When do non-equilibrium forms of disordered energy qualify as heat? \textcolor{blue}{We address this question in the context of cyclically operating heat engines in contact with a non-equilibrium energy reservoir that defies the zeroth law of thermodynamics. To consistently address the latter as a heat bath requires the existence of a precise mapping to an equivalent cycle with an equilibrium bath at a time-dependent effective temperature. We identify the most general setup for which this can generically be ascertained and thoroughly discuss an analytically tractable, experimentally relevant scenario}: a Brownian particle confined in a \textcolor{blue}{periodically} modulated harmonic potential and coupled to some non-equilibrium bath of variable activity. We deduce formal limitations for its thermodynamic performance, including maximum efficiency, efficiency at maximum power, and maximum efficiency at fixed power. They can guide the design of new micro-machines and clarify how much these can outperform passive-bath designs, which has been a debated issue for recent experimental realizations. To illustrate the general principles for practical quasi-static and finite-rate protocols, we further analyze a specific realization of such an active heat engine based on the paradigmatic Active Brownian Particle (ABP) model. This reveals some non-intuitive features of the explicitly computed dynamical effective temperature, illustrates various conceptual and practical limitations of the effective-equilibrium mapping, and clarifies the operational relevance of various coarse-grained measures of dissipation.
We present the exact analysis of a chiral Brownian motor and heat pump. Optimization of the construction predicts, for a nanoscale device, frequencies of the order of kHz and cooling rates of the order of femtojoule per second.
A Brownian particle moving in the vicinity of a generic potential minimum under the influence of dissipation and thermal noise from two different heat baths is shown to act as a minimal heat engine, generating a systematic torque onto the physical object at the origin of the potential and an opposite torque onto the medium generating the dissipation.
Heat fluctuation probability distribution function in Brownian transducers operating between two heat reservoirs is studied. We find, both analytically and numerically, that the recently proposed Fluctuation Theorem for Heat Exchange [C. Jarzynski and D. K. Wojcik, Phys. Rev. Lett. 92, 230602 (2004)] has to be modified when the coupling mechanism between both baths is considered. We also extend such relation when external work is present. Our work fixes the domain of applicability of the theorem in more realistic operating systems.
A Brownian information engine is a device extracting a mechanical work from a single heat bath by exploiting the information on the state of a Brownian particle immersed in the bath. As for engines, it is important to find the optimal operating condition that yields the maximum extracted work or power. The optimal condition for a Brownian information engine with a finite cycle time $τ$ has been rarely studied because of the difficulty in finding the nonequilibrium steady state. In this study, we introduce a model for the Brownian information engine and develop an analytic formalism for its ste…
Understanding the statistical behavior of the heat in stochastic systems gives us insight about the thermodynamics of such systems. Using the recently proposed Relativistic Stochastic Thermodynamics, we investigate the statistics of the heat of a Relativistic Ornstein-Uhlenbeck particle, comparing with the classical cases. The results are exact through numerical integration of the Fokker-Planck of the joint distribution, and are validated by numerical simulations.
We report on a lossless information engine that converts nearly all available information from an error-free feedback protocol into mechanical work. Combining high-precision detection at resolution of 1 nm with ultrafast feedback control, the engine is tuned to extract the maximum work from information on the position of a Brownian particle. We show that the work produced by the engine achieves a bound set by a generalized second law of thermodynamics, demonstrating for the first time the sharpness of this bound. We validate a generalized Jarzynski equality for error-free feedback-controlled i…
Using a time-averaging technique we obtain exactly the probability distribution for position and velocity of a Brownian particle under the influence of two heat baths at different temperatures. These baths are expressed by a white noise term, representing the fast dynamics, and a colored noise term, representing the slow dynamics. Our exact solution scheme accounts for inertial effects, that are not present in approaches that assume the Brownian particle in the over-damped limit. We are also able to obtain the contribution associated with the fast noise that are usually neglected by other appr…
Brownian information engines can extract work from thermal fluctuations by utilizing information. So far, the studies on Brownian information engines consider the system in a thermal bath; however, many processes in nature occur in a nonequilibrium setting, such as the suspensions of self-propelled microorganisms or cellular environments called an active bath. Here, we introduce an archetypal model for Maxwell-demon type cyclic Brownian information engine operating in a Gaussian correlated active bath. The active engine can extract more work than its thermal counterpart, exceeding the bound se…
We apply the concept of a frequency-dependent effective temperature based on the fluctuation-dissipation ratio to a driven Brownian particle in a nonequilibrium steady state. Using this system as a thermostat for a weakly coupled harmonic oscillator, the oscillator thermalizes according to a canonical distribution at the respective effective temperature across the entire frequency spectrum. By turning the oscillator from a passive "thermometer" into a heat engine, we realize the cyclic extraction of work from a single thermal reservoir, which is feasible only due to its nonequilibrium nature.
The energetics of a Brownian heat engine and heat pump driven by position dependent temperature, known as the Büttiker-Landauer heat engine and heat pump, is investigated by numerical simulations of the inertial Langevin equation. We identify parameter values for optimal performance of the heat engine and heat pump. Our results qualitatively differ from approaches based on the overdamped model. The behavior of the heat engine and heat pump, in the linear response regime is examined under finite time conditions and we find that the efficiency is lower than that of an endoreversible engine worki…
Shortcuts to isothermality provide a powerful method to speed up quasistatic thermodynamic processes within finite-time manipulation. We employ the shortcut strategy to design and optimize Brownian heat engines, and formulate a geometric description of the energetics with the thermodynamic length. We obtain a tight and reachable bound of the output power, which is reached by the optimal protocol to vary the control parameters with a proper constant velocity of the thermodynamic length. Our results generalize the previous optimization in the highly underdamped and the overdamped regimes to the …
A microscopic heat engine is modeled as a Brownian particle in a sawtooh potential (with load) moving through a highly viscous medium driven by the thermal kick it gets from alternately placed hot and cold heat reservoirs. We found closed form expression for the current as a function of the parameters characterizing the model. Depending on the values these model parameters take, the engine is also found to function as a refrigerator. Expressions for the efficiency as well as for the refrigerator performance are also reported. Study of how these quantities depend on the model parameters enabled…
Micro-to-nano scale thermal devices that operate under large thermal fluctuations, are an active field of research where instead the average values, the full distributions of thermodynamic quantities are important. Here we study a model of stochastic heat engine consisting of a harmonically trapped Brownian particle driven by the time-periodic strength of the confinement, within two thermal baths of different temperatures. The particle follows two isotherms correspond to two baths and connected by two micro-adiabates. The microadiabaticity is implemented by conserving the phase space volume of…
The energy partitioning during activation and relaxation events under steady-state conditions for a Brownian particle driven by multiple thermal reservoirs of different local temperatures is investigated. Specifically, we apply the formalism derived in a previous article [G. T. Craven and A. Nitzan, J. Chem. Phys. 148, 044101 (2018)] to examine the thermal transport properties of two sub-ensembles of Brownian processes, distinguished at any given time by the specification that all the trajectories in each group have, at that time, energy either above (upside) or below (downside) a preselected …
Using the Onsager-Machlup functional integral approach, we obtain the work distribution function and the distribution of the dissipated heat of a Brownian particle subjected to a confining harmonic potential and an oscillatory driving force. In the long time limit, the width of the work distribution function initially increases with the frequency of the driving force and finally saturates to a fixed value for large values of the angular frequency. Using the results from the work distribution part, we next obtain the distribution of the dissipated heat for the equilibrium initial condition. Usi…
We model a Brownian heat engine as a Brownian particle that hops in a periodic ratchet potential where the ratchet potential is coupled with a linearly decreasing background temperature. It is shown that the efficiency of such Brownian heat engine is far from Carnot efficiency even at quaistatic limit. At quasistatic limit, the efficiency of the heat engine approaches the efficiency of endoreversible engine $η=1-\sqrt{{T_{c}/T_{h}}}$ \cite{c18}. On the other hand, the maximum power efficiency of the engine approaches $η^{MAX}=1-({T_{c}/T_{h}})^{1\over 4}$. Moreover, the dependence of the curre…
We study the Quantum Brownian motion of a charged particle moving in a harmonic potential in the presence of an uniform external magnetic field and linearly coupled to an Ohmic bath through momentum variables. We analyse the growth of the mean square displacement of the particle in the classical high temperature domain and in the quantum low temperature domain dominated by zero point fluctuations. We also analyse the Position Response Function and the long time tails of various correlation functions. We notice some distinctive features, different from the usual case of a charged quantum Browni…
Active Brownian engines rectify energy from reservoirs composed of self-propelling non-equilibrium molecules into work. We consider a class of such engines based on an underdamped Brownian particle trapped in a power-law potential. The energy they transform has thermodynamic properties of heat only if the non-equilibrium reservoir can be assigned a suitable effective temperature consistent with the second law and thus yielding an upper bound on the engine efficiency. The effective temperature exists if the total force exerted on the particle by the bath is not correlated with the particle posi…
Two losing games, when alternated in a periodic or random fashion, can produce a winning game. This paradox occurs in a family of stochastic processes: if one combines two or more dynamics where a given quantity decreases, the result can be a dynamic system where this quantity increases. The paradox could be applied to a number of stochastic systems and has drawn the attention of researchers from different areas. In this paper we show how the phenomenon can be used to design Brownian or molecular motors, i.e., thermal engines that operate by rectifying fluctuations. We briefly review the liter…
The governing equations of Brownian rigid bodies that both translate and rotate are of interest in fields such as self-assembly of proteins, anisotropic colloids, dielectric theory, and liquid crystals. In this paper, the partial differential equation that describes the evolution of concentration is derived from the stochastic differential equation of a sphere experiencing Brownian motion in a viscous medium where a potential field may be present. The potential field may be either interactions between particles or applied externally. The derivation is performed once for particles whose orienta…
We explore the transport features of a Brownian particle that walks in a periodic ratchet potential that is coupled with a spatially varying temperature background. Since the viscous friction of the medium decreases as the temperature of the medium increases, any reasonable exploration regarding the thermodynamic features of the Brownian engine should take into account the role of temperature on the viscosity of the fluid. In this work, we study this effect of temperature by considering a viscous friction that decreases exponentially as the background temperature increases. Our result depicts …
We investigate directed thermal heat flux across 1D homogenous nonlinear lattices when no net thermal bias is present on average. A nonlinear lattice of Fermi-Pasta-Ulam-type or Lennard-Jones-type system is connected at both ends to thermal baths which are held at the same temperature on temporal average. We study two different modulations of the heat bath temperatures, namely: (i) a symmetric, harmonic ac-driving of temperature of one heat bath only and (ii) a harmonic mixing drive of temperature acting on both heat baths. While for case (i) an adiabatic result for the net heat transport can …
We discuss the reversibility of Brownian heat engine. We perform asymptotic analysis of Kramers equation on Büttiker-Landauer system and show quantitatively that Carnot efficiency is inattainable even in a fully overdamping limit. The inattainability is attributed to the inevitable irreversible heat flow over the temperature boundary.
We discuss two-dimensional diffusion of a Brownian particle confined to a periodic asymmetric channel with soft walls modeled by a parabolic potential. In the channel, the particle experiences different noise intensities, or temperatures, in the transversal and longitudinal directions. The model is inspired by the famous Feynman's ratchet and pawl. Using ideas of the Fick-Jacobs approximation we derive the mean velocity of the particle, the effective diffusion coefficient, and the stationary probability density in the potential unit cell. The derived results are exact for small channel width. …
We perform an extensive analysis of passive as well as active micro-heat engines with different single-particle stochastic models. Using stochastic thermodynamics we calculate thermodynamic work, heat, entropy production and efficiency of passive and active Brownian heat engines analytically as well as numerically and compare them. We run the heat engines with a protocol for which the average thermodynamic quantities are calculated exactly for an arbitrary cycle time. We also discuss about the group of protocols for which exact non-quasistatic calculations can be done, completely in the passiv…
Quantum Brownian motion, described by the Caldeira-Leggett model, brings insights to understand phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classic…
Brownian heat engines use local temperature gradients in asymmetric potentials to move particles against an external force. The energy efficiency of such machines is generally limited by irreversible heat flow carried by particles that make contact with different heat baths. Here we show that, by using a suitably chosen energy filter, electrons can be transferred reversibly between reservoirs that have different temperatures and electrochemical potentials. We apply this result to propose heat engines based on mesoscopic semiconductor ratchets, which can quasistatically operate arbitrarily clos…
Recent advances in experimental control of colloidal systems have spurred a revolution in the production of mesoscale thermodynamic devices. Functional "textbook" engines, such as the Stirling and Carnot cycles, have been produced in colloidal systems where they operate far from equilibrium. Simultaneously, significant theoretical advances have been made in the design and analysis of such devices. Here, we use methods from thermodynamic geometry to characterize the optimal finite-time, nonequilibrium cyclic operation of the parametric harmonic oscillator contact with a time-varying heat bath, …
We discuss the effect of subdividing the ratchet potential on the performance of a tiny Brownian heat engine that is modeled as a Brownian particle hopping in a viscous medium in a sawtooth potential (with or without load) assisted by alternately placed hot and cold heat baths along its path. We show that the velocity, the efficiency and the coefficient of performance of the refrigerator maximize when the sawtooth potential is subdivided into series of smaller connected barrier series. When the engine operates quasistatically, we analytically show that the efficiency of the engine can not appr…
The theory of constructing instantaneous equilibrium (ieq) transition under arbitrary time-dependent temperature and potential variation for a Brownian particle is developed. It is shown that it is essential to consider the underdamped dynamics for temperature-changing transitions. The ieq is maintained by a time-dependent auxiliary position and momentum potential, which can be calculated for given time-dependent transition protocols. Explicit analytic results are derived for the work and heat statistics, energy, and entropy changes for harmonic and non-harmonic trapping potential with arbitra…
We study analytically the probability distribution of the heat released by an ensemble of harmonic oscillators to the thermal bath, in the nonequilibrium relaxation process following a temperature quench. We focus on the asymmetry properties of the heat distribution in the nonstationary dynamics, in order to study the forms taken by the Fluctuation Theorem as the number of degrees of freedom is varied. After analysing in great detail the cases of one and two oscillators, we consider the limit of a large number of oscillators, where the behavior of fluctuations is enriched by a condensation tra…
When a Brownian particle in contact with a heat bath at a constant temperature is controlled by a time-dependent harmonic potential, its distribution function can be rigorously derived from the Kramers equation with the consideration of the inertial effect of the Brownian particle. Based on this rigorous solution and the concept of shortcuts to adiabaticity, we construct a stochastic heat engine by employing the time-dependent harmonic potential to manipulate the Brownian particle to complete a thermodynamic cycle. We find that the efficiency at maximum power of this stochastic heat engine is …
The Carnot cycle imposes a fundamental upper limit to the efficiency of a macroscopic motor operating between two thermal baths. However, this bound needs to be reinterpreted at microscopic scales, where molecular bio-motors and some artificial micro-engines operate. As described by stochastic thermodynamics, energy transfers in microscopic systems are random and thermal fluctuations induce transient decreases of entropy, allowing for possible violations of the Carnot limit. Despite its potential relevance for the development of a thermodynamics of small systems, an experimental study of micro…
We study the non-equilibrium thermodynamics of single Brownian macromolecules immersed in water solvent. They are under both a hydrodynamic interaction and a feedback control on their movement by an external agent. The macromolecules are described by a Langevin equation with a multiplicative noise. Work done by the macromolecules on the water solvent is dissipated as heat. Thus, the heat is expressed as the integration of an interacting force between the macromolecules and the water solvent along the position space trajectories of the macromolecules. This integration is stochastic due to the B…
We analyze the equations governing the evolution of distributions of the work and the heat exchanged with the environment by a manipulated stochastic system, by means of a compact and general derivation. We obtain explicit solutions for these equations for the case of a dragged Brownian particle in a harmonic potential. We successfully compare the resulting predictions with the outcomes of experiments, consisting in dragging a micron-sized colloidal particle through water with a laser trap.
Work can be extracted from a single bath beyond the limit set by the second law by performing measurement on the system and utilising the acquired information. As an example we studied a Brownian particle confined in a two dimensional harmonic trap in presence of magnetic field, whose position co-ordinates are measured with finite precision. Two separate cases are investigated in this study - (A) moving the center of the potential and (B) varying the stiffness of the potential. Optimal protocols which extremise the work in a finite time process are explicitly calculated for these two cases. Fo…
We investigate stochastic thermodynamics of a two-particles Langevin system. Each particle is in contact with a heat bath at different temperatures $T_1$ and $T_2~(<T_1)$, respectively. Particles are trapped by a harmonic potential and driven by a linear external force. The system can act as an autonomous heat engine performing work against the external driving force. Linearity of the system enables us to examine thermodynamic properties of the engine analytically. We find that the efficiency of the engine at maximum power $η_{MP}$ is given by $η_{MP} = 1-\sqrt{T_2/T_1}$. This universal form h…
We investigate the stochastic behavior of a two-temperature Langevin system with non-Markovian thermal reservoirs. The model describes an overdamped Brownian particle in a quadratic potential and coupled to heat baths at different temperatures. The reservoirs are characterized by Gaussian white and colored noises and a dissipation memory kernel. The stationary states present non-trivial average rotational motion influenced by stochastic torques due to harmonic, friction and fluctuating thermal forces. However, the Markovian limit leads to a vanishing average torque produced by fluctuating ther…
Achieving the Carnot efficiency at finite power is a challenging problem in heat engines due to the trade-off relation between efficiency and power that holds for general heat engines. It is pointed out that the Carnot efficiency at finite power may be achievable in the vanishing limit of the relaxation times of a system without breaking the trade-off relation. However, any explicit model of heat engines that realizes this scenario for arbitrary temperature difference has not been proposed. Here, we investigate an underdamped Brownian Carnot cycle where the finite-time adiabatic processes conn…
We propose a new look at the heat bath for two Brownian particles, in which the heat bath as a `system' is both perturbed and sensed by the Brownian particles. Non-local thermal fluctuation give rise to bath-mediated static forces between the particles. Based on the general sum-rule of the linear response theory, we derive an explicit relation linking these forces to the friction kernel describing the particles' dynamics. The relation is analytically confirmed in the case of two solvable models and could be experimentally challenged. Our results point out that the inclusion of the environment …
We study the influence of entanglement on the relation between the statistical entropy of an open quantum system and the heat exchanged with a low temperature environment. A model of quantum Brownian motion of the Caldeira-Leggett type - for which a violation of the Clausius inequality has been stated by Th.M. Nieuwenhuizen and A.E. Allahverdyan [Phys. Rev. E 66, 036102 (2002)] - is reexamined and the results of the cited work are put into perspective. In order to address the problem from an information theoretical viewpoint a model of two coupled Brownian oscillators is formulated that can al…
We introduce an alternative route for obtaining reliable cyclic engines, based on interacting Brownian particles under time-periodic drivings. General expressions for the thermodynamic fluxes, such as power and heat, are obtained using the framework of Stochastic Thermodynamics. Several protocols for optimizing the engine performance are considered, by looking at system parameters such as the output forces and their phase-difference. We study both work-to-work and heat-to-work engines. Our results suggest that carefully designed interactions between particles can lead to more efficient engines…
We review a series of experimental studies of the thermodynamics of nonequilibrium processes at the microscale. In particular, in these experiments we studied the fluctuations of the thermodynamic properties of a single optically-trapped microparticle immersed in water and in the presence of external random forces. In equilibrium, the fluctuations of the position of the particle can be described by an effective temperature that can be tuned up to thousands of Kelvins. Isothermal and non-isothermal thermodynamic processes that also involve changes in a control parameter were implemented by cont…
We investigate a model for a Stirling-like engine consisting of a passive Brownian particle confined by a harmonic potential and interacting with a suspension of active Brownian particles that self-propel in a viscous solvent, which cyclically operates under isothermal conditions by means of temporal variations of the trap stiffness and the self-propulsion speed of the active particles. We derive an effective stochastic equation of motion of the trapped Brownian particle, which includes a friction memory kernel as well as thermal and active fluctuating forces due to its coupling with the activ…
Methods of stochastic thermodynamics and hydrodynamics are applied to the a recently introduced model of active particles. The model consists of an overdamped particle subject to Gaussian coloured noise. Inspired by stochastic thermodynamics, we derive from the system's Fokker-Planck equation the average exchanges of heat and work with the active bath and the associated entropy production. We show that a Clausius inequality holds, with the local (non-uniform) temperature of the active bath replacing the uniform temperature usually encountered in equilibrium systems. Furthermore, by restricting…
We experimentally realize quasistatic adiabatic processes using a single optically-trapped micro- sphere immersed in water whose effective temperature is controlled by an external random electric field. A full energetic characterization of adiabatic processes that preserve either the position dis- tribution or the full phase space volume is presented. We show that only in the latter case the exchanged heat and the change in the entropy of the particle vanish when averaging over many repetitions. We provide analytical expressions for the distributions of the fluctuating heat and en- tropy, whic…
An alternative equilibrium stochastic dynamics for a Brownian particle in inhomogeneous space is derived. Such a dynamics can model the motion of a complex molecule in its conformation space when in equilibrium with a uniform heat bath. The derivation is done by a simple generalization of the formulation due to Zwanzig for a Brownian particle in homogeneous heat bath. We show that if the system couples to different number of bath degrees of freedom at different conformations then the alternative model is derived. We discuss results of an experiment by Faucheux and Libchaber which probably has …
The construction of efficient thermal engines operating at finite times constitutes a fundamental and timely topic in nonequilibrium thermodynamics. We introduce a strategy for optimizing the performance of Brownian engines, based on a collisional approach for unequal interaction times between the system and thermal reservoirs. General (and exact) expressions for thermodynamic properties and their optimized values are obtained, irrespective of the driving forces, asymmetry, the temperatures of reservoirs and protocol to be maximized. Distinct routes for the engine optimization, including maxim…
Typical heat engines exhibit a kind of homotypy: The heat exchanges between a cyclic heat engine and its two heat reservoirs abide by the same function type; the forward and backward flows of an autonomous heat engine also conform to the same function type. This homotypy mathematically reflects in the existence of hidden symmetries for heat engines. The heat exchanges between the cyclic heat engine and its two reservoirs are dual under the joint transformation of parity inversion and time-reversal operation. Similarly, the forward and backward flows in the autonomous heat engine are also dual …
We retrospect three abstract models for heat engines which include a classic abstract model in textbook of thermal physics, a primary abstract model for finite-time heat engines, and a refined abstract model for finite-time heat engines. The detailed models of heat engines in literature of finite-time thermodynamics may be mapped into the refined abstract model. The future developments based on the refined abstract model are also surveyed.
