the microcompartment is its ability to sequester some molecules and be porous to others to maintain both an **out**-**of**-**equilibrium** state and a certain level **of** identity. This control **of** transport and sequestration can be achieved by a membrane or more simply using phase partitioning as in the case **of** droplets 38 . Membrane- based microcompartments—for example stabilized by lipid bilay- ers—have the drawback that during growth and division, both the volume and the membrane must grow 10 . For droplets, the volume growth is directly linked to the surface growth and a single process is therefore sufﬁcient for proliferation. Droplets do have a biolo- gical relevance: liquid structures are found in living cells in the form **of** P-granules formed by phase separation in liquid–liquid system providing means to compartmentalize reactions in the cytoplasm 59 . Considering that life has emerged from the most simple system, droplets appear to be relevant models. Using the conceptual basis that the systems found in the laboratory need not be chemically similar to the actual molecular assemblies **of** living cells 10 but that the key point is to mimick the functions and the essential properties **of** living systems, our droplets engineered from a phase separation in a ﬂuorinated oil/water mixture are direct analogon **of** the coacervate droplets and therefore bear a relevance in the context **of** the build-up **of** minimal functional micro- compartments having life-like properties.

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Universidad de Santiago de Chile, Facultad de Ciencia, Departamento de F´ısica, Santiago, Chile
3 Universit´ e Grenoble Alpes and CNRS, LIPHY, F-38000 Grenoble, France
(Dated: April 5, 2017)
We study the mechanical fluctuations **of** a micrometer sized silicon cantilever subjected to a strong heat flow, thus having a highly non-uniform local temperature. In this non-**equilibrium** steady state, we show that fluctuations are equivalent to the thermal noise **of** a cantilever at **equilibrium** around room temperature, while its mean local temperature is several hundred **of** degrees higher. Changing the mechanical dissipation by adding a coating to the cantilever, we recover the expected rise **of** fluctuations with the mean temperature. Our work demonstrates that inhomogeneous dissipation mechanisms can decouple the amplitude **of** thermal fluctuations from the average temperature. This property could be useful to understand **out**-**of**-**equilibrium** fluctuating systems, or to engineer low noise instruments.

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speed **of** propagation. Then we show that, with high probability, only the
evolution **of** a restricted chain inside a suitable ergodic component matters. This reduction is performed via a general result on Markov processes which we derive in section 3. The ergodic component is chosen in such a way that the log-Sobolev constant for the restricted chain is much smaller than t. This second reduction is new and it is at this stage that the restriction on q appears and that all the difficulties **of** the non-**equilibrium** dynamics appear. Its implementation requires the estimate **of** the spectral gap **of** the process restricted to the ergodic component (see section 6) and the study **of** the persistence **of** zeros **out** **of** **equilibrium** (see section 4).

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Lieb-Robinson bounds Even if it has been possible to obtain crucial results for both the time evolution and thermalization in different experimental situations, much work remains to be done to interpret these complicated phenomena. Many **of** the theoretical tools used to describe systems at **equilibrium** cannot be adapted to the **out**-**of**-**equilibrium**. The research on general principles to understand the time evolution is one **of** the most exciting and difficult problems in modern physics. One **of** the few results existing in this field are general bounds on the dynamics **of** local observables, as correlations. They con- strain the time evolution **of** specific observables inside precise portions **of** the space-time plane. The most famous one is the Lieb-Robinson bound, which applies to lattice models interacting via finite-range potentials, such as nearest-neighbor or contact terms. The authors find a bound on the commutator between two local operators defined over two disjoints set **of** the lattice. The bound divides the space-time plane in two regions. One where the commutator can be significant and the other where it is exponentially small. It defines then a position-dependent activation time t ⋆ (R) ∼ R/v

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statistics (hni ≃62.01) is the same as that given by the Boltzmann statistics (hni ≃62.51) within a 1% accuracy; as a consequence, a classical description seems rea- sonable. Moreover, when the system is brought **out** **of** thermal **equilibrium** by external energy injection, the effective temperature **of** each normal mode is higher, thus increasing the occupation number **of** each mode and making the classical approximation even more reasonable. Furthermore, for example, the vibrational properties **of** proteins are very well described by molecular dynamics simulations performed in a classical context. Therefore, the first important question addressed in the present work is whether a phonon condensation phenomenology can also be retrieved in a classical framework and **out** **of** thermal **equilibrium**. To this aim, by resorting to a dequantization method, we have worked **out** a classical version **of** the original Fröhlich model, finding that—remarkably—in a classical context too, Bose-like phonon condensation is possible [16] . This possibility requires us to consider a biomolecule as an open system—that is, far from thermal **equilibrium** with its environment—through which energy flows under the simultaneous actions **of** an external energy supply and **of** dissipation due to radiative, dielectric, and viscous energy losses. We find that the classical Bose-like condensation in the lowest vibrational mode occurs when the energy input rate exceeds some threshold value. Then, this a priori nonobvious result motivates an experimental effort to find if the theoretically predicted phenomenon is actually possible in the physical world. Two independent experiments, in geographically distinct laboratories, have confirmed the existence **of** **out**-**of**-**equilibrium** collective oscillations for a model protein. This is a proof **of** concept **of** which the most significant implication is that, in compliance with a theoretical prediction [8] , a crucial prerequisite is fulfilled to excite intermolecular long-range electrodynamic interactions. In turn, as already mentioned above, these interactions could affect the biomolecular dynamics by contributing to drive the high efficiency and rapidity **of** mutual encounters **of** the partners **of** biochemi- cal reactions in living matter, encounters that do not always appear to be the result **of** Brownian diffusion only.

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broken in a way that leads very naturally to fluctuation relations like the Jarzynski equality or even the underlying fluctuation theorem.
We identify another new symmetry **of** the MSRJD generating functional, which is valid in but also **out** **of** **equilibrium**. At the level **of** observables, it generates equations **of** motion coupling correlations and responses. These Schwinger-Dyson equations provide a nice way to express all sorts **of** responses in terms **of** correlation functions without applying any extra field. This has direct applications in computer simulations where the computation **of** linear responses using weak perturbations (to stay in the linear regime) is not an easy task; besides requiring two simulations (one with and one without the perturbation) it also requires a lot **of** statistical averaging to get a good signal-to-noise ratio.

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G. Lantz 1,2 , B. Mansart 1 , D. Grieger 3 , D. Boschetto 4 , N. Nilforoushan 1 , E. Papalazarou 1 , N. Moisan 1 ,
L. Perfetti 5 , V.L.R. Jacques 1 , D. Le Bolloc’h 1 , C. Laulhe ´ 6,7 , S. Ravy 1,6 , J.-P. Rueff 6,8 , T.E. Glover 9 , M.P. Hertlein 9 , Z. Hussain 9 , S. Song 10 , M. Chollet 10 , M. Fabrizio 3 & M. Marsi 1
The study **of** photoexcited strongly correlated materials is attracting growing interest since their rich phase diagram often translates into an equally rich **out**-**of**-**equilibrium** behaviour. With femtosecond optical pulses, electronic and lattice degrees **of** freedom can be transiently decoupled, giving the opportunity **of** stabilizing new states inaccessible by quasi-adiabatic

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studied 8–11 , which aim at explicitly summing the series **of** Feynman diagrams numerically, for example for the self-energy. Concerning quantum impurity models, there has been an intense activity in the recent years in the development **of** new continuous (mostly imaginary) time quantum Monte-Carlo techniques, based on an expansion in U (or around the strong coupling limit). These new algorithms are **of** huge practical value in solving the self- consistent impurity problems that arise from the dynam- ical mean-field theory **of** correlated bulk systems 12–15 , even though they still suffer from the sign problem. They have been extended to the non-**equilibrium** case in a rel- atively straightforward way, simply adapting the Monte- Carlo method to the Keldysh formalism 16–20 . However, these **out**-**of**-**equilibrium** versions suffer from a severe dy- namical sign problem, compared to their **equilibrium** counterparts, which has severely limited their usage in practice. In particular, they can not reach the long- time steady-state limit in several regimes **of** parameters. Moreover, the approach **of** Ref. 19 and 20 has only been shown to work with sufficient accuracy for an Anderson impurity with particle-hole symmetry, i.e. a very special point **of** the phase diagram. More recently, bold diagram- matic Monte-Carlo for impurity models have also been extended to the Keldysh context and used in combina- tion with the master equation for the density matrix 21,22 to reach longer time. Finally, testing these approaches in large systems, even at moderate interaction, remains also an open question.

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1. Introduction
Understanding the mechanisms involved in interacting systems far from **equilibrium** is one **of** the major problems in condensed matter physics. We study in this paper quantum dots under nonequilibrium conditions. Quantum dots are realized by confining electrons in a small spatial region weakly coupled to two leads. They give rise to a Kondo e↵ect at low temperature [1] when the dot is occupied by a single electron, and hence acquires a spin which is antiferromagnetically coupled to the spin **of** the conduction electrons in the leads. Predicted at a theoretical level at the end **of** the 80ies [2, 3], the Kondo e↵ect in quantum dots has been observed experimentally since the end **of** the 90ies [4-6]. **Out** **of** **equilibrium** the Kondo e↵ect in quantum dots has been studied by a variety **of** analytical and numerical techniques. In most cases, these techniques combine many-body methods to treat strong interactions with non-**equilibrium** Green function techniques to take non-**equilibrium** conditions into account. The system is driven **out** **of** **equilibrium** either by the application **of** a dc bias voltage between the two leads or by irradiation with an electromagnetic field. Driving the system **out** **of** **equilibrium** leads to a decoherence **of** the Kondo many-body singlet state and induces a crossover from the Fermi liquid strong coupling regime (local Fermi liquid) to the weak coupling regime. Despite all the e↵ort developed these last years along this direction, many open questions remain about how electron correlations and nonequilibrium e↵ects interfere in those mesoscopic systems.

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Steady States (NESS) is thus **of** deep importance. We present here first studies **of** such NESS in the context **of** systems with long range interactions. The most prominent result is the finding **of** **out** **of** **equilibrium** phase transitions.
These first studies have been done in the context **of** two dimensional flows. This is indeed essential in this case, as in many applications **of** fluid dynamics, one **of** the most important problem is the prediction **of** the very high Reynolds’ large-scale flows. The highly turbulent nature **of** such flows, for instance ocean circulation or atmosphere dynamics, renders a probabilistic description desirable, if not necessary. At **equilibrium**, a statistical mechanics explanation **of** the self-organization **of** geophysical flows has been proposed by Robert-Sommeria and Miller (RSM). **Out** **of** **equilibrium**, there are several practical and fundamental problems to understand: How the invariants are selected by the presence **of** weak forces and dissipation? What are the associated fluctuations? Are all forcings compatible with RSM equilibria?

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present at **equilibrium** in the Kondo regime. We show how our approximation scheme improves the result for the Kondo temperature T K upon earlier predictions.
We present in Sec. IV our numerical results—both at **equilibrium** and **out** **of** **equilibrium**—for the density **of** states, the 共linear and differential兲 conductance, and the bias- induced decoherence rate. A self-consistent treatment is re- quired in order to determine the expectation values involved in the Green’s function **of** the dot. At **equilibrium**, the density **of** states in the particle-hole symmetric case shows a three- peak structure at low temperature with a Kondo resonance peak in the local-moment regime. This result constitutes an advantage **of** our approximation scheme compared to the La- croix approximation. The unitary limit for the linear conduc- tance G = 2e 2 /h is analytically recovered at zero temperature

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1. Introduction
Saltation is known as the main mode **of** sediment transport in air and characterizes the motion **of** the parti- cles jumping along the sand surface in ballistic trajectories (Bagnold, 1941). When a suﬃciently strong wind blows over a sand bed, aeolian transport **of** sand is initiated and an **equilibrium** between air ﬂow and trans- ported particle is eventually achieved. This **equilibrium** state is characterized by an **equilibrium** or saturated mass ﬂux, which is an increasing function **of** the wind strength. Up to now, most researches were focused on the description **of** the **equilibrium** state **of** transport where erosion and deposition processes balance exactly (Bagnold, 1941; Durán et al., 2011; Kok et al., 2012; Shao, 2008). A further interesting and important issue is the description **of** the **out**-**of**-**equilibrium** transport. If we consider a situation where we have a sudden increase or decrease in wind velocity, the mass ﬂow rate does not adapt instantaneously to its new **equilibrium** value. This transient lasts a certain characteristic time (or occurs over a certain characteristic distance) for the new grains extracted from the bed to reach the air velocity and for the particle concentration to relax toward its new **equilibrium** value (Durán et al., 2011; Valance et al., 2015). During this transient process, the air speed within the transport layer has also to decrease to its **equilibrium** value because **of** the negative feedback **of** transport on the wind. Bagnold (1941) was the ﬁrst to evidence that the saturation process takes a ﬁnite distance. When brought slightly **out** **of** **equilibrium**, the particle-laden air ﬂow is shown to relax to its **equilibrium** state via an exponential behavior which can be modeled by a linear ﬁrst-order diﬀerential equation for the mass ﬂux Q (Andreotti et al., 2010; Pähtz et al., 2013, 2015):

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9
we are in a **out**-**of**-**equilibrium** dissipative system. Others aspects can therefore play a role, like dissipation, which is probably different in the cluster and liquid phase, or the very particular thermalization imposed by vertical vibration. This effect could be especially important near the transition where a critical behavior can be expected. Actually, an exponent α < 1 is measured in another **out**- **of**-**equilibrium** system: the slow compaction **of** granular packing by tapping [4], but the meaning **of** such law is still under debate in this case. This issue deserves further studies by changing the boundary shape (from square to circular) or by doing the experiment in a viscous fluid to modify the dissipation or by exciting the grains by vibra- tion using colored noise with a broad frequency range.

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Despite these successes, several problems remain open. The theory has been developed using mainly a class **of** toy models, the so-called “spherical p-spin glass models”, whose dynamical equations also correspond to the so-called schematic limit **of** the Mode-Coupling Theory equations [ 3 , 16 ]. While these models share most **of** the basic phenomelonogy **of** real glasses [ 6 , 7 ], it remains highly desirable to develop a theory specific to particle systems. This has been achieved recently by considering the infinite-dimensional limit [ 17 – 20 ], but only in the special case **of** **equilibrium** dynamics. The extension to the **out**-**of**-**equilibrium** regime is yet to be done but it would have potentially very interesting applications, for example: (i) the study **of** the glass transition in active matter, where the behaviour **of** the glass transition line is seen to depend on the details **of** the interaction potential [ 21 , 22 ]; (ii) the dynamical scaling in the vicinity **of** the jamming transition [ 23 – 26 ]; (iii) the dynamical behaviour in rheological experiments, where several interesting phenomena such as plasticity, yielding, and non-Newtonian flow curves appear [ 27 ]. These phenomena cannot be fully captured by the simplest p-spin glass models. Extensions **of** the Mode-Coupling Theory, which, as said above, in the schematic limit correspond to the p-spin dynamics, to **out**-**of**-**equilibrium** situations have been obtained. These extensions obtained partial successes in the flow [ 28 , 29 ], active [ 30 ], and aging [ 31 ] regimes, but failed in other cases [ 32 ], calling for alternative approaches.

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DOI: 10.1103/PhysRevE.85.031108 PACS number(s): 05.10.Gg, 77.22.Ej
I. INTRODUCTION
Despite the success **of** theoretical calculations on the equi- librium Casimir force [ 1 – 3 ], aspects **of** the **out**-**of**-**equilibrium** behavior still are poorly understood and are the subject **of** controversy and debate. A number **of** approaches have been adopted to compute thermal fluctuation induced forces **out** **of** **equilibrium** in simple models **of** soft matter systems and binary liquids. For example, the stress tensor has been used to compute the force [ 4 – 8 ] in a variety **of** nonequilibrium contexts. While it is clear that, in such systems, computations using the stress tensor yield the average value **of** the force at thermal **equilibrium**, it seems, nevertheless, that more information is needed regarding the dynamics **of** the field theory representing the critical or fluctuating field [ 9 ], in particular, how the value **of** the field at a surface changes when the surface is moved. An alternative approach is to define an energetic interaction **of** the field with a surface and then to define forces via the principle **of** virtual work [ 10 , 11 ]. Yet another is to define the force at a surface by a local kinetic argument, for instance, by using the ideal gas form for the pressure as a function **of** the local density field [ 12 , 13 ]. There are notable differences in **out**-**of**-**equilibrium** forces computed using the approaches above, and relatively few systems have been studied explicitly. However, studies **of** free Gaussian field theories undergoing model A (nonconserved) dynamics have been carried **out**. Model A dynamics for the field is basically a diffusion equation driven by white noise, and the thermal Casimir forces are found to tend toward the **equilibrium** value with diffusive scaling [ 6 , 7 , 10 , 11 ]. As well as studying the approach to **equilibrium** for dynamics obeying detailed balance, one can examine what happens when the noise is nonthermal, for instance, colored. In this case, the steady state Casimir interaction, under model A-type dynamics, tends to acquire an additional screening due to temporal correlations in the noise [ 4 , 10 , 11 ]. We also note that the Parisi-Wu stochastic

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result: the current vanishes like a power law in the large voltage regime, which means that the differential conductance is negative in this domain. Also, among the exact **out**-**equilibrium** results for the IRLM, we want to point the perturbative approach (infra-red, or low-voltage regime) **of** Ref. [ 9 ], where and analysis **of** ≈ 1500 diagrams was carried **out**, as well as the ultra-violet (high-voltage regime) expansion [ 59 ], which captures the negative differential conductance for low values **of** the interaction. As it was mentioned above, an analytical solution **of** the IRLM **out**-**of**-**equilibrium** for arbitrary values **of** interaction is not accessible. At this point, numerical tech- niques come into play, like the numerical renormalization group [ 50 , 60 , 61 ] and the time-dependent density matrix renormalization group [ 62 , 63 , 64 , 2 ]. One im- portant advance was the agreement between analytical results and numerics at the self-dual point [ 2 ]. It was shown in particular that t-DMRG simulations can probe the lattice formulation **of** the IRLM in a regime – called scaling regime – where the model behaves as its continuum counter part. In this regime, transport properties like the steady current become, after some suitable rescaling, a function **of** a single combination **of** the voltage bias and tunneling amplitude to the dot. 4

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tems far-from-**equilibrium**, Z η/s tends to vanish. Thus,
in systems **out**-**of**-**equilibrium**, higher order viscous cor- rections effectively reduce the value **of** η/s entering the second order viscous hydrodynamic equations, an effect first pointed **out** by Lublinsky and Shuryak [18]. As can be seen on Fig. 1 (grey dashed line), this simple renor- malization brings the solution **of** the lowest non trivial truncation quite close to the exact solution. That is, with this correction, second order viscous hydrodynam- ics reproduces accurately the exact solution **of** the kinetic theory.

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(Received 6 December 2019; revised 25 June 2020; accepted 27 June 2020; published 5 August 2020) As the places where most **of** the fuel **of** the cell, namely, ATP, is synthesized, mitochondria are crucial organelles in eukaryotic cells. The shape **of** the invaginations **of** the mitochondria inner membrane, known as a crista, has been identified as a signature **of** the energetic state **of** the organelle. However, the interplay between the rate **of** ATP synthesis and the crista shape remains unclear. In this work, we investigate the crista membrane deformations using a pH-dependent Helfrich model, maintained **out** **of** **equilibrium** by a diffusive flux **of** protons. This model gives rise to shape changes **of** a cylindrical invagination, in particular to the formation **of** necks between wider zones under variable, and especially oscillating, proton flux.

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Dynamics **of** UV short pulse laser induced plasmas from a ceramic material “titanium carbide”, a hydrodynamical **out** **of** **equilibrium** investigation
The present work is motivated by the numerous applications **of** short lasers-ceramics interaction. It aims at applying a newly developed model to investigate the dynamic **of** laser induced plasmas from a ceramic material into a helium gas under atmospheric pressure. To have a better understanding **of** the link between the material properties, the plume characteristics and its interaction with the laser beam a thorough examination **of** the entire ablation processes is conducted. Comparison with the behavior **of** laser induced plumes under the same conditions from a pure material is shown to have a key role in shedding the light on what monitors the plume expansion in the background environment. Plume temperatures, velocities, ionization rates as well as elemental composition have been presented and compared under carefully chosen relevant conditions. This study is **of** interest for laser matter applications depending on the induced plasmas dynamics and composition.

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The anomalously low D47 values exhibited by modern seep carbonates may arise from rapid mineralization, which may allow them to retain an isotopic ﬁngerprint **of** processes controlling solution disequilibrium, as well as surface kinetic processes. Quantitative models have shown that minerals forming rapidly will have a higher propensity to exhibit geochemical signatures **out** **of** **equilibrium** with the ambient environment 58 , and recent observations have shown this can apply to clumped isotope signatures 40 , although recent laboratory-based rapid precipitation experiments can yield **equilibrium** clumped isotope signatures 59 . In cold seep settings, mineral precipitation rates are difﬁcult to quantify, so that although there are many reports **of** AOM rates, authigenic carbonate precipitation rates are rarely available. Where carbonate precipitation and AOM rates have been quantiﬁed in the Nile Delta 60,61 and Mediterranean Sea 62,63 precipitation rates broadly overlap with AOM rates (see Supplementary Fig. 1). Therefore environments exhibiting

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