Two-temperature Langevin dynamics in a parabolic potential

Active magneto gyrator: Memory-induced trapped diamagnetism

We analytically explore the dynamics of a charged active particle coupled to two thermal baths kept at two different temperatures in two dimensions. The particle is confined to an asymmetric harmonic potential and a magnetic field of constant magnitude is applied perpendicular to the plane of motion of the particle. For such a system, as opposed to a Brownian gyrator, the potential asymmetry and temperature gradient are not the key factors for the gyration, as long as finite activity and magnetic field are present. The system shows only a paramagnetic behavior in the absence of either potential asymmetry or temperature gradient. However, by tuning the temperature gradient or potential asymmetry, the system as a function of the duration of activity can exhibit paramagnetic, diamagnetic, or coexistence of both the phases. Interestingly, the magnetic moment vanishes for parameters for which the system possesses a nonequilibrium steady state and hence, a magnetic transition is observed through these nonmagnetic points. Further, when the system is suspended in a viscoelastic medium characterized by a finite memory, it exhibits a magnetic transition in the activity-memory parameter space through a nonmagnetic line. This nonmagnetic line is sensitive to temperature gradient and potential asymmetry. It interestingly forms a closed loop with a diamagnetic phase inside the loop and the entire regime outside as paramagnetic. This results in the emergence of a trapped diamagnetic phase existing only within a finite regime of activity-memory parameter space. This phase eventually disappears as the temperature gradient increases (or decreases) depending on the sign of the potential asymmetry. Moreover, it is observed that by tuning the system parameters, one can obtain zero magnetic moment even for parameter ranges that defy the equilibrium condition of the system.

Ergodicity breaking and restoration in models of heat transport with microscopic reversibility

The behavior of lattice models in which time reversibility is enforced at the level of trajectories (microscopic reversibility) is studied analytically. Conditions for ergodicity breaking are explored, and a few examples of systems characterized by an additional conserved quantity besides energy are presented. All the systems are characterized by ergodicity restoration when put in contact with a thermal bath, except for specific choices of the interactions between the atoms in the system and the bath. The study shows that the additional conserved quantities return to play a role in nonequilibrium conditions. The similarities with the behavior of some mesoscale systems, in which the transition rates satisfy detailed balance but not microscopic reversibility, are discussed.

Variance Resonance in Weakly Coupled Harmonic Oscillators Driven by Thermal Gradients

We study two harmonic oscillators with high quality factors, driven by equilibrium and off equilibrium thermal noise, the latter mimicked by establishing a temperature gradient. The two oscillators are coupled via a third reciprocal harmonic interaction. We deepen the case of a weak coupling between the two oscillators, and show the emergence of a "spike" in the displacement variance of the colder oscillator, when the respective elastic constants approach each other. Away from the peak, the displacement variance of each oscillator only reflects the value of the local temperature. We name this phenomenon the variance resonance, or alternatively covariance resonance, in the sense that it comes about as one element of the covariance matrix describing both oscillators. In fact, all of the elements of the covariance matrix show some distinctive behavior. The oscillator at the lower temperature, therefore, oscillates as if driven by a higher temperature, resonating with the other one. By converse, the variance of the hotter oscillator develops a deep dent, or depression, around the same region. We could not reproduce this behavior if either the coupling constant is not small compared to those of the two oscillators, or if the quality factors are not large enough. In fact, in such instances the system tends to resemble one which is in equilibrium at the average temperature, regardless of the relative strengths of the elastic constants of the two oscillators. Our results could have various applications including for example precision measurement systems, when not all parts of the apparatuses are at the same temperature.

Destructive effect of fluctuations on the performance of a Brownian gyrator

The Brownian gyrator (BG) is often called a minimal model of a nano-engine performing a rotational motion, judging solely upon the fact that in non-equilibrium conditions its torque, specific angular momentum  and specific angular velocity  have non-zero mean values. For a time-discretised (with time-step δt) model we calculate here the previously unknown probability density functions (PDFs) of  and . We show that for finite δt, the PDF of  has exponential tails and all moments are therefore well-defined. At the same time, this PDF appears to be effectively broad - the noise-to-signal ratio is generically bigger than unity meaning that  is strongly not self-averaging. Concurrently, the PDF of  exhibits heavy power-law tails and its mean is the only existing moment. The BG is therefore not an engine in the common sense: it does not exhibit regular rotations on each run and its fluctuations are not only a minor nuisance - on contrary, their effect is completely destructive for the performance. Our theoretical predictions are confirmed by numerical simulations and experimental data. We discuss some plausible improvements of the model which may result in a more systematic rotational motion.

Large deviations of the stochastic area for linear diffusions

The area enclosed by the two-dimensional Brownian motion in the plane was studied by Lévy, who found the characteristic function and probability density of this random variable. For other planar processes, in particular ergodic diffusions described by linear stochastic differential equations (SDEs), only the expected value of the stochastic area is known. Here we calculate the generating function of the stochastic area for linear SDEs, which can be related to the integral of the angular momentum, and extract from the result the large deviation functions characterizing the dominant part of its probability density in the long-time limit, as well as the effective SDE describing how large deviations arise in that limit. In addition, we obtain the asymptotic mean of the stochastic area, which is known to be related to the probability current, and the asymptotic variance, which is important for determining from observed trajectories whether or not a diffusion is reversible. Examples of reversible and irreversible linear SDEs are studied to illustrate our results.

Dynamical large deviations of linear diffusions

Linear diffusions are used to model a large number of stochastic processes in physics, including small mechanical and electrical systems perturbed by thermal noise, as well as Brownian particles controlled by electrical and optical forces. Here we use techniques from large deviation theory to study the statistics of time-integrated functionals of linear diffusions, considering three classes of functionals or observables relevant for nonequilibrium systems which involve linear or quadratic integrals of the state in time. For these, we derive exact results for the scaled cumulant generating function and the rate function, characterizing the fluctuations of observables in the long-time limit, and study in an exact way the set of paths or effective process that underlies these fluctuations. The results give a complete description of how fluctuations arise in linear diffusions in terms of effective forces that remain linear in the state or, alternatively, in terms of fluctuating densities and currents that solve Riccati-type equations. We illustrate these results using two common nonequilibrium models, namely, transverse diffusions in two dimensions involving a nonconservative rotating force, and two interacting particles in contact with heat baths at different temperatures.

Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator

The nonequilibrium Fokker-Planck dynamics in an arbitrary force field f[over ⃗](x[over ⃗]) in dimension N is revisited via the correspondence with the non-Hermitian quantum mechanics in a real scalar potential V(x[over ⃗]) and in a purely imaginary vector potential [-iA[over ⃗](x[over ⃗])] of real amplitude A[over ⃗](x[over ⃗]). The relevant parameters of irreversibility are then the N(N-1)/2 magnetic matrix elements B_{nm}(x[over ⃗])=-B_{mn}(x[over ⃗])=∂_{n}A_{m}(x[over ⃗])-∂_{m}A_{n}(x[over ⃗]), while it is enlightening to explore the corresponding gauge transformations of the vector potential A[over ⃗](x[over ⃗]). This quantum interpretation is even more fruitful to study the statistics of all the time-additive observables of the stochastic trajectories, since their generating functions correspond to the same quantum problem with additional scalar and/or vector potentials. Our main conclusion is that the analysis of their large deviations properties and the construction of the corresponding Doob conditioned processes can be drastically simplified via the choice of an appropriate gauge for each purpose. This general framework is then applied to the special time-additive observables of Ornstein-Uhlenbeck trajectories in dimension N, whose generating functions correspond to quantum propagators involving quadratic scalar potentials and linear vector potentials, i.e., to quantum harmonic oscillators in constant magnetic matrices. As simple illustrative example, we finally focus on the Brownian gyrator in dimension N=2 to compute the large deviations properties of the entropy production of its stochastic trajectories and to construct the corresponding conditioned processes having a given value of the entropy production per unit time.

Active responsive colloids driven by intrinsic dichotomous noise

We study the influence of intrinsic noise on the structure and dynamics of responsive colloids (RCs), which actively change their size and mutual interactions. The colloidal size is explicitly resolved in our RC model as an internal degree of freedom (DOF) in addition to the particle translation. A Hertzian pair potential between the RCs leads to repulsion and shrinking of the particles, resulting in an explicit responsiveness of the system to self-crowding. To render the colloids active, their size is internally driven by a dichotomous noise, randomly switching ("breathing") between growing and shrinking states with a predefined rate, as motivated by recent experiments on synthetic active colloids. The polydispersity of this dichotomous active responsive colloid (D-ARC) model can be tuned by the parameters of the noise. Utilizing stochastic computer simulations, we study crowding effects on the spatial distributions, relaxation times, and self-diffusion of dense suspensions of the D-ARCs. We find a substantial influence of the "built-in" intrinsic noise on the system's behavior, in particular, transitions from unimodal to bimodal size distributions for an increasing colloid density as well as intrinsic noise-modified diffusive translational dynamics. We conclude that controlling the noise of internal DOFs of a macromolecule or cell is a powerful tool for active colloidal materials to enable autonomous changes in the system's collective structure and dynamics towards the adaptation of macroscopic properties to external perturbations.

Inertialess gyrating engines

A typical model for a gyrating engine consists of an inertial wheel powered by an energy source that generates an angle-dependent torque. Examples of such engines include a pendulum with an externally applied torque, Stirling engines, and the Brownian gyrating engine. Variations in the torque are averaged out by the inertia of the system to produce limit cycle oscillations. While torque generating mechanisms are also ubiquitous in the biological world, where they typically feed on chemical gradients, inertia is not a property that one naturally associates with such processes. In the present work, seeking ways to dispense of the need for inertial effects, we study an inertia-less concept where the combined effect of coupled torque-producing components averages out variations in the ambient potential and helps overcome dissipative forces to allow sustained operation for vanishingly small inertia. We exemplify this inertia-less concept through analysis of two of the aforementioned engines, the Stirling engine, and the Brownian gyrating engine. An analogous principle may be sought in biomolecular processes as well as in modern-day technological engines, where for the latter, the coupled torque-producing components reduce vibrations that stem from the variability of the generated torque.

Tunable Brownian magneto heat pump

We propose a mesoscopic Brownian magneto heat pump made of a single charged Brownian particle that is steered by an external magnetic field. The particle is subjected to two thermal noises from two different heat sources. When confined, the particle performs gyrating motion around a potential energy minimum. We show that such a magneto-gyrator can be operated as both a heat engine and a refrigerator. The maximum power delivered by the engine and the performance of the refrigerator, namely the rate of heat transferred per unit external work, can be tuned and optimised by the applied magnetic field. Further tunability of the key properties of the engine, such as the direction of gyration and the torque exerted by the engine on the confining potential, is obtained by varying the strength and direction of the applied magnetic field. In principle, our predictions can be tested by experiments with colloidal particles and complex plasmas.

Active responsive colloids coupled to different thermostats

We introduce a model of active responsive colloids (ARCs) in which an internal degree of freedom (DoF) of a single colloidal particle is "activated" by coupling it to a different thermostat than for the translational DoFs. As for the responsive internal DoF, we consider specifically the size (diameter) of the spherical particles, which is confined by a harmonic parent potential being either entropic or energetic in nature. The ARCs interact via a repulsive Hertzian pair potential, appropriate to model hydrogels or elastic colloids, and are studied for various densities using Brownian dynamics simulations. We tune the internal activity in the nonequilibrium steady state by scanning through a wide range of internal temperatures, both smaller ("colder") and larger ("hotter") than the translational temperature. The results show a rich and intriguing behavior for the emergent property distributions, colloidal pair structure, and the diffusive translational dynamics controlled by the internal activity, substantially depending on whether the internal DoF moves in an entropic or energetic potential. We discuss theoretical thermal limits and phenomenological models which can explain some of the nonequilibrium trends qualitatively. Our study indicates that particle dynamical polydispersity as well as the structure and dynamics of dense macromolecular suspensions can be vastly tuned by internal activity in terms of internal "hot" or "cold" fluctuating states.

Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator

The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally relevant model of such a system-the so-called Brownian gyrator-a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under nonequilibrium conditions, while no net rotation takes place under equilibrium ones. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some "fingerprint" properties specific to the behavior in nonequilibrium. We show that indeed one can conclusively distinguish between equilibrium and nonequilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a nonequilibrium dynamics.

Energy harvesting from anisotropic fluctuations

We consider a rudimentary model for a heat engine, known as the Brownian gyrator, that consists of an overdamped system with two degrees of freedom in an anisotropic temperature field. Whereas the hallmark of the gyrator is a nonequilibrium steady-state curl-carrying probability current that can generate torque, we explore the coupling of this natural gyrating motion with a periodic actuation potential for the purpose of extracting work. We show that path lengths traversed in the manifold of thermodynamic states, measured in a suitable Riemannian metric, represent dissipative losses, while area integrals of a work density quantify work being extracted. Thus, the maximal amount of work that can be extracted relates to an isoperimetric problem, trading off area against length of an encircling path. We derive an isoperimetric inequality that provides a universal bound on the efficiency of all cyclic operating protocols, and a bound on how fast a closed path can be traversed before it becomes impossible to extract positive work. The analysis presented provides guiding principles for building autonomous engines that extract work from anisotropic fluctuations.

Tethered tracer in a mixture of hot and cold Brownian particles: can activity pacify fluctuations?

We study how an interacting mixture of components with differing levels of activity can affect the fluctuations of an embedded object such as a tracer. In particular, we consider a simple model of a tracer that is harmonically bound within a mixture of hot and cold Brownian particles, which, like a mixture of active and passive particles, can phase separate. By measuring the fluctuations of the tracer, we find that this collective behavior gives rise to an effective temperature for the tracer. Additionally, we find that there is an increased tendency for cold particles to accumulate on the surface of the tracer due to the hot particles, potentially dampening its fluctuations and decreasing its effective temperature. These results suggest that the phase separation of a mixture of hot/cold or active/passive particles may have strong effects on the fluctuations of an embedded object. We discuss potential implications of these results for experiments on fluctuations of nuclear envelope affected by the activity in the chromatin.

Landscape-Flux Framework for Nonequilibrium Dynamics and Thermodynamics of Open Hamiltonian Systems Coupled to Multiple Heat Baths

We establish a nonequilibrium dynamic and thermodynamic formalism in the landscape-flux framework for open Hamiltonian systems in contact with multiple heat baths governed by stochastic dynamics. To systematically characterize nonequilibrium steady states, the nonequilibrium trinity construct is developed, which consists of detailed balance breaking, nonequilibrium potential landscape, and irreversible probability flux. We demonstrate that the temperature difference of the heat baths is the physical origin of detailed balance breaking, which generates the nonequilibrium potential landscape characterizing the nonequilibrium statistics and creates the irreversible probability flux signifying time irreversibility, with the latter two aspects closely connected. It is shown that the stochastic dynamics of the system can be formulated in the landscape-flux form, where the reversible force drives the conservative Hamiltonian dynamics, the irreversible force consisting of a landscape gradient force and an irreversible flux force drives the dissipative dynamics, and the stochastic force adds random fluctuations to the dynamics. The possible connection of the nonequilibrium trinity construct to nonequilibrium phase transitions is also suggested. A set of nonequilibrium thermodynamic equations, applicable to both nonequilibrium steady states and transient relaxation processes, is constructed. We find that an additional thermodynamic quantity, named the mixing entropy production rate, enters the nonequilibrium thermodynamic equations. It arises from the interplay between detailed balance breaking and transient relaxation, and it also relies on the conservative dynamics. At the nonequilibrium steady state, the heat flow, entropy flow, and entropy production are demonstrated to be thermodynamic manifestations of the nonequilibrium trinity construct. The general nonequilibrium formalism is applied to a class of solvable systems consisting of coupled harmonic oscillators. A more specific example of two harmonic oscillators coupled to two heat baths is worked out in detail. The example may facilitate connection with experiments.

Inertial effects on the Brownian gyrator

The recent interest into the Brownian gyrator has been confined chiefly to the analysis of Brownian dynamics both in theory and experiment despite the applicability of general cases with definite mass. Considering mass explicitly in the solution of the Fokker-Planck equation and Langevin dynamics simulations, we investigate how inertia can change the dynamics and energetics of the Brownian gyrator. In the Langevin model, the inertia reduces the nonequilibrium effects by diminishing the declination of the probability density function and the mean of a specific angular momentum, j_{θ}, as a measure of rotation. Another unique feature of the Langevin description is that rotation is maximized at a particular anisotropy while the stability of the rotation is minimized at a particular anisotropy or mass. Our results suggest that the Langevin dynamics description of the Brownian gyrator is intrinsically different from that with Brownian dynamics. In addition, j_{θ} is proven to be essential and convenient for estimating stochastic energetics such as heat currents and entropy production even in the underdamped regime.

Autonomous Brownian gyrators: A study on gyrating characteristics

We study the nonequilibrium steady-state (NESS) dynamics of two-dimensional Brownian gyrators under harmonic and nonharmonic potentials via computer simulations and analyses based on the Fokker-Planck equation, while our nonharmonic cases feature a double-well potential and an isotropic quartic potential. In particular, we report two simple methods that can help understand gyrating patterns. For harmonic potentials, we use the Fokker-Planck equation to survey the NESS dynamical characteristics; i.e., the NESS currents gyrate along the equiprobability contours and the stationary point of flow coincides with the potential minimum. As a contrast, the NESS results in our nonharmonic potentials show that these properties are largely absent, as the gyrating patterns are very distinct from those of corresponding probability distributions. Furthermore, we observe a critical case of the double-well potential, where the harmonic contribution to the gyrating pattern becomes absent, and the NESS currents do not circulate about the equiprobability contours near the potential minima even at low temperatures.

Engineered swift equilibration of a Brownian gyrator

In the context of stochastic thermodynamics, a minimal model for nonequilibrium steady states has been recently proposed: the Brownian gyrator (BG). It describes the stochastic overdamped motion of a particle in a two-dimensional harmonic potential, as in the classic Ornstein-Uhlenbeck process, but considering the simultaneous presence of two independent thermal baths. When the two baths have different temperatures, the steady BG exhibits a rotating current, a clear signature of nonequilibrium dynamics. Here, we consider a time-dependent potential, and we apply a reverse-engineering approach to derive exactly the required protocol to switch from an initial steady state to a final steady state in a finite time τ. The protocol can be built by first choosing an arbitrary quasistatic counterpart, with few constraints, and then adding a finite-time contribution which only depends upon the chosen quasistatic form and which is of order 1/τ. We also get a condition for transformations which, in finite time, conserve internal energy, useful for applications such as the design of microscopic thermal engines. Our study extends finite-time stochastic thermodynamics to transformations connecting nonequilibrium steady states.

Three-body problem for Langevin dynamics with different temperatures

A mixture of Brownian particles at different temperatures has been a useful model for studying the out-of-equilibrium properties of systems made up of microscopic components with differing levels of activity. This model was previously studied analytically for two-particle interactions in the dilute limit, yielding a Boltzmann-like two-particle distribution with an effective temperature. Like the Newtonian two- and three-body problems, we ask here whether the two-particle results can be extended to three-particle interactions to get the three-particle distributions. By considering the special solvable case of pairwise quadratic interactions, we show that, unlike the two-particle distribution, the three-particle distribution cannot in general be Boltzmann-like with an effective temperature. We instead find that the steady-state distribution of any two particles in a triplet depends on the properties of and interactions with the third particle, leading to some unexpected behaviors not present in equilibrium.

Approach to equilibrium and nonequilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths

A Hamiltonian-based model of many harmonically interacting massive particles that are subject to linear friction and coupled to heat baths at different temperatures is used to study the dynamic approach to equilibrium and nonequilibrium stationary states. An equilibrium system is here defined as a system whose stationary distribution equals the Boltzmann distribution, the relation of this definition to the conditions of detailed balance and vanishing probability current is discussed both for underdamped as well as for overdamped systems. Based on the exactly calculated dynamic approach to the stationary distribution, the functional that governs this approach, which is called the free entropy S_{free}(t), is constructed. For the stationary distribution S_{free}(t) becomes maximal and its time derivative, the free entropy production S[over ̇]_{free}(t), is minimal and vanishes. Thus, S_{free}(t) characterizes equilibrium as well as nonequilibrium stationary distributions by their extremal and stability properties. For an equilibrium system, i.e., if all heat baths have the same temperature, the free entropy equals the negative free energy divided by temperature and thus corresponds to the Massieu function which was previously introduced in an alternative formulation of statistical mechanics. Using a systematic perturbative scheme for calculating velocity and position correlations in the overdamped massless limit, explicit results for few particles are presented: For two particles localization in position and momentum space is demonstrated in the nonequilibrium stationary state, indicative of a tendency to phase separate. For three elastically interacting particles heat flows from a particle coupled to a cold reservoir to a particle coupled to a warm reservoir if the third reservoir is sufficiently hot. This does not constitute a violation of the second law of thermodynamics, but rather demonstrates that a particle in such a nonequilibrium system is not characterized by an effective temperature which equals the temperature of the heat bath it is coupled to. Active particle models can be described in the same general framework, which thereby allows us to characterize their entropy production not only in the stationary state but also in the approach to the stationary nonequilibrium state. Finally, the connection to nonequilibrium thermodynamics formulations that include the reservoir entropy production is discussed.

Nonequilibrium probability flux of a thermally driven micromachine

We discuss the nonequilibrium statistical mechanics of a thermally driven micromachine consisting of three spheres and two harmonic springs [Y. Hosaka et al., J. Phys. Soc. Jpn. 86, 113801 (2017)JUPSAU0031-901510.7566/JPSJ.86.113801]. We obtain the nonequilibrium steady state probability distribution function of such a micromachine and calculate its probability flux in the corresponding configuration space. The resulting probability flux can be expressed in terms of a frequency matrix that is used to distinguish between a nonequilibrium steady state and a thermal equilibrium state satisfying detailed balance. The frequency matrix is shown to be proportional to the temperature difference between the spheres. We obtain a linear relation between the eigenvalue of the frequency matrix and the average velocity of a thermally driven micromachine that can undergo a directed motion in a viscous fluid. This relation is consistent with the scallop theorem for a deterministic three-sphere microswimmer.

Two-temperature Brownian dynamics of a particle in a confining potential

We consider the two-dimensional motion of a particle in a confining potential, subject to Brownian orthogonal forces associated with two different temperatures. Exact solutions are obtained for an asymmetric harmonic potential in the overdamped and underdamped regimes. For more general confining potentials, a perturbative approach shows that the stationary state exhibits some universal properties. The nonequilibrium stationary state is characterized with a nonzero orthoradial mean current, corresponding to a global rotation of the particle around the center. The rotation is due to two broken symmetries: two different temperatures and a mismatch between the principal axes of the confining asymmetric potential and the temperature axes. We confirm our predictions by performing a Brownian dynamics simulation. Finally, we propose to observe this effect on a laser-cooled atomic gas.

Overdamped stochastic thermodynamics with multiple reservoirs

After establishing stochastic thermodynamics for underdamped Langevin systems in contact with multiple reservoirs, we derive its overdamped limit using timescale separation techniques. The overdamped theory is different from the naive theory that one obtains when starting from overdamped Langevin or Fokker-Planck dynamics and only coincides with it in the presence of a single reservoir. The reason is that the coarse-grained fast momentum dynamics reaches a nonequilibrium state, which conducts heat in the presence of multiple reservoirs. The underdamped and overdamped theory are both shown to satisfy fundamental fluctuation theorems. Their predictions for the heat statistics are derived analytically for a Brownian particle on a ring in contact with two reservoirs and subjected to a nonconservative force and are shown to coincide in the long-time limit.

Theoretical description of effective heat transfer between two viscously coupled beads

We analytically study the role of nonconservative forces, namely viscous couplings, on the statistical properties of the energy flux between two Brownian particles kept at different temperatures. From the dynamical model describing the system, we identify an energy flow that satisfies a fluctuation theorem both in the stationary and in transient states. In particular, for the specific case of a linear nonconservative interaction, we derive an exact fluctuation theorem that holds for any measurement time in the transient regime, and which involves the energy flux alone. Moreover, in this regime the system presents an interesting asymmetry between the hot and cold particles. The theoretical predictions are in good agreement with the experimental results already presented in our previous article [Imparato et al., Phys. Rev. Lett. 116, 068301 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.068301], where we investigated the thermodynamic properties of two Brownian particles, trapped with optical tweezers, interacting through a dissipative hydrodynamic coupling.

Nonequilibrium statistical mechanics of mixtures of particles in contact with different thermostats

We introduce a novel type of locally driven systems made of two types of particles (or a polymer with two types of monomers) subject to a chaotic drive with approximately white noise spectrum, but different intensity; in other words, particles of different types are in contact with thermostats at different temperatures. We present complete systematic statistical mechanics treatment starting from first principles. Although we consider only corrections to the dilute limit due to pairwise collisions between particles, meaning we study a nonequilibrium analog of the second virial approximation, we find that the system exhibits a surprisingly rich behavior. In particular, pair correlation function of particles has an unusual quasi-Boltzmann structure governed by an effective temperature distinct from that of any of the two thermostats. We also show that at sufficiently strong drive the uniformly mixed system becomes unstable with respect to steady states consisting of phases enriched with different types of particles. In the second virial approximation, we define nonequilibrium "chemical potentials" whose gradients govern diffusion fluxes and a nonequilibrium "osmotic pressure," which governs the mechanical stability of the interface.