

then all the above relations, including the Clausius relation, would not hold anymore due to the presence of an ”anomalous’” entropy production term, $\Sigma_{an}$. However it is quite difficult to imagine Eq. (5) in macroscopic thermodynamics, since the very definition of macroscopic entropy production is the difference between $dS$ and $\delta Q/T$ de Groot and Mazur (1984). On the contrary, equations similar to (5) have appeared in the literature in a stochastic thermodynamic treatment of systems with feedback and model of self-propelled particles Kim and Qian (2004); Munakata and Rosinberg (2012); Ganguly and Chaudhuri (2013); Munakata and Rosinberg (2014); Mandal et al. (2017).

II.2 Mesoscopic level

When a small system is considered, a stochastic description is necessary in order to incorporate fluctuations. At thermodynamic equilibrium the stochastic evolution must be consistent with micro-reversibility. More precisely, the couple “system plus thermostat” describes all degrees of freedom of the world and therefore it has to satisfy an exact symmetry under time-reversal: when the heat bath is replaced by an effective stochastic bath force, time-reversal is mapped into the equivalence of probabilities of a trajectory and its time-reversal, which coincides with detailed balance if the process is Markovian.

When an external, non-conservative, force is applied to the system, one may expect that the stochastic bath force is not changed (for instance if the bath is very large and is weakly affected by the external force). This amounts to say that the non-equilibrium model contains the sum of two forces which both concur to change the energy of the system: the external force does work, the bath force brings heat Sekimoto (2010). A notion of entropy production rate $\sigma$ of a trajectory $\omega(t)$, for Markovian stochastic systems, has been introduced in Lebowitz and Spohn (1999) and revisited in Seifert (2005). It can be summarized as

[TABLE]

where $\overline{\omega}$ is the time-reversal of the phase-space variables (typically positions are unchanged and velocities are reflected), $s(t)=-\ln p[\omega(t)]$ is the microscopic Gibbs entropy in the point $\omega(t)$ in phase space and $\delta s_{m}$ is the so-called entropy production of the surrounding medium Seifert (2005). In the rest of the paper we consider, for simplicity, the infinitesimal version of Eq. (6), i.e. $\delta\sigma=ds+\delta s_{m}$. An average over noise realizations and initial conditions is expected to give back the macroscopic quantities, i.e. $\delta\Sigma=\langle\delta\sigma\rangle$ and $dS=\langle ds\rangle$, such that Eq. (1) implies $\langle\delta s_{m}\rangle=-\delta Q/T$. Indeed in many models at constant temperature, one has $\delta s_{m}=-\delta q/T$ with $\delta q$ the heat injected by the bath force, which satisfies $\langle\delta q\rangle=\delta Q$. The total entropy production $\int_{0}^{t}\delta\sigma(t^{\prime})$ satisfies the Fluctuation-Relation at any time $t>0$ and this guarantees that $\delta\Sigma=\langle\delta\sigma\rangle$ is non-negative Lebowitz and Spohn (1999). In a stationary state $\langle\delta s_{m}\rangle\geq 0$ then follows.

As a useful example, let us consider the evolution of a colloidal particle of mass $m$, position and velocity $x(t),u(t)$ in one dimension, under the action of an external potential $\phi(x)$ and of a non-conservative external force $f_{nc}(t)$.

[TABLE]

with $dW(t)$ the Wiener infinitesimal increment (with variance $dt$). Defining energy as $e=mu^{2}/2+\phi(x)$, it is easy to see that heat (going from the bath into the system) reads

[TABLE]

where we have defined the work $\delta w=uf_{nc}dt$, and $\circ$ denotes products which must be integrated according to the Stratonovich rule.

For this model, it is possible to compute the conditional probability appearing in (6) and therefore compute $\delta s_{m}$. The result depends upon the parity of $f_{nc}$ under time-reversal Chetrite and Gawedzki (2008). In simple cases, for instance when magnetic fields are not involved Pradhan and Seifert (2010), such a force is assigned even parity under time-reversal. In this case one gets (see Appendix)

[TABLE]

which is $\geq 0$ on average, leading to the usual Clausius relation.

On the contrary if $f_{nc}$ is odd, for instance if the coarse-graining has delivered a force which is proportional to odd powers of the velocity of external bodies, or if magnetic fields are involved Pradhan and Seifert (2010), the relation (10) does not hold anymore. In such cases, things seem to improve when the so-called conjugated dynamics is considered, by changing the sign of odd external non-conservative forces when computing the probability of inverse paths appearing in the denominator of Eq. (6) Chernyak et al. (2006); Chetrite and Gawedzki (2008); Spinney and Ford (2012); Munakata and Rosinberg (2014): basically this amounts to change the parity of the force and get back the result in Eq. (10). The problem of such an artificial prescription, however, is that the conjugated dynamics cannot be realized in experiments and therefore an empirical evaluation (i.e. without a detailed knowledge of the equation of motions) of the conjugated probability is not available, neither it is possible to experimentally observe the associated fluctuation relation.

III Active particles: the coarse-grained heat and Clausius relation

The analogy between stochastic and macroscopic thermodynamics, Eq. (10), rests upon two main ingredients: 1) the heat bath must be modeled as a stochastic force which - if non-conservative forces are removed - satisfies detailed balance with respect to the equilibrium probability distribution ($\delta\sigma\equiv 0$) and 2) the non-conservative forces are even under time-reversal, a fact which is expected to be realized when the microscopic forces are not velocity-dependent (e.g. there are no Lorentz forces) and the coarse-graining does not change or mix their parity. Many models of active particles abandon such basic facts (in particular detailed balance Cates (2012)), with the aim of describing the relevant variables (such as positions or orientations of the micro-swimmers) which evolve on scales much slower than those affected by thermal agitation. An interesting example of model of active swimmers where this procedure can be analyzed is one where self-propulsion takes the form of an Ornstein-Uhlenbeck process: its non-zero correlation time represents the persistence of motion due to activity.

Here we introduce the model at a space-time scale fine enough to describe the real velocity $u$ of the particle and thermal fluctuations:

[TABLE]

where $T_{b}$ is the environmental (solvent) temperature and the active force satisfies

[TABLE]

with $dW_{2}$ another (independent) Wiener increment with variance $dt$. Here we consider for simplicity the $1$-particle case in one dimension, with a potential $\phi(x)$ which does not depend upon time. Later we generalize some of the results to many interacting particles and with a time-dependent potential.

Note that, when $\phi=0$, $\langle x^{2}\rangle\sim 2(D_{a}+T_{b}/\gamma)t$ for large times. Based upon such a bare diffusivity, the “active bath temperature” $T_{a}=\gamma D_{a}$ is usually defined 222In Marconi et al. (2017) a “mass-less” active temperature $T_{\tau}=D_{a}/\tau$ was defined. In this paper we show that it is not necessary, if an “effective mass” $\mu=\gamma\tau$ is used, as in Mandal et al. (2017).. Of course there is no thermostat at temperature $T_{a}$, such a temperature is only useful to define a relevant energy scale.

Since active micro-swimmers are usually dispersed in viscous liquids, it is much more common to find the overdamped version of the model Maggi et al. (2015), which describes the position of the particle on a time-scale slower than the relaxation time due to inertia:

[TABLE]

III.1 Heat dissipation into the solvent

Interpreting $f_{a}$ as an external force derived - through the coarse-graining of the full microscopic dynamics - from forces which do not depend upon velocities, it is reasonable to consider it even. According to the recipe of stochastic thermodynamics discussed above, Eq. (6) applied to Eq. (11) or Eq. (13), see Appendix, one gets Eq. (10), with

[TABLE]

which is the heat absorbed from the reservoir, satisfying in the steady state the Clausius relation at constant temperature, i.e.

[TABLE]

The interpretation is obvious, the active force $f_{a}(t)$ acts as an external non-conservative force and transfers energy in the system which is dissipated into the bath. This can be measured by ordinary calorimetry in the solvent Speck (2016). As discussed above, such a measurement is in principle very difficult in experiments with living micro-swimmers, since released heat is affected by many other non-equilibrium biological functions. A promising direction could be the use of artificial active particles Bechinger et al. (2016).

III.2 Removing the solvent from the description

Since $T_{b}$ is orders of magnitude smaller than active temperatures, it is very useful - also for computational purposes - to remove it from Eq. (13), keeping only

[TABLE]

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 Zamponi et al. (2005); Speck and Seifert (2007); Crisanti et al. (2012).

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

[TABLE]

where we have introduced the effective mass $\mu=\gamma\tau$ and the space-dependent viscosity correction $\Gamma(x)=1+\frac{\tau}{\gamma}\phi^{\prime\prime}(x)$.

As highlighted by the two versions in Eqs. (17b)-(17c), the evolution of the effective velocity $u$ is affected by the conservative force $-\phi^{\prime}(x)$ 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 $f_{nc}=-\tau\phi^{\prime\prime}(x)u$ 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.

III.2.1 Equilibrium bath with a non-conservative force: conjugated entropy production

This interpretation is considered in Mandal et al. (2017). The authors propose to define heat as the energy injected by the force $-\gamma udt+\gamma\sqrt{2D_{a}}dW$, as if it were an equilibrium bath

[TABLE]

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 Sec. II.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 Eq. (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)

[TABLE]

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

[TABLE]

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 $\phi(x)\sim x^{2}$. Since in the harmonic case Eq. (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 Eq. (20) hardly deserves the name “Clausius relation”, as it does not give the same important information about the sign of the average heat.

III.2.2 Equilibrium bath with a non-conservative force: standard entropy production

In Fodor et al. (2016) the authors consider formula (6) (without conjugation for the reversed dynamics) applied to the dynamics in Eq. (17c). However all terms giving exact deterministic differentials are thrown away, leading to an approximate formula (see Appendix)

[TABLE]

In the steady state the neglected terms have zero average, and indeed only the average formula is reported in Fodor et al. (2016). Fluctuations and large deviations functions, however, may keep the memory of those terms Visco (2006); Puglisi et al. (2006); Bonetto et al. (2006).

Another difficulty of formula (21) is its connection with heat. In the end of their paper, the authors manage to show that Eq. (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 Eq. (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 Fodor et al. (2016). 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 Visco (2006); Puglisi et al. (2006); Bonetto et al. (2006).

III.2.3 Non-equilibrium bath

If the standard recipe of stochastic thermodynamics, Eq. (6), is used without neglecting any term, one gets (see Appendix)

[TABLE]

with the “active bath heat” defined as

[TABLE]

the “active bath force” as

[TABLE]

and the “local active temperature” $\theta(x)=T_{a}/\Gamma(x)$. This interpretation 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 Marconi et al. (2017) for details.

Averaging Eq. (22) at fixed position $x$, one gets

[TABLE]

which in the steady state ($dS=0$) is identical to the Clausius relation. In Marconi et al. (2017), the ensemble average of the above relation over the steady state probability distribution $p(x,u)$ has been considered. Interestingly, the local average $\dot{\tilde{q}}(x)$ of the dissipated heat flux reads

[TABLE]

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. Eq. (27) shows a straightforward way to measure such “active heat”.

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 $\phi(x)\sim x^{2}$ which is a special case where $\theta(x)$ is uniform Marconi et al. (2017).

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 Eq. (17) reads

[TABLE]

with $\Gamma_{ij}=\delta_{ij}+\frac{\tau}{\gamma}\partial_{j}\partial_{i}\phi$ and indexes running over all particles and all Cartesian components and the Einstein summation convention is assumed. The potential $\phi$ 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:

[TABLE]

Computation of the entropy production leads, therefore, to

[TABLE]

with $\theta_{i}({\mathbf{R}})=T_{a}/\lambda_{i}(\mathbf{R})$ the $i$-th component of the local active temperature and

[TABLE]

Notice that eq.(30) 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 $\phi(r)=\phi({\mathbf{r}})$, we have the following Cartesian representation of the matrix $D_{ij}(r)=D_{r}(r)\hat{r}_{i}\hat{r}_{j}+D_{t}(r)(\delta_{1j}-\hat{r}_{i}\hat{r}_{j})$ with $D_{r}(r)=1+\frac{\tau}{\gamma}\phi^{\prime\prime}(r)$ and $D_{t}(r)=1+\frac{\tau}{\gamma}\phi^{\prime}(r)/r$. The two temperatures are $\theta_{r}(r)=1/D_{r}(r)$ and $\theta_{t}(r)=1/D_{t}(r)$.

III.3 Time-dependent potential

When an external transformation is considered, i.e. a time-dependent potential $\phi(x,t)$ is taken into account, Eq. (17c) is replaced with

[TABLE]

Also, in this case, we get (see Appendix) the validity of the mesoscopic Clausius relation Eq. (22) with the active heat Eq. (23). Time-dependent potentials are at the basis, for instance of realizations of heat engines Zakine et al. (2017).

III.3.1 Very slow transformations

Imagine a very slow transformation from a $\phi(x,t_{1})$ to a new $\phi(x,t_{2})$: this means transforming the original non-equilibrium steady state (“NESS”, at $t<t_{1}$) to a new non-equilibrium steady state (for $t\gg t_{2}$). As discussed above, in the initial and final NESS there is heat going steadily to the bath, even without the transformation. Therefore for very slow transformations $\Delta Q\to-\infty$ and the Clausius relation becomes useless. For this reason Oono-Paniconi Oono and Paniconi (1998), then Hatano-Sasa Hatano and Sasa (2001), Bertini et al. Bertini et al. (2015) and Maes Maes and Netocný (2014) have found expressions for the so-called “excess heat”, i.e. heat which is released for the sole purpose of the transformation: this heat is obtained removing the “housekeeping heat” (necessary for the steady states) from the total $\Delta Q$. All the mentioned proposals have been given for overdamped systems, where certain symmetries are more clear but also less general. Active particles have some kind of inertia or persistence which cannot be disregarded and therefore do not comply with such an assumption. It would be interesting to see the above simple ideas applied to the model in Eq. (16) with a slow transformation of the potential.

IV Conclusions

In this paper, we have reviewed and compared three different prescriptions to extend the Clausius formula to active systems, i.e. to particles able to self-propel by means of metabolic processes or chemical reactions and to dissipate energy by a frictional mechanism. Those relations between the heat dissipation, entropy and work appear at the mesoscopic level, where fluctuations are taken into account by means of a stochastic description, but the contribution to these fluctuations coming from the molecular bath is neglected, leaving a certain freedom in defining heat and entropy production.

While all three methods are admissible and do not contradict any general principle, our point of view gives indications that only one of these prescriptions can be considered as a stochastic version of the original Clausius heat theorem, that is Eq. (22) with “active heat” defined in (23). The stochastic ”active” version of Clausius formula we have derived coincides with the one recently presented by using ensemble averaged quantities Marconi et al. (2017).

Acknowledgements.

We warmly akcnowledge useful discussions and communications with L. Cerino, A. Sarracino and F. van Wijland.

Appendix A Entropy production

We consider here a generalization of the dynamics in Eq. (8) with $f_{nc}(x,u,t)$ representing any kind of time-dependent term: it can be external or internal (that is function also of system’s degrees of freedom), odd or even under time-reversal, and we consider both the normal or the conjugated dynamics for the probability of the time-reversed path. In particular we assume $dx=udt$ and

[TABLE]

to generate the dynamics of the forward and reversed paths, respectively. The Wiener increments $dW(t)$ have variance $dt$. When the standard entropy production is computed, the dynamics is the same, i.e. $\overline{f}_{nc}(x,u,t)=f_{nc}(x,u,t)$. On the contrary for the conjugated entropy production $\overline{f}_{nc}(x,u,t)=-f_{nc}(x,u,t)$.

Following Eq. (6) (factorized for the Markovian dynamics), the infinitesimal entropy discharged into the surrounding medium reads

[TABLE]

In the first passage we have used the fact that the time-reversal of $du_{t}$ is $-u_{t}-(-u_{t+dt})=du_{t}$. In the second passage we have neglected terms which goes to zero faster than $dt$ and we have replaced $du_{t}(u_{t}+u_{t+dt})/2$ with $du_{t}\circ u_{t}$.

The cases considered in this paper are the following:

• •

The standard case in Eq. (8) where $f_{nc}(t)$ is even and external (i.e. it does not depend upon $x,v$): $\overline{f}_{nc}(t)=f_{nc}(t)$. In this case the last two terms in Eq. (37) become of higher order in $dt$ and one gets:

[TABLE]

which immediately gives Eq. (10).

• •

The case considered in Mandal et al. (2017), where $f_{nc}(x,u,t)=-\tau u\phi^{\prime\prime}(x)$ and (“conjugated entropy production”) $\overline{f}_{nc}(x,u,t)=\tau u\phi^{\prime\prime}(x)$. In this case the last term becomes of higher order in $dt$, while the term $du^{2}$ cannot be discarded (as it contains $dW^{2}\sim dt$), and therefore one gets

[TABLE]

(where we have used $m\equiv\mu=\gamma\tau$), that is Eq. (19).

• •

The case considered in Marconi et al. (2017) and in Fodor et al. (2016), where $f_{nc}(x,u,t)=-\tau u\phi^{\prime\prime}(x)$ and (according to the standard definition of stochastic entropy production) $\overline{f}_{nc}(x,u,t)=-\tau u\phi^{\prime\prime}(x)$. In this case the third term in Eq. (37) is of higher order in $dt$. All the other terms must be kept, giving

[TABLE]

If no terms are neglected, it gives exactly Eq. (22).

• •

If in Eq. (40) the exact differentials ($u_{t}\circ du_{t}=du_{t}^{2}/2$, $u_{t}\phi^{\prime}(x_{t})dt=d\phi(x_{t})$ and $u_{t}\phi^{\prime}(x_{t})\phi^{\prime\prime}(x_{t})dt=d[\phi^{\prime}(x_{t})]^{2}/2$) are removed, then only one terms remains:

[TABLE]

ie. Eq. (21).

• •

If a time-dependent potential is considered, then a second non-conservative force appears $f_{nc,2}(x,t)=-\tau\partial_{x}\partial_{t}\phi(x,t)$. We stress that the dependence upon time of $\phi(x,t)$ is external, i.e. (keeping the standard recipe of stochastic thermodynamics) the probability of the reversed dynamics is generated by the same equation, that is no change of sign is attributed to $\partial_{t}$. Basically we have $\overline{f}_{nc,2}(x,t)=-\tau\partial_{x}\partial_{t}\phi(x,t)$. Introducing $f_{nc,2}$ in Eq. (35) leads to the appearance of two new addends in the brackets $[...]$ of Eq. (37): one totally new term $-\tau u_{t}\partial^{2}\phi(x_{t},t)f_{nc,2}(x_{t},t)$ coming from the product $f_{nc}f_{nc,2}$; one surviving term $-\gamma u_{t}f_{nc,2}(x_{t},t)$ in the third addend. No new terms appear inside the fourth and fifth addend. In conclusion one gets

[TABLE]

which gives again Eq. (22).