Rémi Goerlich

Luís Barbosa Pires

Giovanni Manfredi
email: giovanni.manfredi@ipcms.unistra.fr

Paul-Antoine Hervieux

Cyriaque Genet
email: genet@unistra.fr

Harvesting information to control non-equilibrium states of active matter

We propose to use a correlated noise bath to drive an optically trapped Brownian particle that mimics active biological matter. Thanks to the flexibility and precision of our setup, we are able to control the different parameters that drive the stochastic motion of the particle with unprecedented accuracy, thus reaching strongly correlated regimes that are not easily accessible with real active matter. In particular, by using the correlation time (i.e., the "color") of the noise as a control parameter, we can trigger transitions between two non-equilibrium steady states with no expended work, but only a calorific cost. Remarkably, the measured heat production is directly proportional to the spectral entropy of the correlated noise, in a fashion that is reminiscent of Landauer's principle. Our procedure can be viewed as a method for harvesting information from the active fluctuations.

Introduction -Chemical and biological non-equilibrium processes reveal the special role played by fluctuations at mesoscopic scales, raising fascinating questions that form a major topic of current transdisciplinary research [START_REF] Rao | Nonequilibrium thermodynamics of chemical reaction networks: Wisdom from stochastic thermodynamics[END_REF][START_REF] Fang | Nonequilibrium physics in biology[END_REF][START_REF] Davies | Does new physics lurk inside living matter?[END_REF]. The combined development of stochastic thermodynamics and optical trapping experiments offers an appropriate framework to describe such processes with the required focus put on fluctuations [START_REF] Bustamante | The nonequilibrium thermodynamics of small systems[END_REF][START_REF] Ciliberto | Experiments in stochastic thermodynamics: Short history and perspectives[END_REF]. Recently, active fluctuations have been artificially injected inside optical traps by adding correlated (i.e. colored) noises to the trapping potential [START_REF] Park | Rapid-prototyping a Brownian particle in an active bath[END_REF][START_REF] Paneru | Transport and diffusion enhancement in experimentally realized non-gaussian correlated ratchets[END_REF][START_REF] Militaru | Escape dynamics of active particles in multistable potentials[END_REF]. The engineering of the noise inside the trap offers a new playground where, for instance, optomechanical models of active matter can be studied in close relation with theory [START_REF] Speck | Stochastic thermodynamics for active matter[END_REF][START_REF] Marconi | Heat, temperature and clausius inequality in a model for active brownian particles[END_REF][START_REF] Sandford | Pressure and flow of exponentially self-correlated active particles[END_REF][START_REF] Martin | Statistical mechanics of active Ornstein-Uhlenbeck particles[END_REF].

In this Letter, we follow this approach to decipher the energetics engaged between non-equilibrium states and correlated baths [START_REF] Dabelow | Irreversibility in active matter systems: Fluctuation theorem and mutual information[END_REF][START_REF] Speck | Modeling of biomolecular machines in nonequilibrium steady states[END_REF]. To do so, we bring an optically confined microparticle in an active-matter-like non-equilibrium steady state (NESS). The injected correlated noise yields the constant external source of heat needed to maintain, through its consumption, the particle in a chosen NESS. We monitor the non-Brownian diffusion of the trapped bead, and probe large deviations from thermal equilibrium, which correspond to strongly correlated regimes unexplored so far to our knowledge and not accessible with experiments that use real, biological active matter [START_REF] Argun | Non-boltzmann stationary distributions and nonequilibrium relations in active baths[END_REF][START_REF] Maggi | Generalized energy equipartition in harmonic oscillators driven by active baths[END_REF][START_REF] Maggi | Memory-less response and violation of the fluctuationdissipation theorem in colloids suspended in an active bath[END_REF].

We further demonstrate that it is possible to drive the bead from one NESS to another through the sole change of the correlation time (i.e., the "color") of the active fluctuations. For instance, transitions between NESS are known to occur when biological matter undergoes a change in mechanical properties, such as during mitosis, and thereby a modification of the intracellular noise spectrum [START_REF] Hurst | Intracellular softening and increased viscoelastic fluidity during division[END_REF][START_REF] Taubenberger | The mechanics of mitotic cell rounding[END_REF]. Modulating the color of the noise without changing its amplitude makes it possible to achieve heat production at constant energy input. We show that this capacity to induce NESS-to-NESS transitions at zero energetic cost is rooted in the informational content of active fluctuations. Indeed, correlated noise carries information that can be quantified by the spectral entropy H s [START_REF] Zaccarelli | Order and disorder in ecological time-series: Introducing normalized spectral entropy[END_REF], the counterpart of Shannon's entropy in the frequency domain, which depends on the noise correlation time. Remarkably, we find that, for a constant noise amplitude quantified by an effective temperature T ef f , the heat ∆Q generated through the transition is proportional to the entropy of the correlated noise: ∆Q = k B T ef f ∆H s , in a fashion that is reminiscent of Landauer's principle. This relation is tested robust over a wide range of heat production and for different noise amplitudes. Effectively therefore, our protocol harvests information from the colored noise and turns it into heat released to the surrounding fluid throughout the NESS-to-NESS transition. Our result makes explicit the deep connection between information and non-equilibrium thermodynamics, which is central to molecular motors and living systems capable of extracting energy from their fluctuating environments to accelerate their average motion, e.g. [START_REF] Astumian | Fluctuation driven transport and models of molecular motors and pumps[END_REF][START_REF] Ezber | Dynein harnesses active fluctuations of microtubules for faster movement[END_REF][START_REF] Linke | Synthetic biology approaches to dissecting linear motor protein function: towards the design and synthesis of artificial autonomous protein walkers[END_REF].

FIG. 1. Schematic view of the experimental realization. The particle is trapped by optical gradient forces using a focused 785 nm laser beam ("optical potential" on the figure). Its stochastic motion is driven by two random baths: a thermal (white noise) bath associated with the surrounding fluid at room temperature T = 296 K (blue) and an active (colored noise) bath generated by an additional 800 nm laser exerting on the bead an actively fluctuating radiation pressure force (red).

Experimental realization -Our experiment consists in optically trapping a single 3 µm dielectric bead with a 785 nm Gaussian laser beam. The optical potential created by the gradient forces at the waist of the beam is harmonic, with a stiffness that is proportional to the intensity of the laser. The bead is immersed in water at ambient temperature and undergoes random motion due to the thermal fluctuations consistent with an Ornstein-Uhlenbeck process. An additional radiation pres-sure force is applied to the bead by a second laser, whose intensity is digitally controlled through time by an acousto-optic modulator. Any waveform can be sent on the bead, including noise with arbitrary spectrum, up to ≈ 10 5 Hz. A schematic view of the experimental realization is shown in Fig. 1 and the experimental setup is described in Appendix A.

Active particles are characterized by a persistence that can be mimicked by an exponentially correlated Gaussian noise [START_REF] Szamel | Self-propelled particle in an external potential: Existence of an effective temperature[END_REF][START_REF] Fodor | How far from equilibrium is active matter?[END_REF][START_REF] Bechinger | Active particles in complex and crowded environments[END_REF]. This noise can be generated by an Ornstein-Uhlenbeck process [START_REF] Jung | Dynamical systems: A unified colorednoise approximation[END_REF][START_REF] Miguel | A colored-noise approach to Brownian motion in position space. corrections to the smolucski equation[END_REF]

dη t = -ω c η t dt + √ 2αω c dW t , (1)
where dW t is a δ-correlated Wiener process. Equation (B1) yields a noise η t with a correlation η t η s = αe -|t-s|ωc , where ω c is the inverse of the correlation time τ c . The variance of the noise η t is given by the amplitude factor α [29].

We emphasize the great flexibility of the above model, which enables us to tune independently the amplitude (variance) and the correlation of the active noise. Thus, we are able to probe a wide range of regimes, from white (small τ c ) to strongly correlated (large τ c ) noises.

The noise η t is generated numerically using Eq. (B1) and sent to the bead via the radiation pressure of the secondary laser beam -see Appendix B. Hence, the overall motion of the bead is subjected to three forces: the deterministic optical trapping force, the stochastic white noise arising from the thermal bath, and the stochastic colored noise applied by the radiation pressure. The position x t of the bead obeys the Langevin equation [START_REF] Maggi | Generalized energy equipartition in harmonic oscillators driven by active baths[END_REF][START_REF] Szamel | Self-propelled particle in an external potential: Existence of an effective temperature[END_REF]

ẋt = -ω 0 x t + √ 2Dξ t + 2D a η t , (2)
where ω 0 = κ/γ is the inverse of the relaxation time in the optical trap, with κ being the stiffness of the trap and γ the Stokes viscous drag. The stochastic variable ξ t is a white noise that models the fluid thermal bath, with thermal diffusion coefficient D = k B T /γ, where k B is the Boltzmann constant and T the temperature of the fluid. D a is the active diffusivity associated with the colored noise η t , which incorporates therefore the optomechanical coupling between the bead and the noisy radiation pressure. Note that the model equation (D1) describes a non-Markovian stochastic process that does not respect the fluctuation-dissipation theorem, as the fluctuating force η t possesses an intrinsic correlation time, whereas the friction term γ ẋt is instantaneous. Hence, the fluctuations of the active bath are not compensated by the dissipation at the same rate [START_REF] Maggi | Memory-less response and violation of the fluctuationdissipation theorem in colloids suspended in an active bath[END_REF], as discussed and probed in detail in Appendix E.

Out-of-equilibrium properties -One remarkable feature of active matter is that, due to the correlation properties of the bath, it diffuses in non-Brownian fashion [START_REF] Sandford | Pressure and flow of exponentially self-correlated active particles[END_REF][START_REF] Maggi | Generalized energy equipartition in harmonic oscillators driven by active baths[END_REF][START_REF] Wu | Particle diffusion in a quasi-twodimensional bacterial bath[END_REF][START_REF] Sandford | Memory effects in active particles with exponentially correlated propulsion[END_REF]. The appropriate observable to estimate the departure from Brownian motion is the mean square displacement MSD(∆) ≡ δx 2 (∆) = (x t+∆ -x t )

2 , where ∆ is a lag time. In general in the short-time limit, the bead will explore the available space inside the trap according to the free diffusive motion MSD ∼ ∆ β , with β = 1 for normal diffusion and β = 1 for anomalous diffusion [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF][START_REF] Mutothya | Characterising stochastic motion in heterogeneous media driven by coloured non-Gaussian noise[END_REF].

In Fig. 2 (a), we present the MSD for two processes, obeying Eq. (D1) with the radiation pressure injecting in the trap either a colored noise, or a white noise of equal amplitude. In the latter case, two white noises (thermal and radiation pressure) drive the bead and the data can be fitted with the standard Ornstein-Uhlenbeck MSD δx 2 (∆) = 2D ef f (1 -exp(-ω 0 ∆))/ω 0 . Using the experimental value for the relaxation time ω -1 0 in the trap, we determine an effective diffusion coefficient D ef f = 0.29 µm 2 /s. The colored noise data is fitted with a modified expression that can be found in [START_REF] Maggi | Generalized energy equipartition in harmonic oscillators driven by active baths[END_REF][START_REF] Chaki | Entropy production and work fluctuation relations for a single particle in active bath[END_REF][START_REF] Chaki | Effects of active fluctuations on energetics of a colloidal particle: Superdiffusion, dissipation and entropy production[END_REF]. This expression takes into account the noise correlation and where the unique fitting parameter is the active diffusion coefficient D a in this way extracted. The superdiffusive nature of the process is clearly visible at short times, with a slope β = 1.5. By tuning the correlation time τ c and the amplitude η 2 t = α of the noise η t injected inside the trap, we can probe different regimes of correlated noise. Of course, all regimes are characterized by a departure from the thermal, white noise equipartition condition x 2 t = D/ω 0 , as a signature of the non-equilibrium nature of the system [START_REF] Maggi | Generalized energy equipartition in harmonic oscillators driven by active baths[END_REF][START_REF] Fodor | How far from equilibrium is active matter?[END_REF].

When the added colored noise is almost white, i.e. when τ c ω 0 1, the bead is driven by two, uncorrelated white noises -see Eq. (D1) -yielding a close-to-equilibrium relation x 2 t ≈ (D + D a )/ω 0 between the motional variance x 2 t and the resulting white noise diffusion coefficients D + D a [START_REF]-time τ γ/κ limit of the MSD is driven by the same temperature as the long-time of the MSD where the motional dynamics is dominated by the trapping process, and by the same temperature fixing[END_REF]. The variance x 2 t is determined experimentally from the recorded trajectories. Importantly, this case leads to a modified equipartition with an effective temperature k B T ef f = κ(D + D a )/ω 0 that directly depends on the choice of the active diffusion coefficient D a [START_REF] Szamel | Self-propelled particle in an external potential: Existence of an effective temperature[END_REF][START_REF] Łuczka | Non-markovian stochastic processes: Colored noise[END_REF]. This weakly correlated regime has been studied both theoretically [START_REF] Sandford | Pressure and flow of exponentially self-correlated active particles[END_REF] and experimentally using colloidal beads confined in a bath of swimming E. coli bacteria [START_REF] Maggi | Generalized energy equipartition in harmonic oscillators driven by active baths[END_REF].

But our ability to manipulate both the color and the amplitude of the noise leads us to investigate strong deviations from equilibrium in the regime of long correlations times that have not been explored experimentally so far. For such long correlation times τ c ω 0 1, we clearly observe in Fig. 2 (b) that the measured variances depart from the white noise D + D a linearity. In this strongly correlated case, the relation between variance and diffusivity can be evaluated from the long-time limit of the MSD with δx 2 (∆

τ c ) = 2 x 2 t ≈ ω -1 0 (D + αD a /(ω 0 + ω c )) (see Appendix D)
. This law fits well the experimental data [red line in Fig. 2 (b)] and the discrepancy between the two blue and red curves quantifies the deviation from equipartition. An important consequence of the explicit dependency of x 2 t on ω c is that one cannot define a unique effective temperature to describe the system both in the short and long time limits [START_REF] Dieterich | Single-molecule measurement of the effective temperature in non-equilibrium steady states[END_REF].

Transition protocols between two NESS -The correlation time of the noise may further be used as a control parameter in a protocol that brings the system from one NESS to another, without changing either the confining potential or the temperature. The simplest possible protocol is a step-like sudden change of the correlation time τ c (referred to as "STEP protocol" hereafter), while keeping the noise amplitude constant. We apply the STEP protocol on the system at a low repetition = γ/κ ≈ 1.2 ms. A naive equipartition result (straight blue line) that would assimilate the correlated noise to a white noise of same amplitude clearly departs from the experimental results, except for very short correlation times. The correct variance estimation for the correlated noise is given by the red curve drawn by taking the Da value extracted from the correlated noise MSD. It fits well the experimental data (see Appendix D for further details). rate, so that it reaches a steady state in between each change of τ c . The ergodic hypothesis, carefully verified (see Appendix C), leads us to build an ensemble of ≈ 1.1 × 10 4 independent trajectories experiencing the same protocol. The main quantity of interest here will be the variance of the response of the bead x 2 t [START_REF] Goerlich | Noise and ergodic properties of brownian motion in an optical tweezer: Looking at regime crossovers in an ornsteinuhlenbeck process[END_REF]. In Fig. 3, we show the results of a STEP protocol where the correlation time is suddenly changed from τ c = 0.8 ms to τ c = 40 ms, while ω -1 0 = 2.1 ms. A number of realizations of the noise variable η t are displayed in Fig. 3 (a): the change in correlation time at t = 0 is clearly visible. In Fig. 3 (b), we represent the variance of the positions of the bead, which undergoes a threefold increase when the correlation time is t of the positions of the bead; the shaded area represents the 99.7% confidence interval. (c) Main plot: Cumulative excess heat in units of kBT . Inset: Cumulative housekeeping heat before the STEP (blue line) and after the STEP (red dotted line). In the inset, the time origin was reset so as to correspond to the beginning of the steady state for each regime.Sign conventions for the heat terms are used in agreement with [START_REF] Sekimoto | Langevin Equation and Thermodynamics[END_REF].

changed. The small dip right after the STEP is due to the fact that the non-Markovian trajectory x t is an integral of the noise η t , hence needs a finite time to probe the total amplitude of η t .

The above STEP protocol has the same effect (increase in the variance x 2 t ) as a protocol where the noise remains white, but its amplitude (temperature) increases [START_REF] Martínez | Adiabatic processes realized with a trapped Brownian particle[END_REF][START_REF] Martinez | Colloidal heat engines: a review[END_REF][START_REF] Albay | Shift a laser beam back and forth to exchange heat and work in thermodynamics[END_REF]. The important difference is that, in our case, we do not change the amplitude of the noise, but only act on its spectrum by modifying the correlation time. In this sense, the protocol can seem costless from the experimentalist's point of view, as no additional power has to be provided to the laser source at the transition time. As a comparison, we estimated the equivalent power needed to induce the same increase in variance as in Fig. 3 (b) through a change in the noise amplitude, i.e., by changing the diffusivity D a . The result is that one would need a laser intensity of 70 mW, whereas we used only 36 mW in our color-based protocol.

From a thermodynamic point of view, our color-based protocol necessarily produces heat, which is then released in the thermal bath. Following Sekimoto's treatment [START_REF] Sekimoto | Langevin Equation and Thermodynamics[END_REF][START_REF] Sekimoto | Stochastic energetics[END_REF], which was recently applied to active matter [START_REF] Dabelow | Irreversibility in active matter systems: Fluctuation theorem and mutual information[END_REF], the cumulative stochastic heat can be written as (see Appendix F):

q(t) = 1 2 t 0 κ dx 2 dt dt -γ t 0 2D a η t ẋt dt . (3)
The first term accounts for the excess heat released during a transient evolution of the distribution, and vanishes for steady states [START_REF] Seifert | Stochastic thermodynamics, fluctuation theorems and molecular machines[END_REF]. The second term, expressed in terms of the crosscorrelation η ẋ , can be evaluated analytically by injecting ẋ as from Eq. (D1) and is shown to grow linearly in time at steady state (see Appendix F). This is the housekeeping heat, representing the constant expense needed to maintain the system in its NESS.

Both heat terms are shown in Fig. 15(c). The main graph represents the excess heat dissipated during the transient, with an energy release of ≈ 10k B T . In the inset, we display the housekeeping heat during the steady states, both before and after the protocol, showing that changing the correlation time affects the heat dissipation rate. Interestingly, the powers at play dQ HK /dt ∼ 2 k B T /ms in our optical trapping experiments are close to those involved in biological processes such as axonal transport, where kinesin consumes around [0.4 -0.8] k B T /ms [START_REF] Woehlke | Walking on two heads: the many talents of kinesin[END_REF][START_REF] Ariga | Nonequilibrium energetics of molecular motor kinesin[END_REF][START_REF] Ariga | Noiseinduced acceleration of single molecule kinesin-1[END_REF].

Harvesting information from the noise -The protocol described in the above paragraphs seems to raise a paradox: after the transition, the heat released during the process appears to increase (see Fig. 15), while no further energy was injected in the system since only the spectrum of the noise was changed and not its amplitude. As the first principle of thermodynamics is of course not violated, this means that the coupling between the correlated noise bath and the bead has increased, so that energy can be transferred more efficiently from the former to the latter. Such a modulation of the coupling is exactly what happens when the correlation time of the active fluctuations is modified.

Nevertheless, it is illuminating to analyze this situation from an informational point of view. Indeed, a correlated noise carries more information and has lower entropy than a white, or a less correlated, noise. This information content can be measured by the spectral entropy H s , which is just the Shannon entropy in the frequency domain [START_REF] Zaccarelli | Order and disorder in ecological time-series: Introducing normalized spectral entropy[END_REF]:

H s = -k B N i=1 P (ω i ) ln P (ω i ), (4)
where

P (ω i ) = S η (ω i )/ i S η (ω i )
, and S η (ω i ) denotes the power spectral density of the signal η at frequency ω i . The spectral entropy vanishes for a monochromatic signal and reaches its maximum k B ln N for white noise. Any correlated noise has an intermediate value of H s , which can be used to measure its information content. We measured H s for various correlation times, but also several noise amplitudes. Each amplitude is labeled by an effective temperatures T ef f , determined by sending a white noise of same amplitude as each colored noise cases. The spectral entropy H s is then compared to the generated excess heat. More precisely, for each effective temperature, we define ∆H s = H s (τ c ) -H s (τ ref ), and similarly for ∆Q ex , where τ ref = 0.5 ms is taken as a reference correlation time. Hence, ∆H s and ∆Q ex represent respectively the informational expenditure and the corresponding energetic cost to go from the reference case to the colored case with correlation time τ c through the STEP protocol described above.

The data from several measurements are plotted in Fig. 4 and obey very neatly the relation ∆Q ex /k B T ef f = ∆H s . This is a striking proof that the excess heat produced in the process corresponds exactly, in units of k B T ef f , to the injected information. Note that this expression is highly nontrivial: the left-hand side is a thermodynamic quantity related to the diffusive motion of the trapped bead, while the righthand side captures the informational content of the colored bath generated by the laser beam. This expression is fully in line with Landauer's principle [START_REF] Landauer | Irreversibility and heat generation in the computing process[END_REF][START_REF] Frank | Physical foundations of Landauer's principle[END_REF] where the physical nature of information is seen in the context of active matter. This remarkable result highlights the informational nature of our process, resolving what could appear as a paradoxically costless protocol.

Conclusion -In this work, we studied experimentally the diffusive motion of an optically trapped particle subjected to both a white thermal noise due to the surrounding fluid and a correlated noise generated by a digitally controlled fluctuating radiation pressure. This configuration constitutes a very accurate, controllable model of active biological matter.

Three major results were obtained: (i) thanks to the flexibility of our setup, we could explore an unprecedented range of regimes, most notably those characterized by long correlation times and strong amplitudes, which are unattainable in experiments with real active matter; (ii) by using the correlation time as a control parameter, we devised a protocol that drives the system from one non-equilibrium steady state to another, at zero nominal energetic cost; (iii) finally, we showed that the excess heat released during such a protocol is proportional to the spectral entropy of the colored noise, a relationship that is akin to Landauer's principle of equivalence between information and energy cost [START_REF] Plenio | The physics of forgetting: Landauer's erasure principle and information theory[END_REF][START_REF] Lutz | Information: From maxwell's demon to landauer's eraser[END_REF]. Effectively, the protocol harvests information from the colored noise, turns it into heat necessary for the transition between the two non-equilibrium states, and finally releases it to the surrounding environment. The ubiquity of non-equilibrium steady states in biological systems, including changes in the spectrum of the bath through time (for example during mitosis [START_REF] Hurst | Intracellular softening and increased viscoelastic fluidity during division[END_REF][START_REF] Taubenberger | The mechanics of mitotic cell rounding[END_REF]) suggests exciting applications for the present findings.

Appendix A: Experimental setup and calibration Our experimental setup consists in optically trapping, in a harmonic potential, a single dielectric bead (3 µm polystyrene sphere) in a fluidic cell filled with dionized water at room temperature. The harmonic potential is induced by focusing inside the cell a linearly polarized Gaussian beam (785 nm, CW 110 mW laser diode, Coherent OBIS) through a high numerical aperture objective (Nikon Plan Apo VC, 60×, NA= 1.20 water immersion, Obj1 on Fig. 5). An additional force in the form of radiation pressure is applied to the sphere using an additional high-power laser (800 nm, CW 5 W Ti:Sa laser, Spectra Physics 3900S). The intensity of this radiation pressure beam is controlled by an acousto-optic modulator (Gooch and Housego 3200s, AOM on Fig. 5) using a digital-to-analogue card (NI PXIe 6361) and a PYTHON code. FIG. 5. Simplified view of the optical trapping setup. The sphere is suspended in water inside the Sample cell inserted between the two objectives Obj1 and Obj2. The 785 nm trapping beam is drawn in red, the 800 nm radiation pressure beam in purple. The intensity of this beam controlled by the acousto-optic modulator (AOM). The instantaneous position of the trapped bead is probed using the auxiliary 639 nm laser beam, drawn in orange, whose scattered signal is sent to a high-frequency photodiode.

The instantaneous position x t of the sphere along the optical axis is measured by recording the light scattered off the sphere of a low-power 639 nm laser (CW 30 mW laser diode, Thorlabs HL6323MG), sent on the bead via a second objective (Nikon Plan Fluor Extra Large Working Distance, 60×, NA= 0.7, Obj2 on the figure). The scattered light is collected by Obj1 and recorded by a photodiode (100 MHz, Thorlabs Det10A). The recorded signal (in V/s) is amplified using a low noise amplifier (SR560, Stanford Research) and then acquired by an analog-to-digital card (NI PCI-6251). The signal is filtered through a 0.3 Hz high-pass filter at 6 dB/oct to remove the DC component and through a 100 kHz low-pas filter at 6 dB/oct to prevent from aliasing. The scattered intensity varies linearly with the position of the trapped bead x t for small enough displacements and we make sure to work in the linear response regime of the photodiode so that the recorded signal is linear with the intensity, resulting in a voltage trace well linear with x(t). The calibration of the recorded voltage is done by fitting the motional Lorentzian spectrum of the sphere -see Fig. 6from which is extracted a calibration coefficient expressed in m/V, generally ∼ 10 -7 m/V. The noise added by the radiation pressure laser modifies the dynamics of the bead, but without changing the properties of the trapping potential. As presented in Fig. 6, we carefully verify that an added white noise that mimics a higher temperature, only leads to an increase in the power spectral density amplitude (as expected when increasing the kinetic temperature of the bead) without modifying neither its Lorentzian profile nor the roll-off frequency of the trap, left unchanged at ≈ 150Hz.

Appendix B: External radiation pressure force acting as a bath: from noise generation to active protocols

We generate the external noise following the sequence described in Fig. 7. Using a PYTHON code with a build-in random noise generator, we can easily generate a white noise for which the choice of the distribution function of the noise has no influence on x(t) (by the virtue of the central limit theorem with all noise events being independent). To maximize the usable dynamical range, we thus simply use a uniformly distributed noise.

In contrast, for a colored noise, both correlation and distribution matter. In this case, we use a Gaussian exponentially correlated noise, given as the solution of the Ornstein-Uhlenbeck process

dη t = -ω c η t dt + √ 2αω c dW t , (B1)
where ω c is the inverse characteristic time of the noise and √ α its amplitude. The variance of such a process is η 2 t = α. This correlated noise will then enter into the Langevin equation as an external random force field

γ ẋ = -κx t + γ √ 2Dξ t + F ext (t) (B2)
with D = k B T /γ the diffusion coefficient in the thermal bath. The external radiation pressure force F ext = F 0 + δF (t) is centered around a mean value F ext = F 0 and with a zero mean noise part δF (t) = 0. The average term F 0 vanishes trivially when looking at the centered process x t -x t = x t -F 0 /κ, which is always the case in our experiments. Random, the external force acts as a secondary bath. It can thus be recast as F ext (t) = γ √ 2D a η t , i.e. on the same footing as the thermal force F th (t) = γ √ 2Dξ t where D is in m 2 /s and ξ (the time derivative of a Wiener process) is in

√

Hz. This leads us to introduce an active diffusion coefficient D a having the same dimension as D, associated with the noise η t solution of Eq. (B1). This gives the noise of dimension

[η t ] ≡ [ √ α] ≡ √
Hz, just like the thermal noise term ξ t . An important asset of our work is the flexibility of our scheme for controlling the color and the amplitude of the noise, which demands to keep ω c and α independent. To do so, we first generate numerically with the PYTHON code, under a sampling frequency of 20kHz, the noise η t of variance α, scaled to the desired amplitude ∈ [0, 1] which corresponds, in Volts, to the range fixed by the radio-frequency generator driving the acousto-optic modulator (AOM). The AOM response is calibrated to yield a linear relation between the input voltage and output laser intensity in the first order diffracted beam. The numerically generated noise is thereby encoded into a radiation-pressure laser intensity noise sent to the bead. This noise intensity acting on the bead depends both on the gain of the AOM-diffracted beam and on the choice of the radiation-pressure laser intensity.

In our experiments, the amplitude of the noise is independent of its color, which is different from the choice made in [START_REF] Maggi | Generalized energy equipartition in harmonic oscillators driven by active baths[END_REF], where α scales as the square root of the inverse correlation time of the noise. This choice necessarily induces the interplay between both correlation times and amplitudes that we want to avoid. The actual amplitude of the noise experienced by the bead, which depends not only on the radiationpressure laser intensity but also on the optomechanical coupling between this laser beam and the trapped sphere, will be taken into account in F ext via the active diffusion coefficient D a . This implies that choosing α = 1 is the simplest option. However, we keep the α term for clarity, as a purely dimensional constant.

FIG. 8. (a)

Power spectral density associated with different noises ηt (for each colored curves, the corresponding correlation time is indicated within brackets in the legend). These spectra are measured directly from the laser intensity signal that is sent on the trapped bead as a radiation pressure. (b) Mean squared displacement of the sphere for each of the noises presented in panel (a). The DC case corresponds to the absence of additional noise, with Fext = F0. We observe that the white noise case and the first colored noise case (correlation time 0.1ms) are almost identical. Superimposing the fit performed with the analytical expression for the mean squared displacement (see below, Sec. D) enables one to extract the active diffusion coefficients Da for each case.

On Fig. 8 (a), we show the power spectral densities of different noises, from white to colored. At high frequencies (> 3000 Hz, see Fig. 6), the signal is dominated by the electronic noise of the experiment, limiting the spectral bandwidth of interest from a fraction of Hz to a few kHz. The blue curve in Fig. 8 corresponds to a white noise generated over the desired bandwidth where we see its flat spectrum covering all the response region of the bead, up to the 20 kHz limit of the generation sampling frequency. The other curves are the dif-ferent colored noises, with correlation times spanning from 0.1 to 50 ms. On Fig. 8 (b), we show the mean squared displacement (MSD) associated with each noise. The black curve shows the DC case with F ext = F 0 where no noise is added. The blue curve corresponds to the white noise drive (blue spectrum of Fig. 8 (a)), slightly above the thermal MSD as a consequence of the increase in effective temperature. The orange curve gives the first colored noise case (orange spectrum of Fig. 8 (a)). We can note that the responses to a white noise and to a colored noise of correlation time 0.1 ms are similar, this colored noise being "almost white". This implies that longer (> 0.1 ms) correlation times are needed to make a clear difference between white and colored cases (as seen for the next colored noise with correlation time 0.5 ms). The other curves are the MSD corresponding to the different noises of Fig. 8 (a). To study protocols, we need an ensemble of independent trajectories all experiencing the same parameter changes. In our experiments, this is a change in correlation time ω c (t) that we modulate in a step-like way from an initial ω i c to a final ω f c values. With one single bead in the optical trap, the ensemble is drawn out of a long time series, for which the ergodic hypothesis is crucial and was carefully checked as discussed in Sec. C. We produce one long noise sequence η t where a large number of correlation time changes are produced following a ω i c /ω f c square modulation at a low enough repetition rate (a few tens of Hertz). This modulation sequence is sent to the bead via the radiation pressure laser. The corresponding trajectory x t of the bead relaxes to one steady-state between each change on ω c . This long trajectory is cut and reshaped into an ensemble of trajectories {x i t } that, each, experience a steplike change in correlation time. In order to build the actual noise protocol driving these trajectories, we generate in parallel two independent sequences of η t time-series with different (but constant) correlation times that are then interspersed synchronously with the ω i c /ω f c square modulation impacting the motional trajectory x t . The detailed procedure is described in Fig. 9.

Our system, consisting of an optically trapped bead thermally diffusing within active fluctuations, is described by a couple of stochastic differential equations that determine the evolution of the position of the bead within the trap according to:

ẋt = -ω 0 x t + √ 2Dξ t + 2D a η t (D1)
where the active noise η t , solution of the Ornstein-Uhlenbeck process

dη t = -ω c η t dt + √ 2αω c dW t , (D2) FIG. 10. (a)
The ergodic parameter (i.e. the normalized variance (∆) presented in our previous work [START_REF] Goerlich | Noise and ergodic properties of brownian motion in an optical tweezer: Looking at regime crossovers in an ornsteinuhlenbeck process[END_REF]) is shown as a function of the time lag ∆ for both the white noise driven process (blue circles) and the colored noise driven process (red triangles) along with the analytical prediction for the white noise case (black dashed line). The red and blue hazes measure the 95% confidence interval. (b) The same ergodic parameter (∆) plotted as a function of the total time τ for both the white-noise driven process (blue circles) and the colored-noise driven process (red triangles) along with the analytical prediction for the white noise case (black dashed line).

is an exponentially correlated Gaussian variable.

We can derive the noise power spectrum density by Fourier transforming Eq. (D2)

-iωη[ω] = -ω c η[ω] + √ αω c ξ[ω] (D3)
where ω c is the correlation pulsation. Taking the squared norm leads to the active noise power spectral density (PSD)

η[ω]η * [ω] = |η[ω]| 2 = αω c ω 2 c + ω 2 . ( D4
)
On Fig. 11 (a), we plot the PSD directly measured from the laser output signal used to induce the noisy radiation pressure, both in the case of a white noise and colored noise. As expected, the spectrum of the white noise is flat on all the studied bandwidth, whereas the spectrum of the colored noise is following a Lorentzian profile, well captured by a fit following Eq. (D4).

The PSD of the motion x t is evaluated from Eq. (D1) as: noting that the implicit averaging performed in this square cancels the two cross-product of the uncorrelated noises η[ω]ξ * [ω] and the complex conjugate. In Eq. (D5), ω 0 = κ/γ is the inverse of the characteristic relaxation time of the system, and η[ω]η * [ω] is given by Eq. (D4) and ξ

x[ω]x * [ω] = 1 ω 2 0 + ω 2 (2Dξ[ω]ξ * [ω] + 2D a η[ω]η * [ω]) , (D5)
[ω]ξ * [ω] = 1. Hence x[ω]x * [ω] = |x[ω]| 2 = 1 ω 2 0 + ω 2 2D + 2D a αω c ω 2 c + ω 2 (D6)
On Fig. 11 (b), we plot the measured spectra of x t both for a white and colored external drive, with the analytical result of Eq. (D6) using the value of D a obtained from the fit of the MSD for the colored noise (see Fig. 8 above) and using the D a → 0 limit for the white noise case. A very good agreement between the theory and the experimental data is clearly seen, confirming that our model captures well the real diffusive dynamics of the trapped bead.

We can also compute the correlation function of the colored noise driven process from the Wiener-Khintchine theorem as:

C xx (∆) = 1 2π +∞ -∞ |x[ω]| 2 e -iω∆ dω = 1 2π +∞ -∞ 2De -iω∆ dω ω 2 0 + ω 2 + 1 2π +∞ -∞ 2D a αω c e -iω∆ dω (ω 2 0 + ω 2 ) (ω 2 c + ω 2 ) , (D7)
where both integrals can be computed via contour integration. For the first one,

f [ω] = D π e -iω∆ ω 2
0 +ω 2 has one simple pole in the upper-half complex plane in iω 0 , leading to compute one residue

+∞ -∞ f [ω]dω = 2iπRes{f [ω], iω 0 } = lim ω→ω0 2De -iω∆ ω + iω 0 = D ω 0 e -ω0∆ .
(D8)

Similarly, the second integral with g

[ω] = Daαωce -iω∆ π(ω 2 0 +ω 2 )(ω 2 c +ω 2 )
is evaluated by separating it through partial fraction decomposition in g

[ω] = Daαωce -iω∆ π(ω 2 c -ω 2 0 ) 1 ω 2 0 +ω 2 -1 ω 2 c +ω 2 ≡ g 1 [ω] + g 2 [ω] leading to two integrals with simple poles in iω 0 and iω c +∞ -∞ g[ω]dω = 2iπRes{g 1 [ω], iω 0 } + 2iπRes{g 1 [ω], iω c } = D a αω 2 c e -ω0∆ ω 0 (ω 2 c -ω 2 0 ) - e -ωc∆ ω c (ω 2 c -ω 2 0 ) = D a αω c ω 0 (ω 2 c -ω 2 0 ) e -ω0∆ - ω 0 ω c e -ωc∆ .
(D9) These evaluations are combined to provide the expression for the correlation function of the diffusion process:

C xx (∆) = D ω 0 e -ω0∆ + D a αω c ω 0 (ω 2 c -ω 2 0 ) e -ω0∆ - ω 0 ω c e -ωc∆ (D10)
On Fig. 12 we represent the normalized correlation function C xx for both white and colored noise drives where we superimpose the analytical result, using the value of D a obtained, as indicated above, from the fit of the MSD for the colored noise. Here too, we use the D a → 0 limit for the white noise. Again, the good agreement between the exponential decays and the analytical models is observed.

The MSD of a colloid diffusing in a thermal environment obeys an Ornstein-Uhlenbeck process and is thus characterized by the white noise MSD: where D is the diffusion coefficient in the thermal bath, expressed in m 2 /s, and ω 0 = κ/γ is the inverse of the characteristic relaxation time of the bead in the trap.

δx 2 (∆) ≡ (x(t + ∆) -x(t)) 2 = 2D ω 0 1 -e -ω0∆ (
In contrast, the MSD of an active particle obeys Eq. (D1) and can be computed as δx 2 (∆) = 2 x 2 t -2C xx (∆) where the variance is the stationary variance of the process and x 2 t is taken as the limit lim t→∞

[ x 2 t ] = D ω0 + Daαωc ω0(ωc+ω0)
). This leads us to calculate directly:

δx 2 (∆) = 2D ω 0 + 2D a α ω 0 (ω c + ω 0 ) + 2D ω 0 e -ω0∆ + 2D a αω c ω 0 (ω 2 c -ω 2 0 ) e -ω0∆ - ω 0 ω c e -ωc∆ = 2D ω 0 1 -e -ω0∆ + 2D a αω c ω 0 (ω 2 c -ω 2 0 ) 1 -e -ω0∆ - ω 0 ω c (1 -e -ωc∆ )
(D12) that constitutes the result used in the main text.

The long-time limit can be easily derived as:

lim ∆→∞ [ δx 2 (∆) ] = 2D ω 0 + 2D a α ω 0 (ω c + ω 0 ) . ( D13
)
To asses the break of the equipartition relation, it is enough to show the absence of linearity between the variance and the effective diffusion coefficient associated with the active process. For different noise colors, we thus measure and fit the MSD to extract each corresponding D a . We then plot in Fig. 2, panel (b) of the main tex the variance against the total diffusion coefficient D + D a . As seen in the figure, we clearly observe that the variance does not follow the intuitive ω -1 0 linearity. In striking contrast, it rather follows one half of Eq. (D13) -i.e., D ω0 + Daα ω0(ωc+ω0) -where the explicit ω c term prevents us from defining a unique effective diffusion coefficient (or effective temperature) and where D a comes from the fit of the MSD displayed in Fig. 8.

Appendix E: Micro Rheology and Fluctuation Dissipation Theorem

We probe the non-equilibrium nature of the active fluctuations and the validity of the Fluctuation Dissipation Theorem (FDT) by comparing the dynamical responses of our system to Active MicroRheological (AMR) and Passive MicroRheological (PMR) excitations, respectively [START_REF] Chaki | Effects of active fluctuations on energetics of a colloidal particle: Superdiffusion, dissipation and entropy production[END_REF][START_REF] Turlier | Equilibrium physics breakdown reveals the active nature of red blood cell flickering[END_REF].

At thermal equilibrium, under detailed balance conditions, the linear response of the system to a small perturbation is connected to equilibrium correlations of fluctuations through the FDT according to which:

∂C xx (t) ∂t = 2k B T R(t), (E1)
where C xx (t) = x(t)x(0) is the motional autocorrelation function and R(t) is the response function of the system. This equation can be more conveniently derived in the frequency domain. If we consider the motion of the bead driven by a noise of unit variance φ t (that takes in our case the form ξ t + D a /Dη t ), the Fourier transform of the corresponding Langevin equation writes as:

-iωγx[ω] = -κx[ω] + 2k B T γφ[ω], (E2)
where κ is the stiffness of the potential, γ the Stokes friction drag, and √ 2k B T γφ[ω] is a generic random force. The equation can be written in terms of a mechanical susceptibility χ[ω] as

x[ω] = χ[ω] 2k B T γφ[ω] (E3)
where χ[ω] can be decomposed into real and imaginary parts as:

χ[ω] = ω 0 γ(ω 2 0 + ω 2 ) +i ω γ(ω 2 0 + ω 2 ) ≡ χ [ω]+iχ [ω]. (E4)
If we compare the imaginary part χ [ω] with the power spectral density obtained, in the case of a thermal, white noise drive, by the square modulus of position Fourier transform |x[ω]| 2 = 2D/(ω 2 0 + ω 2 ), we obtain the expression of the FDT in the Fourier space: If now one adds a small sinusoidal perturbation on the bead by means of an external force (which corresponds to radiation pressure in our experiments), the FDT can be tested experimentally by measuring the response function. Under the sinusoidal ac drive of the AMR mode at pulsation ω ac , the PSD takes the following form [START_REF] Schnoering | Thermally limited force microscopy on optically trapped single metallic nanoparticles[END_REF] |x

χ [ω] = ω|x[ω]| 2 2k B T (
[ω]| 2 ac = 1 ω 2 0 + ω 2 2D + F 2 ac 2γ 2 δ(ω -ω ac ) (E6)
where F ac is the Fourier force component of the drive, while the unperturbed PSD of the PMR mode writes

|x[ω]| 2 = 2D ω 2 0 + ω 2 . (E7)
By computing the ratio |x[ω ac ]| 2 /|x[ω ac ]| 2 ac at the pulsation ω ac for a bead on which a "white noise" radiation pressure is exerted (hence maintaining it close to thermal equilibrium but at an effective temperature T ef f higher than room tempertaure), we can extract the value of F ac by taking the mean value of all realizations. This value can then be used to calibrate the response function χ On Fig. 13 (a), we show the external drive in the timedomain and the motional response of the bead inside the trap for two different modulation frequencies. By repeating the procedure for frequencies ranging from 10 Hz to 6 kHz, the response of the bead is characterized over all the useful bandwidth (see Fig. 13 (b)). On Fig. 14, the values of ω|x[ω]| 2 /2k B T and χ [ω] are plotted together for the probed frequencies, for both the white noise and colored noise driven processes. We clearly observe that in both cases, the response functions associated with the mechanical susceptibilities fall back on the same trend. This trend is exaclty the one associated with the white noise driven PSD as expected from Eq. (E5). In contrast, the spectral density of the colored noise driven process significantly departs from the FDT in Eq. (E5), and more particularly for the low frequencies of the active fluctuation spectrum. This is in agreement with other observations made recently in active systems [START_REF] Turlier | Equilibrium physics breakdown reveals the active nature of red blood cell flickering[END_REF], where the active mechanical processes mostly appear at low frequency, while the FDT is recovered for the thermally dominated highfrequency part. We note here that a simple observation of the breaking of the FDT can already be seen in our model described by Eq. (D1), where the fluctuating forces associated with the added noises ξ t and η t now possess an intrinsic correlation time due to the correlated nature of η t , while the friction kernel γ is taken as instantaneous Γ(t, t ) = γδ(t, t ). This choice has been shown to be valid in the experimental case [START_REF] Maggi | Memory-less response and violation of the fluctuationdissipation theorem in colloids suspended in an active bath[END_REF], where the fluctuations of the active bath are not compensated by a dissipation with the same rate. In the limit of vanishing correlation times, the FDT is recovered as the noise is white (δcorrelated) and its only effect is an effective change in temperature, as was already observed using a different experimental technique [START_REF] Martínez | Effective heating to several thousand kelvins of an optically trapped sphere in a liquid[END_REF][START_REF] Mestres | Realization of nonequilibrium thermodynamic processes using external colored noise[END_REF].

tem is brought to a Non-Equilibrium Steady-State (NESS) where the stationary stochastic laser drive maintains -through the action of radiation pressure -the system out of its equilibrium state at a given temperature and stiffness κ. Following the standard methods of stochastic energetics [START_REF] Sekimoto | Langevin Equation and Thermodynamics[END_REF][START_REF] Sekimoto | Stochastic energetics[END_REF], we write, from Eq. (D1), for our process γ ẋt -γ √ 2Dξ t dt dx = -κx t + γ 2D a η t dx. (F1)

The left-hand side is interpreted as the heat exchanged with the thermal bath δq = -γ ẋ -√ 2Dξ t dt dx. Since the active force is stochastic, it produces no work. In fact, including a random force in the expression of work induces a violation of the Crooks relation [START_REF] Gomez-Solano | Steady-state fluctuation relations for systems driven by an external random force[END_REF]. The internal energy stays related to the potential energy dU = -κx 2 t dx and the remaining term γ √ 2D a η t dx is the energy exchanged with the active bath. Interpreted as a heat term [START_REF] Dabelow | Irreversibility in active matter systems: Fluctuation theorem and mutual information[END_REF], it can be evaluated from the right-hand side of Eq. (F1) as

δq(t) = κ dx 2 dt dt -γ 2D a η t ẋt dt, ( F2
)
which can be integrated to give the stochastic heat evaluation q(t) = Finally, we compute the ensemble average heat, which will be expressed in terms of variance and cross-correlations Q(t) ≡ q(t) = 1 -e -s(ω0+ωc) ds -t = 2γD a α ω 0 ω 0 + ω c -1 t + 2γD a ω 0 (ω 0 + ω c ) 2 (1 -e -t(ω0+ωc) ), (F6)

to give, after an exponential decorrelation at short times (just after the noise is turned on, a short-time regime that is never probed in our experiments), a linear heat expenditure with negative (since ω 0 > 0) slope 2γD a α ω0 ω0+ωc -1 that account for the heat needed to maintain the system in its NESS. We note again that, in the white noise limit on an infinite bandwidth, ω c → ∞ makes this quantity vanish.

Therefore, if we discard the decorrelation after the noise is turned on, we obtain the following expression for the cumulative heat: where the two quantities are reminiscent of the excess (EX) and housekeeping (HK) heat terms [START_REF] Seifert | Stochastic thermodynamics, fluctuation theorems and molecular machines[END_REF]. In Fig. 15, we display the time evolution of these two quantities for the τ c (t) STEP protocol described above. Figure 15 (a) shows the heat necessary to maintain the NESS both before and after the change of τ c . As we see, changing the correlation time changes the rate of heat dissipation. In Fig. 15 (b), the time-evolution of the excess heat discarded through the transient is plotted for both increasing or decreasing STEP of τ c . It is remarkable to stress that the quantity of heat ≈ 10k B T exchanged is the same for both cases.

Q(t) =
Note that an alternative expression for the heat can be found in the context of active matter and active Ornstein-Uhlenbeck processes. These different approaches lead all to similar results, since Sekimoto's definition of heat as δq = -γ ẋ -γ

√ 2Dξ t dt dx (F8)
is uniquely defined [START_REF] Park | Rapid-prototyping a Brownian particle in an active bath[END_REF][START_REF] Sekimoto | Langevin Equation and Thermodynamics[END_REF]. The differences stem from the way to evaluate this quantity. In the description of non-reciprocal systems [START_REF] Loos | Irreversibility, heat and information flows induced by non-reciprocal interactions[END_REF], the steady-state heat is computed as a sum of correlation between variables and velocities. In our case of unidirectional coupling, this simplifies to a term ∼ η t ẋt which is the term we also obtain. Another definition is based on the deviation from the fluctuation dissipation relation, the Harada-Sasa relation [START_REF] Harada | Equality connecting energy dissipation with a violation of the fluctuation-response relation[END_REF] used in [START_REF] Park | Rapid-prototyping a Brownian particle in an active bath[END_REF][START_REF] Chaki | Effects of active fluctuations on energetics of a colloidal particle: Superdiffusion, dissipation and entropy production[END_REF], which, similarly to our calculations, gives a linear heat production in the steady state.

FIG. 2 . 1 0= 1 . 2

 2112 FIG. 2. (a) Mean square displacement measured experimentally for a white noise added to the existing thermal fluctuations inside the fluid (blue line), showing a short-time diffusive limit (∼ ∆ 0.94 ), and a correlated noise with τc = 1 ms (red line), yielding a short-time superdiffusive limit (∼ ∆ 1.5 ); the superimposed dashed lines are the MSD expected from theory, with a Ornstein-Uhlenbeck MSD evaluated for the white noise case (taking T = 296 K, ω -1 0 = 1.2 ms and extracting a thermal diffusion coefficient D ef f = 0.29 µm 2 /s) and a modified MSD in the correlated-noise case detailed in Appendix D. The best fit yields an active diffusion coefficient Da = 1.44 µm 2 /s, to be compared with the thermal diffusion coefficient D. The shaded regions accounts for the uncertainties associated with the fitting error in determining ω0 and the systematic error in the sphere radius determination. (b) The relation between motional variance x 2 t and the D + Da diffusivity is measured on a 99.7% confidence level (blue dots and 3-σ bars) for different correlation times ranging from τc = 0.1 ms to τc = 100 ms, with relaxation time inside the optical trap ω -1 0

FIG. 3 .

 3 FIG. 3. STEP protocol using the noise correlation time as control parameter. (a) Several digital realizations of the noise variable ηt, undergoing a change in the correlation time at t = 0, from τc = 0.8 ms before the STEP to τc = 40 ms after. (b) Corresponding experimental ensemble variance x 2t of the positions of the bead; the shaded area represents the 99.7% confidence interval. (c) Main plot: Cumulative excess heat in units of kBT . Inset: Cumulative housekeeping heat before the STEP (blue line) and after the STEP (red dotted line). In the inset, the time origin was reset so as to correspond to the beginning of the steady state for each regime.Sign conventions for the heat terms are used in agreement with[START_REF] Sekimoto | Langevin Equation and Thermodynamics[END_REF].

FIG. 4 .

 4 FIG. 4. Measured excess heat between two NESS (in units of kBT ef f ) plotted as a function of the calculated spectral entropy, for three values of the noise amplitude, each characterized by an effective temperature T ef f -experimental details given in Appendix G. The dashed line has a slope equal to unity.

FIG. 6 .

 6 FIG. 6. Power spectral density Sx[ω] of the bead position, recorded with no additional noise, i.e. solely driven by thermal fluctuations (blue circles) and with additional white noise injected inside the trap (red circles) using the auxiliary radiation pressure laser made noisy via the AOM. In both cases, the dashed lines correspond to Lorentzian fits, with shaded regions indicate the limits of the fits. The vertical line mark the position of the trap roll-off frequency at ∼ 10 2 Hz. Note the onset of the electronic noise floor at high frequencies.

FIG. 7 .

 7 FIG.7. Schematic representation of the signal and data acquisition processing implemented in our experiments. The noise ηt is digitally generated, scaled to a voltage V (t) that can be sent to the acoutooptic modulator (AOM), producing a diffracted beam whose intensity varies linearly with this input voltage. The laser beam diffracted through the AOM exerts a radiation pressure on the optically trapped sphere, whose position is recorded as detailed in Sec. A. A small part of the laser beam is also measured and used to monitor and evaluate the noise ηt as it enters the trap.

FIG. 9 .

 9 FIG. 9. (a) Temporal noise series ηt modulated between two correlation times (two colors) following a 20 Hz square modulation function (top) and the resulting trajectory xt (bottom). (b) Each long trajectory is reshaped as an ensemble. In the upper panel, the {η i t } ensemble clearly displays at t = 0 the instantaneous change in correlation with constant amplitude. In the lower panel, the ensemble of trajectories {x i t } of the bead show the progressive change in the motional variance that results from the step-like change of correlation times of the bath.

FIG. 11 .

 11 FIG. 11. (a) Power spectral densities (PSD) of the white noise (blue curve) and colored noise (red curve) measured on a 10% fraction of the laser beam signal that is sent inside the trap to act on the bead as a noisy radiation pressure. A Lorentzian fit (black dashed curve) is superimposed on the colored noise spectrum. (b) Motional PSD Sx[ω] = 2|x[ω]| 2 plotted as a function of the frequency for the white noise driven process (blue) and the colored noise driven process (red) along with the associated theoretical PSD (black dashed curves).

FIG. 12 .

 12 FIG. 12. Correlation function Cxx(∆) = x(t + ∆)x(t) plotted as a function of the lag ∆ for the white (blue curve) and colored (red curve) noise-driven processes, both normalized to the zero-delay ∆ = 0 correlation function Cxx(0). The theoretical expressions derived in Sec. D are displayed as dashed black curves for both cases.

  E5) where the spectrum |x[ω]| 2 is the Fourier transform of autororrelation function C xx (t) (Wiener-Khinchine theorem).

  FIG. 13. (a) Active micro-rheological (AMR) experiment where the sinusoidal forcing of the system is monitored in the time domain. The recorded trajectories of the bead inside the trap are superimposed to the sinusoidal traces of the force for two different modulation frequencies. (b) Power spectral densities displayed together for different modulation frequencies of the external force drive. The Fourier components of each harmonic forcing are clearly seen as peaks in the PSD.

FIG. 14 .

 14 FIG. 14. We compare the measured values of χ [ω] for white (open circles) and correlated (open squares) noise for different modulation frequencies ωac and small sinusoidal perturbations with the stationary correlation spectra plotted as ω|x[ω]| 2 /2kBT for white (blue curve) and correlated (orange curve) noises. One immediately remarks the breaking of the FDT for the colored-noise driven process.

2D a x s ηs ds = -γ t 0

 0 s ẋs ds.(F4)The first term is connected to the evolution of the variance. It vanishes in the steady-state and only accounts for the heat released during a transient evolution of the distribution. The second term can be computed analytically by injecting the differential equation Eq. (D1) for ẋt : x t ηt = -ω 0 x t η t + √ 2D a α. The first term can be computedx t η t = t 0 2D a η t η s e -ω0(t-s) ds = α t 0 2D a e -ωc|t-s|-ω0(t-s) ds = √ 2D a α ω 0 + ω c 1 -e -t(ω0+ωc) ,2Da -ω 0 x s η s + 2D a α ds = 2γD a α t 0 ω 0 ω 0 + ω c

≡

  Q EX (t) + Q HK (t), (F7)

FIG. 15 .

 15 FIG. 15. (a) Measured heat necessary to keep the system in a NESS (in units of kBT ) both for τc = 0.8 ms (blue line, before the STEP) and for τc = 40 ms (red dotted line, after the STEP). (b) Released heat measured through the transient both for a increasing correlation time (blue circles) and equivalently for a decreasing correlation time (red triangles).

Appendix D: Power spectral density, autocorrelation, mean squared displacement and equipartition breaking

ACKNOWLEDGMENTS We thank Samuel Albert, Minghao Li, and Laurent Mertz for discussions. This work of the Interdisciplinary Thematic Institute QMat, as part of the ITI 2021 2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR-10-IDEX-0002) and by SFRI STRATUS project (ANR-20-SFRI-0012), and by ANR Equipex Union (ANR-10-EQPX-52-01), the Labex NIE projects ANR-11-LABX-0058-NIE, and USIAS (ANR-10-IDEX-0002-02) under the framework of the French Investments for the Future Program.

Appendix C: Ergodicity

Ergodicity is the equality of time and ensemble averages in the limit of infinite time T and infinite ensemble. But as detailed in our previous work [START_REF] Goerlich | Noise and ergodic properties of brownian motion in an optical tweezer: Looking at regime crossovers in an ornsteinuhlenbeck process[END_REF], appropriate tools exist that can asses the ergodic nature of trajectories {x t } on finite samples and finite times. To do so, we rely on an estimator [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF] corresponding to the variance of the ratio between time averaged mean squared displacement (MSD) and timeensemble averaged MSD. This ratio should becomes Diraclike for long-time (or short time lag ∆ in the MSD). The 0 limit of the variance of this ration for ∆/T → 0 is a necessary and sufficient condition for ergodicity [START_REF] Goerlich | Noise and ergodic properties of brownian motion in an optical tweezer: Looking at regime crossovers in an ornsteinuhlenbeck process[END_REF].

The result, plotted in Fig. 10 (a) with fixed T and varying ∆, decays to zero for short time-lag as expected. With fixed ∆, varying T , the expected decrease towards 0 for long time, with a linear trend in log scale, is also clearly seen in Fig. 10 (b). These two results validate our ergodic assumption for the time-series of position x t , and therefore our treatment when it comes to building trajectory ensembles.

Appendix F: Thermodynamics

The application of stochastic thermodynamics to active matter has already been studied theoretically [START_REF] Speck | Stochastic thermodynamics for active matter[END_REF][START_REF] Marconi | Heat, temperature and clausius inequality in a model for active brownian particles[END_REF][START_REF] Dabelow | Irreversibility in active matter systems: Fluctuation theorem and mutual information[END_REF][START_REF] Chaki | Entropy production and work fluctuation relations for a single particle in active bath[END_REF][START_REF] Chaki | Effects of active fluctuations on energetics of a colloidal particle: Superdiffusion, dissipation and entropy production[END_REF] and experimentally in some cases [START_REF] Argun | Non-boltzmann stationary distributions and nonequilibrium relations in active baths[END_REF]. In this Appendix, we describe in detail how the stochastic heat can be efficiently used to describe and characterize the processes at play in our experiments. We first note that, in our experiments, our sys-

The information content of the injected noise is measured by the spectral entropy H s [START_REF] Zaccarelli | Order and disorder in ecological time-series: Introducing normalized spectral entropy[END_REF], which is precisely the Shannon entropy measured in the frequency domain. To evaluate this quantity and its relation to heat, we perform a series of measurement varying the correlation time of the noise τ c while keeping the other parameters constant, and test the robustness of the result for different sets of parameter (stiffness and noise intensity). For each experiment, an equivalent white noise experiment is also performed, that allows us to extract an effective temperature evaluated through equipartition: T ef f = κ x 2 /k B . We then compare Q EX /k B T ef f to H s . The evaluation of H s is done on the normalized power spectral density of the noise itself. In Fig. 16, we represent the normalized power spectral densities (PSD) for white and colored noises generated at 20 kHz, along with the spectral boundaries used to get rid of the nonphysical part of the signal (high frequency noise of the electronics).

On this PSD, the spectral entropy is then evaluated as:

where


Dextera Domini. The Earliest Inscriptions on Liturgical Gloves 1

In the Middle Ages, the glove was at the same time instrumentum, signum and ornamentum.

On liturgical gloves, one of the seven emblems that distinguished the bishop from the priest, embroidered or enameled medallions were often sewn on the backs. They bore the texts dextera Domini on the right hand and Agnus Dei on the left hand, inscribed around representations of the Blessing Hand and the Mystic Lamb. What role did inscriptions play in this triple function? How did epigraphy on textiles shed light on the relationship to the body and the sacred? The French collection of the 12 th -13 th c. (18 gloves) provides a substantial body of evidence for new answers, by exploring first the material links between glove and inscription, then, the earliest inscribed texts as a different discourse from the liturgical exegesis on the glove, and finally, the pairs of medallions as a dialog on the bishop's role. In the 15 th -16 th c., the monogram of Christ became the most common ornamental element of the episcopal gloves. At this time, this sign was widespread and devotion to the name of Jesus was encouraged in Franciscan preaching, especially by Bernardine of Siena (1380-1444).

However, earlier inscriptions on liturgical gloves proposed a more complex discourse: the two hands of the same pair were well differentiated, and each had its autonomy with a different text. The epigraphic texts were more diverse, referring to God, Christ, the Apostles or the Virgin Mary, with biblical and liturgical expressions and quotations, and the usual colour of the gloves was white. The inscriptions varied from one pair to another, though two recurring formulas were common.

If some liturgists of the 12 th c., such as Honorius of Autun 3 and Hugh of Saint Victor, 4 saw the origin of these gloves in the first centuries of the Church as a prescription of the apostles, 2 This glove is made of purple silk thread, decorated with motifs knitted in jacquard with gold thread. It is 29 cm long and 17 cm wide. It was found during the French Revolution, when the tomb of François d'Estaing was plundered. It is now preserved in the Musée Fenaille of Rodez (Lestoquoit, Rapport de conservation, 1995). Similar gloves have been collected in England, Spain, Italy and France (treasure of the Cathedral of Saint-Bertrand-de-Comminges). This form and decoration seem to have been used since the end of the 14 th century: a similar glove, worn by Bishop William II, is preserved at the New College of Oxford. 3 Honorius of Autun, De Gemma Animae I, CCXIV (PL, t. 172, col. 609). This work is a liturgical treatise with an allegorical view of the liturgy and its practices, written around 1134-1140. This text was very successful (more than 200 manuscripts are preserved). It is composed of four books, and the first one deals with issues of their use is not really documented before the 6 th -7 th c. 5 But when did the inscriptions on the gloves appear? In the absence of written testimonies on the topic we must rely on material evidence, often coming from archaeological discoveries. If, in Germany, examples can be found as early as the second half of the 11 th c., such as the arm reliquary of St Basil in Essen,6 in French collections, the earliest inscriptions date back to the 12 th c. For the 12 th and 13 th c.

we know 18 inscriptions on gloves in France, from Amiens, Troyes, Orleans, Sens, Cahors, Lodève and Conflans: 14 on medallions, 1 embroidered glove, and 3 of uncertain character.

14 of the 18 are still preserved today.

Gloves were bodily and social attributes. Liturgical gloves were one of the seven -or sometimes the nine -emblems that distinguished the bishop from the priest and that symbolized his clerical function during the Middle Ages. Nevertheless, their use was also granted to other ecclesiastics, certain abbots, for instance, as a special privilege. Gloves were part of the clerical identity and the formalism that developed in religious vestments from the 11 th to the 14 th c. Whereas the gloves veiled the anointed hands of the bishop and participated in the effect of "enfouissement du corps" 7 by the accumulation of the layers of clothing, the inscriptions related to the transformation that happened during the ritual. The glove was at the same time instrumentum, signum and ornamentum to use the terms of Michel Pastoureau, because it was materially necessary for the exercise of certain activities, but it was above all a badge of power, an attribute linked to various functions and a social marker; it played an important role in many rituals where, rather than protecting the hand, it replaced, extended, transformed, concealed or embellished. 8 What role did inscriptions play in this triple function? How did epigraphy on textiles shed light on the relationship to the body and the sacred? To answer these questions, from the French collections of the 12 th and 13 th c., I shall address first the material links between glove and inscription; then, the texts inscribed as a the sacrifice, the mass, the ceremonies, the prayers, the church, the ornaments, the ministers, the altar and the priestly garments. different discourse from the liturgical exegesis on the glove; and finally, the pairs of medallions as a dialog on the bishop's role.9

The material links between glove and inscription Of the three parts of a glove -cuff, hand, fingers -the inscriptions were always on the back of the hand. There were two solutions for displaying an inscription or an ornament: either it was knitted, woven or embroidered at the same time as the glove, or it was added afterwards, on a removable and independent medium, sewn to the tops of the gloves. Another solution could have been paintings, because some ornaments were painted with gold on liturgical textiles,10 nevertheless, as far as we know, there is no such example for gloves.

Only one French inscription used the first technique: the glove of Saint Fulcran in the ancient cathedral of Lodève (Hérault), dating from the mid-13 th c.11 The threads of the glove are in white linen, while those of the embroidery are in coloured silk and metallic thread, originally gilded. The technique used is plain interlinking, embroidered with multicolored silk and metallic thread. 12 On the back there are seven lines of writing, on each line the letters are embroidered with three successive colors distributed differently on the lines in a regular sequence: green, yellow, grey-violet (formerly golden). Some letters falling in the change of color are embroidered halfway in one color, halfway in the other: for instance, the R of mater is green and then yellow. This example remains exceptional. Most inscriptions were not directly on the glove, as in this case, but on another support, a circular plate in the shape of a medallion, 13 a lamina, tasellus, fibula, monile, paratura, or even a circulum in gold. These were made of fabric or metal plaques. When they were made of fabric, they were made of silk or wool, one reason why they survive the gloves themselves, which were often made of vegetable fibres such as linen.

In the Treasury of Sainte-Croix Cathedral in Orléans, the medallions attributed to the bishop Robert de Courtenay (1258-1279) were brocaded with gold thread. The first medallion was embroidered with gold thread on a plain samite background and reads ASSVMPTVS EST VT I(N); the second medallion was embroidered on a very fine taffeta, with the inscription + DEXTERA•D(OMI)NI•. 14 The inscriptions have been embroidered too. 15

12 For a complete and very detailed description of the technique, see Cardon, Le Gant de saint Fulcran, 1993, n°10, 36-37. 13 There were also some non-circular examples with rosette or diamond shapes decorated with gemstones and pearls. Braun, Die Liturgische Gewandung, 1907, 374. 14 For a precise description of the stitches used, see Desrosiers, Bedat, Les vêtements liturgiques, 2006, 113-141. 15 The medallions are oval and small: height 5.6/5.4 cm, width 6.2/6 cm. They were discovered during the excavations led by Chenesseau in 1937 inside a sarcophagus among bones and some fragments of liturgical garments along with a crosier, a chalice and a paten of gilded silver, and an emerald. There is no real evidence that this is the burial of Robert de Courtenay. He was first canon of Bourges, then deacon of Chartres before becoming bishop of Orléans. He was a member of the royal family and took part in the Eighth Crusade. See Chenesseau, Les fouilles de la cathédrale d 'Orléans, 1938, 73-94; Brun, Mise au jour en 1937, 1978, 125-142. As they were independent from the glove, the medallion could be adapted in turn to the gloves, as is suggested by the case of Geoffroy de Loudon, bishop of Le Mans (1234-1255), who bequeathed his ornaments to his cathedral, including five pairs of gloves but only two medallions: quinque paria cirothecarum et duas paraturas argenteas deauratas ad opus earumdem cirothecarum. 18 The inscribed medallions could also be reused, recycled in the course of time, as is evident from the chronological differences between the dates of fabrication of the metal disks and of the bishops who wore them. The enameled medallions, as those found in Cahors, could also be employed for other purposes, as on a chasse or a reliquary.

One may wonder whether the addition of the medallion was not a paradox, because as Honorius of Autun explained, gloves must be inconsutiles (seamless) as the robe of Jesus, and 17 These two circular plaques (diameter 4,2 cm) were discovered in a tomb beside a crosier in 1872 during work under the choir of the cathedral of Cahors. They are pierced with sixteen regularly spaced holes, which allow them to be sewn to fabric. The identity of the bishop buried with these plaques is not known.The names of Pons of Antéjac (who died in 1235-36) and Anthony of Luzech (1497-1508) have been suggested but without proof. See CIFM 9, Paris 1984, Lot n°5, 86-88;Taburet-Delahaye, Gloves Plaques, 1996, 144. 18 Barbier de Montault, Les gants pontificaux, 1876, 442. Nadège Bavoux explained that the bishops, abbots and chapter members of many churches, cathedrals, abbeys or collegiate churches contributed, with their own money, to the enrichment of the wardrobe. Bavoux, Histoire du vêtement d 'autel, 2012, 152. because the actions of the prelates must be in accordance with the right faith. 19 Nevertheless, in the De Missarum Mysteriis (1195-1197), the future pope Innocent III specified chirotheca circulum aureum desuper habet. 20 To conclude with material aspect of the gloves, I would like to focus on their contexts of discovery. Most of the glove plaques have been found in a funerary context alongside other insignia. The archbishop of Sens, Pierre de Charny, was buried with his pallium, his gloves, his crosier, his ring and a chalice in August 1274, in the choir of the cathedral. 21 The postmortem clothing of prelates is clearly represented in sculpture, with gloves carved or with real metal plaques. In the Musée des Augustins in Toulouse, carved medallions can be seen on several gisants of bishops, 22 though without inscriptions, or the recess in which a plaque could have been inset. The gloves of Saint Fulcran offer a different example. Although the context is always postmortem, the 13 th century glove never belonged to Fulcran, because he was bishop of Lodève three centuries before, from 949 to 1006. He was considered as a saint, an example of holy bishop. When his cadaver was found intact in his tomb in the 13 th c., it was decided to expose his holy body, clothed in liturgical vestments and insignia, every year on his feast day, sitting in an armchair placed on the high altar of the cathedral, with his right arm raised, blessing the worshippers who came to kiss his feet. 23 The inscriptions: a different exegesis?

The gloves share with other pieces of textile three features highlighted by Lieb Ludger: the flexibility (locomobile and elastic, the textiles adapt to the interior they create or encase), the drawing line (they create limits, boundaries, mostly an interior protected from the eyes of those standing outside) and the symbolicity (they have surfaces particularly suitable for making symbolic statements, either by the material design of the surfaces or by the application of signs of any kind). 24 This third aspect was particularly developed in the medieval period.

With the maniple and the veil, the gloves isolated the bishop's hands when he touched the sacred objects. As both prophylactic artefact, protecting the anointed hands from all contamination, and ornament of the body, the gloves played a key role in the first part of the ritual: worn at the beginning of the mass, they were removed at the offertory. The liturgists of the 12 th and 13 th c. explained in the expositio missae the symbolism of the glove: it veils the hand and allows one to ignore what the other hand is doing, as recommended the Evangelist Matthew (6, 3-4: when you give to the needy, do not let your left hand know what your right hand is doing, so that your giving may be in secret). Through a synecdoche, the hand represents all the actions done and the golden medallion the good deeds that shine before men and glorify God, according to the Gospel of Matthew (5, 16: let your light shine before others, that they may see your good deeds and glorify your Father in heaven). 25 The gloves expressed 23 Vidal, La première Vie de saint Fulcran, 1965. 7-20. The bishops of Lodève used the memory, the life and the body of saint Fulcran to affirm the episcopal power by the other bishops against the counts of Toulouse and Rodez. 24 Lieb, Woven Words, 2019, 209. 25 William Durand devoted a chapter of his Rationale divinorum officiorum to the liturgical gloves (book 3, chapter 12). For the French translation, see: Guillaume Durand, Rationale divinorum officiorum, translation Charles Barthélemy. Paris 1854, vol. 1, 255-257. purity from sin, the performance of good works, and carefulness of procedure; and for William Durand, caution, discretion and the secret. 26 The discourse suggested by the inscriptions on the gloves is different though complementary to this liturgical exegesis. Two expressions in particular were recurrent in these inscriptions: dextera Domini for the right hand (gloves of Orléans, Cahors, Sens) and Agnus Dei for the left hand (gloves of Cahors, Sens, Amiens). This expression Manus Dei/Domini appears mainly in the Old Testament, where it designates the whole of God's action.

The authors of the New Testament on the other hand used Dextera Dei/Domini, which was first of all employed to designate a placement: to be or to sit at the right hand of God, i.e. in heaven. However, the glove inscriptions do not designate such a placement. The inscription equates the hand of the bishop with that of God or even evokes the substitution of one for the other. Not only is the bishop an intermediary between secular and divine, but he is God's representative, through the transfer of powers during the episcopal ordination, notably the It is an active hand. In these cases and others, the biblical and liturgical quotations appeal to the memory of the reader, who is required to participate mentally and figuratively in the construction of the text. 31 Dextera Domini could also sometimes be read on another part of the episcopal garment: the maniple. The two examples of the maniple of St Ulrich, bishop of Augsburg in the middle of the 10 th c. 32 and of St Cuthbert in Durham (England) 33 date before the earliest surviving inscriptions on gloves. Indeed, this piece of liturgical attire became ornamental by the tenth century and was highly visible at the end of the braid, when worn over the celebrant's left arm. Beyond the garments, this biblical expression could also be inscribed on liturgical vessels, specifically on the paten, the small plate used to hold Eucharistic bread for consecration during the mass. 34 The inscriptions on left hand gloves employed the same layouts as those on the right, though they allude not to God but to Christ. The most common inscription was Agnus Dei, for instance on the left hand of the unidentified bishop of Sens. 35 31 Ingrand-Varenne, La brièveté, 2013, 213-234. 32 Coatsworth, Owen-Crocker, Clothing the past, 2018, 333. 33 Hohler,The stole and maniple, 1956, 396-408;Coatsworth, Text and Textile, 2007, 187-207. 34 For instance, the paten found in the tomb of Udo, bishop of Hildesheim who died in 1114. See Wulf, Die Inschriften der Stadt Hildesheim, 2003, Nr. 30 With the Agnus Dei/Dextera Domini formula, there is a concentration of the double aspect of the bishop, at the same time the victim and the blessing, the offering and its acceptance. The Lamb of God recalls the sacrifice, which the bishop mystically slaughters on the altar; but the blessing hand of God symbolizes the omnipotence of God, through whose power the Vicar of Christ makes the bread and wine become the body and the blood of Christ. 37 In this way, the inscriptions anticipate and reiterate the Eucharistic sacrifice. More generally, the Christological significance of the liturgical vestments became increasingly important in the 11 th -12 th century. They carried a message addressed to the bishop himself, the first reader of the inscriptions, an element of "enclothed cognition," to use the expression of Hajo Adam and Adam D. Galinsky, that is to say the vestments had influence and power not only over others, but also on the wearer's psychological processes. These scholars argue that just like physical experiences, the experience of wearing clothes triggers associated abstract concepts and their symbolic meanings for the wearer. In particular, they posit that wearing clothes causes people to "embody" the clothing and its symbolic meaning. Consequently, when a piece of clothing is worn, it exerts an influence on the wearer's psychology. 38 Until he removed his gloves at the offertory, the bishop constantly saw this message, even if the exact moment when he put the gloves and removed them could differ. 39 But to take off the gloves as well as the mitre highlighted the head and the hands, noble parts of the body.

The entirely preserved glove from Essen -without equivalent in France -allows us to take the measure of the duplication that occurs: one hand is covered by another hand (the glove), itself doubled by the image of a blessing hand, around which unfolds the inscription that names the "hand." There is a stacking, a redundancy, almost a "mise en abyme."

In the 13 th century new inscriptions appeared on the gloves, extending their reference beyond God/Christ to the figure of the Virgin 40 who building on devotion to Christ and the Eucharist appears as an essential element of the Incarnation. The byzantine medallions found in Orléans featured the portraits and the names of Christ and of the Virgin. [see fig. 4] In Conflans (Savoy), the gloves that would have served Thomas Becket (1120-1170) were inscribed with 37 Braun, Die Liturgische Gewandung, 1907, 375. 38 Adam, Galinsky Enclothed cognition, 2012, vol. 48, 918-925;Adam, Galinsky, Reflections on enclothed cognition, 2019, vol. 83, 157-159. its influence. 39 Braun, Die Liturgische Gewandung, 1907, 380-381. 40 We have already seen the reference to saint Fulcran. the dialogue AVE MA on the right and RIA GRA(TIA PLENA) on the left. 41 These new texts are more like prayers than the previous ones. This dialog between the hands was not intended to identify a particular person, but a function: that of the bishop. They are part of the liturgical strategyof the sign, which is expressed in a "dress code" and can be considered as a new element of distinction, ornamentation and sacredness. The glove is one of the seven elements that distinguish the bishop from the priest, along with the sandals, the dalmatic, the rational, the mitre, the ring and the staff, while they wear seven garments in common. In Honorius's list, the glove is in the fifth item, after the mitre and before the ring and the staff, whereas for William Durand, it is the eighth of nine items, between the dalmatic and the ring. 42 However the decorated glove was not the exclusive privilege of the clergy. Kings wore them, as the effigies of Henry II (1188) and Richard the Lionheart (1199) in the royal abbey of Fontevraud testify; however inscriptions do not seem to be found on royal gloves in the central Middle Ages. The writing and the glove distinguished the ecclesiastical hierarchy. The gloves and their inscriptions, together with the embroidery, enamels and precious metals, contributed to the sumptuousness of the bishop's costume, an element of the body's adornment in a regime of identity and identification. The addition of the medallion was an extra layer sacralizing the body of the bishop, reinforced by ritual texts, under the impetus of the Gregorian reform. 43

Conclusion:

The inscribed liturgical gloves preserved in the French collections show that the Dextera Domini and the Agnus Dei were the earliest and the most common texts inscribed on the gloved hands of the bishop in the 12 th -13 th c. Many variations existed, and these variants often explained or further developed the meaning of these main formulas. Each hand clearly had its own text, but they spoke to each other as complementary pairs, as right to left. These inscriptions also resonated with the exegesis of contemporary liturgists.

The episcopal gloves were embroidered and exhibit textile art, but the inscriptions could appear also on other materials (metal), with the same idea of richness, in order to glorify God.

On the top of the gloves, the inscribed medallions were removable and were available to be used -in the case of metal plaques -for other purposes. The gloves are themselves an ornamentum,as they helped to differentiate the bishop from the priest. The inscriptions were 41 Rohault de Fleury, La messe, vol. 8, 1889, 194. 42 Honoré d'Autun, De Gemma Animae, I, 209;William Durand, Rational, p. 214. 43 Miller, Clothing the Clergy, 2014, 183. part of the multiplication of allegorical commentaries on the liturgical vestment, using biblical quotations with echoes in the liturgy to amplify the bishop's role. They reflect the sacralisation of the body of the priest and, even more so, of the bishop, as a proxy or equivalent of Christ. They were a signum. To use the expression of St. Paul in the letter to Galatians 3:27, the bishop has "clothed himself with Christ". The gloves were not the only item of ecclesiastical vestment adorned with inscriptions. One can speak of the epigraphic textus of the episcopal body in every sense of the term. It would be interesting to map and to analyze the network of all of these texts on the body of single bishop, with all of its episcopalia.

Der

  Fig.1: Glove of François d'Estaing, bishop of Rodez (Musée Fenaille, Rodez). Picture : ©Musée Fenaille, A. Méravilles. Collections de la Société des Lettres Sciences et Arts de l'Aveyron.

Fig. 2 :

 2 Fig.2: Glove of saint Fulcran, ancient cathedral of Lodève. Picture: from Fils renoués.

  The text completely covers the hand with the litanies of the Virgin Mary: PRECLARA / GRACIOSA / MATER DE / I DIGNA FL / OS VIRGIN / ITATIS VIR / GO REG(IN)A (illustrious, full of grace, worthy Mother of God, flower of virginity, virgin, queen).

Fig. 3 :

 3 Fig.3: Medallions of Robert de Courtenay, bishop. Treasury of Sainte-Croix Cathedral in Orléans. Picture: Jean-Pierre Brouard CIFM/CESCM. The medallions could also be metal disks. Two Byzantine medallions, perhaps from the South of Italy and dating from the end of the 12 th c. or the beginning of the 13 th c., survive from a pair of gloves from the tomb of Raoul Groparmi, another bishop of Orléans (1306-1311).The nomina sacra of Christ and the Virgin Mary are incised in Greek on each side of the figures: IC XCand MP•ΘΥ. 16

Fig. 4 :

 4 Fig. 4: Metal disks found in the tomb of Raoul Groparmi, bishop. Treasury of Sainte-Croix Cathedral in Orléans. Picture: Jean-Pierre Brouard CIFM/CESCM.

Fig. 5

 5 Fig. 5 and 6: Circular plaques found in a tomb of a bishop. Treasury of the cathedral of Cahors. Picture: Jean Michaud CIFM/CESCM.

Fig. 7 :

 7 Fig. 7: Gisant for a bishop, 13 th c. Musée des Augustins, Toulouse. Picture: Estelle Ingrand-Varenne.

  It may seem obvious that the right hand should be qualified as dextera Domini, but what does this really mean? Several different expressions were used in the Bible to speak about the hand of God. Manus Dei/Domini was the most common, but only two gloves are documented as bearing this inscription: the medallion, found in the tomb of Michel de Corbeil, bishop of Sens (from 1194 to 1199), but now disappeared, 27 and the medallion of an abbot of Corbie, preserved in the Museum of Amiens.28

Fig. 8 :

 8 Fig. 8: Medallion of an abbot of Corbie, Museum of Picardie, Amiens. Picture: Collection du Musée de Picardie, Amiens.

  anointing of hands. Another inscribed medallion from Sens, + DEXTERA •DOMINI•FECIT•VIRTUTE(M), found in the tomb of Peter of Corbeil, archbishop who died in 1222, 29 further develops the meaning of the hand through quotation of Psalm 118 (117), 16: Dextera Domini exaltavit me: Dextera Domini fecitvirtutem (the right hand of the Lord has unfolded its power, the right hand of the Lord has exalted me). The inscription, embroidered with gold threads, encircles the representation of the hand of God, surrounded by four stars. [Fig.9: Medallion of Pierre de Corbeil, archbishop. Museum of Sens. Picture: Jean Michaud CIFM/CESCM. This antiphon was usually sung at the offertory of the Mass on Holy Thursday, at the beginning of the Easter Triduum; it was also sung at the offertory of Easter Day. Another example shows that the Dextera Domini is the benediction gesture: a medallion from Sens for an unidentified bishop (at of the 13 th c. or beginning of the 14 th c.) bears the text: + IN NOMINE PATRIS ET. 30

Fig. 10 :

 10 Fig.10: Medallion for an archbishop. Museum of Sens. Picture: Jean Michaud CIFM/CESCM.

  . About the paten and the dextera Domini, see the recent study of Brandt, Die Patene im Hochmittelalter, 2019, 50-58.I thank Clemens M.M. Bayer for this reference.35 CIFM 21. Paris 2000, n°163, 180-181.

Fig. 11 :

 11 Fig.11: Medallion for an archbishop. Museum of Sens. Picture: Jean Michaud CIFM/CESCM. It recalls and repeats the gesture and words of John the Baptist pointing to Christ, in the Gospel of John (1, 29): ecce Agnus Dei (behold the Lamb of God), because of the bishops inevitable performance of pointing or gesturing with gloved hand repeats, thus fulfilling the verbal interjection ecce. The enameled medallion of Cahors develops the quotation further:Agnus Dei qui tollit p(eccata mundi).[see fig.6] This quotation anticipates the phrase that the celebrant will utter during the Eucharistic ritual, particularly during the Elevation, when it is introduced in the 13 th century. The medallion on the glove of the left hand of the bishop of Orleans in the second half of the 13 th c., Robert de Courtenay, is more original, however, and the layout is quite different. The text does not frame the picture, with a circular form, but the letters are part of the image, scattered here and there around the hand of God.36 The hand is not pointing the three fingers to the top but emerges from a cloud in a reverse order. It is very difficult to read, but it seems that the text was: ASSVMPTVS EST VT I, with arrangement of the letters both vertically and horizontally. [see fig.3] This is probably the end of the Gospel of Mark: Dominus Iesus assumptus est in caelum, et sedet ad dexteram Dei (16, 19) which, moreover, evokes the Dextera Domini. In this case, the text alludes to the Ascension of Christ and thus also, perhaps, to the elevation, equating or analogizing the two. This example draws together the meanings of the two most common formulas.

I am very pleased to thank P. Scott Brown for his help with the translation and for his valuable comments.

.Barraud, Des gants, 1867, 209.

Barraud, Des gants, 1867, 203-211. See also the study ofSchwineköper, Der Handschuhim Recht, 1938, and Beaulieu, Les gants liturgiques, 1968, 137-153.

This arm reliquary wears a glove on which is a circular medallion in gilded silver, depicting the right hand of God with the inscription Dextera Dei. This gloved arm reliquary has no comparable example in the arm reliquaries of the Romanesque period. Hermann, Die Inschriften der Stadt Essen, 2011, Nr 27.

To use Odile Blanc's term. SeeBlanc, Parades et parure, 1997, 136-141. This study focuses on the miniatures that adorn sumptuous, late medieval princely manuscripts reflecting courtly costume and fashion. Nevertheless this expression is particularly relevant to describe the layers of liturgical garments that were added during the medieval period.

Pastoureau, Le gant médiéval, 2007, 121-137. For a broad perspective on medieval clothing and its symbolic function as a language of power, see the studies of Gil Bartholeyns, in particular in the same volume:Bartholeyns, L'enjeu du vêtement, 2007, 219-257.

Concerning the German collections of inscribed gloves, see the study of Hannes Fahrnbauer in this volume, and his ongoing PhD thesis on clerical gloves of the Latin Middle Ages (11 th -beginning 14 th centuries).

For instance, Pierre de Charny, bishop of Sens in the 13 th century, wore as burial garment a painted alb. As Eugène Chartraire explained, the golden decoration on the purple ground of his alb looked at first sight like sumptuous embroidery, but it was in fact gold painting. Chartraire,Insignes épiscopaux, 1918, 31.

They are preserved in the treasury of the church Saint-Fulcran in Lodève in the chapel of the relics. The glove is 23.7 cm high by 14 cm wide.The dating was proposed by Robert Favreau based on paleographic criteria, in accordance with analysis of the textile.

CIFM 21. Paris 2000, n°156, 175.