We determine the maximal work extractable via a cyclic Hamiltonian process from a positive-temperature ($T>0$) microcanonical state of a $N\gg 1$ spin bath. The work is much smaller than the total energy of the bath, but can be still much larger than the energy of a single bath spin, e.g. it can scale as ${\cal O}(\sqrt{N\ln N})$. Qualitatively same results are obtained for those cases, where the canonical state is unstable (e.g., due to a negative specific heat) and the microcanonical state is the only description of equilibrium. For a system coupled to a microcanonical bath the concept of free energy does {\it not generally} apply, since such a system|starting from the canonical equilibrium density matrix $ρ_T$ at the bath temperature $T$|can enhance the work extracted from the microcanonical bath without changing its state $ρ_T$. This is impossible for any system coupled to a canonical thermal bath due to the relation between the maximal work and free energy. But the concept of free energy still applies for a sufficiently large $T$. Here we find a compact expression for the {\it microcanonical free-energy} and show that in contrast to the canonical case it contains a {\it linear entropy} instead of the von Neumann entropy.
On the basis of analytical results and molecular dynamics simulations we clarify the nonequilibrium dynamics of a long-range interacting system in contact with a heat bath. For small couplings with the bath, we show that the system can first be trapped in a Vlasov quasi-stationary state, then a microcanonical one follows, and finally canonical equilibrium is reached at the bath temperature. We compare a Langevin mesoscopic thermostat with Hamiltonian reservoirs microscopically coupled with the system and demonstrate the equivalence of the two descriptions. Our identification of the parameters …
It is typically assumed, without justification, that a weak coupling between a system and a bath is a necessary condition for the equivalence of a canonical ensemble and a microcanonical ensemble. For instance, in a canonical ensemble, temperature emerges if the system and the bath are uncoupled or weakly coupled. We investigate the validity region of this weak coupling approximation, using a coupled composite-spin system. Our results show that the spin coupling strength can be as large as the level spacing of the system, indicating that the weak coupling approximation has a much wider region …
The grand canonical ensemble lies at the core of quantum and classical statistical mechanics. A small system thermalizes to this ensemble while exchanging heat and particles with a bath. A quantum system may exchange quantities represented by operators that fail to commute. Whether such a system thermalizes and what form the thermal state has are questions about truly quantum thermodynamics. Here we investigate this thermal state from three perspectives. First, we introduce an approximate microcanonical ensemble. If this ensemble characterizes the system-and-bath composite, tracing out the bat…
The theory of probability shows that, as the fraction $X_n/Y\to 0$, the conditional probability for $X_n$, given $X_n+Y \in h_δ:=[h, h+δ]$, has a limit law $f_{X_n}(x)e^{-ψ_n(h_δ)x}$, where $ψ_n(h_δ) $ equals to $[\partial \ln P(Y \in y_δ)/\partial y]_{y=h}$ plus an additional term, contributed from the correlation between $X_n$ and bath $Y$. By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form …
The canonical ensemble describes an open system in equilibrium with a heat bath of fixed temperature. The probability distribution of such a system, the Boltzmann distribution, is derived from the uniform probability distribution of the closed universe consisting of the open system and the heat bath, by taking the limit where the heat bath is much larger than the system of interest. Alternatively, the Boltzmann distribution can be derived from the Maximum Entropy Principle, where the Gibbs-Shannon entropy is maximized under the constraint that the mean energy of the open system is fixed. To ma…
We discuss the possibility of extracting energy from a single thermal bath using microcanonical Szilárd engines operating in finite time. This extends previous works on the topic which are restricted to the quasistatic regime. The feedback protocol is implemented based on linear response predictions of the excess work. It is claimed that the underlying mechanism leading to energy extraction does not violate Liouville's theorem and preserves ergodicity throughout the cycle. We illustrate our results with several examples including an exactly solvable model.
We consider closed quantum systems (into which baths may be integrated) that are driven, i.e., subject to time-dependent Hamiltonians. As a starting point we assume that, for systems initialized in microcanonical states at some energies, the resulting probability densities of work (work-PDFs) are largely independent of these specific initial energies. We show analytically that this assumption of "stiffness", together with the assumption of an exponentially growing density of energy eigenstates, is sufficient but not necessary for the validity of the Jarzynski relation (JR) for the above microc…
We investigate the limiting cases of Tsallis statistics. The viewpoint adopted is not the standard information-theoretic one, where one derives the distribution from a given measure of information. Instead the mechanical approach recently proposed in [M. Campisi, G.B. Bagci, Phys. Lett. A (2006), doi:10.1016/j.physleta.2006.09.081], is adopted, where the distribution is given and one looks for the associated physical entropy. We show that, not only the canonical ensemble is recovered in the limit of $q$ tending to one, as one expects, but also the microcanonical ensemble is recovered in the li…
Following recent work by Marathe and Parrondo [PRL, 104, 245704 (2010)], we construct a classical Hamiltonian system whose energy is reduced during the adiabatic cycling of external parameters, when initial conditions are sampled microcanonically. Combining our system with a device that measures its energy, we propose a cyclic procedure during which energy is extracted from a heat bath and converted to work, in apparent violation of the second law of thermodynamics. This paradox is resolved by deriving an explicit relationship between the average work delivered during one cycle of operation, a…
In a recent paper [Vaikuntanathan and Jarzynski, Phys. Rev. E {\bf 83}, 061120 (2011), arXiv:1105.1744] a model was introduced whereby work could be extracted from a thermal bath by measuring the energy of a particle that was thermalized by the bath and manipulating the potential of the particle in the appropriate way, depending on the measurement outcome. If the extracted work is $W_1$ and the work $W_{\text{er}}$ needed to be dissipated in order to erase the measured information in accordance with Landauer's principle, it was shown that $W_1\leq W_{\text{er}}$ in accordance with the second l…
We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different mi…
We calculate exactly both the microcanonical and canonical thermodynamic functions (TDFs) for a one-dimensional model system with piecewise constant Lennard-Jones type pair interactions. In the case of an isolated $N$-particle system, the microcanonical TDFs exhibit (N-1) singular (non-analytic) microscopic phase transitions of the formal order N/2, separating N energetically different evaporation (dissociation) states. In a suitably designed evaporation experiment, these types of phase transitions should manifest themselves in the form of pressure and temperature oscillations, indicating cool…
We consider locally thermal states (for two qubits) with certain amount of quantum entanglement present between them. Unlike previous protocols we show how work can be extracted by performing local unitary operations on this state by allowing those two qubits to interact with thermal baths of different temperatures, thereby gradually removing the entanglement between them till they reach a direct product state. Also we demonstrate that, further work can be extracted from this direct product state by performing global unitary operation, thereby establishing that work can be extracted from a sys…
It is well known that a system, S, weakly coupled to a heat bath, B, is described by the canonical ensemble when the composite, S+B, is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true both for classical distributions on the phase space and for quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subsp…
We show that work can be extracted from a two-level system (spin) coupled to a bosonic thermal bath. This is possible due to different initial temperatures of the spin and the bath, both positive (no spin population inversion) and is realized by means of a suitable sequence of sharp pulses applied to the spin. The extracted work can be of the order of the response energy of the bath, therefore much larger than the energy of the spin. Moreover, the efficiency of extraction can be very close to its maximum, given by the Carnot bound, at the same time the overall amount of the extracted work is m…
The stationary state of a quantum particle strongly coupled to a quantum thermal bath is known to be non-gibbsian, due to entanglement with the bath. For harmonic potentials, where the system can be described by effective temperatures, thermodynamic relations are shown to take a generalized Gibbsian form, that may violate the Clausius inequality. For the weakly-anharmonic case a Fokker-Planck type description is constructed. It is shown that then work can be extracted from the bath by cyclic variation of a parameter. These apparent violations of the second law are the consequence of quantum co…
The anisotropic quantum Heisenberg model with Curie-Weiss-type interactions is studied analytically in several variants of the microcanonical ensemble. (Non)equivalence of microcanonical and canonical ensembles is investigated by studying the concavity properties of entropies. The microcanonical entropy s(e,m) is obtained as a function of the energy e and the magnetization vector m in the thermodynamic limit. Since, for this model, e is uniquely determined by m, the same information can be encoded either in s(m) or s(e,m1,m2). Although these two entropies correspond to the same physical settin…
Inspired by the available examples of Microcanonical Szilárd Engines and by the original Szilárd Engine, we devise a system with two degrees of freedom whose ensemble average energy, starting with a microcanical ensemble, decreases after a cyclic variation of its external parameters. We use the Ergodic Adiabatic Theorem to motivate our cycle and numerical simulations to check the decrement in the average energy. We then compare our system to the aforementioned Szilárd Engines, Microcanonical or not, and speculate about symmetry breaking being the cause of energy extraction in cyclic processes,…
The spin-boson model, often used in NMR and ESR physics, quantum optics and spintronics, is considered in a solvable limit to model a spin one-half particle interacting with a bosonic thermal bath. By applying external pulses to a non-equilibrium initial state of the spin, work can be extracted from the thermalized bath. It occurs on the timescale $\T_2$ inherent to transversal (`quantum') fluctuations. The work (partly) arises from heat given off by the surrounding bath, while the spin entropy remains constant during a pulse. This presents a violation of the Clausius inequality and the Thomso…
We construct a finite bath with variable temperature for quantum thermodynamic simulations in which heat flows between a system $\mathcal{S}$ and the bath environment $\mathcal{E}$ in time evolution of an initial $\mathcal{SE}$ pure state. The bath consists of harmonic oscillators that are not necessarily identical. Baths of various numbers of oscillators are considered; a bath with five oscillators is used in the simulations. The bath has a temperature-like level distribution. This leads to definition of a system-environment microcanonical temperature $T_\mathcal{SE}(t)$ which varies with tim…
Recently Allahverdyan and Nieuwenhuizen (cond-mat/0006404) argued that the second law of thermodynamics may be violated in a quantum system as a "consequence of quantum coherence in the presence of the slightly off-equilibrium nature of the bath." By using a standard result about relative entropy, we prove rigorously that the second law is never violated (and, in particular, a perpetual motion of the second kind can never be realized) in quantum systems no matter how strong ``quantum coherence'' is or no matter how far one goes from equilibrium.
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 optimal efficiency of quantum (or classical) heat engines whose heat baths are $n$-particle systems is given by the information geometry and the strong large deviation. We give the optimal work extraction process as a concrete energy-preserving unitary time evolution among the heat baths and the work storage. We show that our optimal work extraction turns the disordered energy of the heat baths to the ordered energy of the work storage, by evaluating the ratio of the entropy difference to the energy difference in the heat baths and the work storage, respectively. By comparing the statistic…
A system in thermal equilibrium with a bath will generally be in an athermal state, if the system-bath coupling is strong. In some cases, it will be possible to extract work from that athermal state, after disconnecting the system from the bath. We use this observation to devise a battery charging and storing unit, simply consisting of a system, acting as the battery, and a bath. The charging cycle---connect, let thermalize, disconnect, extract work---requires very little external control and the charged state of the battery, being a part of global thermal equilibrium, can be maintained indefi…
The spin-boson model, often used in NMR and ESR physics, quantum optics and spintronics, is considered in a solvable limit to model a spin one-half particle interacting with a bosonic thermal bath. By applying external pulses to a non-equilibrium initial state of the spin, work can be extracted from the thermalized bath. It occurs on the timescale T_2 inherent to transversal (`quantum') fluctuations. The work (partly) arises from heat given off by the surrounding bath, while the spin entropy remains constant during a pulse. This presents a violation of the Clausius inequality and the Thomson f…
We study dipolarly coupled three dimensional spin systems in both the microcanonical and the canonical ensembles by introducing appropriate numerical methods to determine the microcanonical temperature and by realizing a canonical model of heat bath. In the microcanonical ensemble, we show the existence of a branch of stable antiferromagnetic states in the low energy region. Other metastable ferromagnetic states exist in this region: by externally perturbing them, an effective negative specific heat is obtained. In the canonical ensemble, for low temperatures, the same metastable states are un…
In our recent letter [1] we discussed that thermodynamics is violated in quantum Brownian motion beyond the weak coupling limit. In his comment, Tasaki [2] derives an inequality for the relative entropy and claims, without making any dynamical assumption, that the Clausius inequality is valid, thus contradicting our statements [1]. Here we point out that the claim is unfunded, since the author did not properly identify the concept of heat. Tasaki also applies the inequality to Thomson's formulation of the second law. This application is invalid as well, since the author did not correctly ident…
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 study work extraction (defined as the difference between the initial and the final energy) in noninteracting and (effectively) weakly interacting isolated fermionic quantum lattice systems in one dimension, which undergo a sequence of quenches and equilibration. The systems are divided in two parts, which we identify as the subsystem of interest and the bath. We extract work by quenching the on-site potentials in the subsystem, letting the entire system equilibrate, and returning to the initial parameters in the subsystem using a quasi-static process (the bath is never acted upon). We selec…
We have studied the properties of a classical $N_S$-body system coupled to a bath containing $N_B$-body harmonic oscillators, employing an $(N_S+N_B)$ model which is different from most of the existing models with $N_S=1$. We have performed simulations for $N_S$-oscillator systems, solving $2(N_S+N_B)$ first-order differential equations with $N_S \simeq 1 - 10$ and $N_B \simeq 10 - 1000$, in order to calculate the time-dependent energy exchange between the system and the bath. The calculated energy in the system rapidly changes while its envelope has a much slower time dependence. Detailed cal…
We revisit the problem of work extraction from a system in contact with a heat bath to a work storage system, and the reverse problem of state formation from a thermal system state in single-shot quantum thermodynamics. A physically intuitive and mathematically simple approach using only elementary majorization theory and matrix analysis is developed, and a graphical interpretation of the maximum extractable work, minimum work cost of formation, and corresponding single-shot free energies is presented. This approach provides a bridge between two previous methods based respectively on the conce…
We study work extraction processes mediated by finite-time interactions with an ambient bath -- \emph{partial thermalizations} -- as continuous time Markov processes for two-level systems. Such a stochastic process results in fluctuations in the amount of work that can be extracted and is characterized by the rate at which the system parameters are driven in addition to the rate of thermalization with the bath. We analyze the distribution of work for the case where the energy gap of a two-level system is driven at a constant rate. We derive analytic expressions for average work and lower bound…
It is demonstrated that Maxwell's demon can be used to allow a machine to extract energy from a heat bath by use of information that is processed by the demon at a remote location. The model proposed here effectively replaces transmission of energy by transmission of information. For that we use a feedback protocol that enables a net gain by stimulating emission in selected fluctuations around thermal equilibrium. We estimate the down conversion rate and the efficiency of energy extraction from the heat bath.
A colloidal particle embedded in a fluid can be used as a microscopic heat engine by means of a sequence of cyclic transformations imposed by an optical trap. We investigate a model for the operation of such kind of Brownian engines when the surrounding medium is viscoelastic, which endows the particle dynamics with memory friction. We analyze the effect of the relaxation time of the fluid on the performance of the colloidal engine under finite-time Stirling cycles. We find that, due to the frequency-dependence of the friction in viscoelastic fluids, the mean power delivered by the engine and …
The nonequilibrium thermodynamics feature of a Brownian motor is investigated by obtaining exact time-dependent solutions. This in turn enables us to investigate not only the long time property (steady-state) but also the short time the behavior of the system. The general expressions for the free energy, entropy production ${\dot e}_{p}(t)$ as well as entropy extraction ${\dot h}_{d}(t)$ rates are derived for a system that is genuinely driven out of equilibrium by time-independent force as well as by spatially varying thermal background. We show that for a system that operates between hot and …
The work content of non-equilibrium systems in relation to a heat bath is often analyzed in terms of expectation values of an underlying random work variable. However, we show that when optimizing the expectation value of the extracted work, the resulting extraction process is subject to intrinsic fluctuations, uniquely determined by the Hamiltonian and the initial distribution of the system. These fluctuations can be of the same order as the expected work content per se, in which case the extracted energy is unpredictable, thus intuitively more heat-like than work-like. This raises the questi…
Recently proposed information-exploiting systems designed to extract work from a single heat bath utilize temporal correlations on an input tape. We study how enforcing time-continuous dynamics, which is necessary to ensure the device is physically realizable, constrains possible designs and drastically diminishes efficiency. We show that these problems can be circumvented by means of applying an external, time-varying protocol. This turns the device from a "passive", free-running machine into an "actively" driven one.
We consider work extraction from $N$ copies of a quantum system. When the same work-extraction process is implemented on each copy, the relative size of fluctuations is expected to decay as $1/\sqrt{N}$. Here, we consider protocols where the copies can be processed collectively, and show that in this case work fluctuations can disappear exponentially fast in $N$. As a consequence, a considerable proportion of the average extractable work $\mathcal{W}$ can be obtained almost deterministically by globally processing a few copies of the state. This is derived in the two canonical scenarios for wo…
The second law of thermodynamics uses change in free energy of macroscopic systems to set a bound on performed work. Ergotropy plays a similar role in microscopic scenarios, and is defined as the maximum amount of energy that can be extracted from a system by a unitary operation. In this analysis, we quantify how much ergotropy can be induced on a system as a result of system's interaction with a thermal bath, with a perspective of using it as a source of work performed by microscopic machines. We provide the fundamental bound on the amount of ergotropy which can be extracted from environment …
From synthetic active devices such as self-propelling Janus colloids to micro-organisms like bacteria, micro-algae, living cells in tissues, active fluctuations are ubiquitous. Thermodynamics of small systems involving thermal as well as active fluctuations are of immense importance. They can be employed to extract thermodynamic work. Here we propose a simple model system that can produce thermodynamic work exploiting active fluctuations. We consider a Brownian particle, trapped by an externally controlled harmonic confinement which time-periodically contracts and expands by modulating its spr…
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…
It is well-known that a small system weakly coupled to a large energy bath in a total microcanonical ensemble will find itself in an (approximately) thermal state and, recently, it has been shown that, if the total state is, instead, a random pure state with energy in a narrow range, then the small system will still be approximately thermal with a high probability (wrt `Haar measure'). We ask what conditions are required for something resembling these 'traditional' and 'modern' thermality results to still hold when system and energy bath are of comparable size. In Part 1, we show that, for giv…
Thermodynamic selection is an indirect competition between agents feeding on the same energy resource and obeying the laws of thermodynamics. We examine scenarios of this selection, where the agent is modeled as a heat-engine coupled to two thermal baths and extracting work from the high-temperature bath. The agents can apply different work-extracting, game-theoretical strategies, e.g. the maximum power or the maximum efficiency. They can also have a fixed structure or be adaptive. Depending on whether the resource (i.e. the high-temperature bath) is infinite or finite, the fitness of the agen…
A new scenario for energy distribution, security and shareability is presented that assumes the availability of quantum information heat engines and a thermal bath. It is based on the convertibility between entropy and work in the presence of a thermal reservoir. Our approach to the informational content of physical systems that are distributed between users is complementary to the conventional perspective of quantum communication. The latter places the value on the unpredictable content of the transmitted quantum states, while our interest focuses on their certainty. Some well-known results i…
The nonequilibrium thermodynamics feature of a Brownian motor operating between two different heat baths is explored as a function of time $t$. Using the Gibbs entropy and Schnakenberg microscopic stochastic approach, we find exact closed form expressions for the free energy, the rate of entropy production and the rate of entropy flow from the system to the outside. We show that when the system is out of equilibrium, it constantly produces entropy and at the same time extract entropy out of the system. Its entropy production and extraction rates decrease in time and saturate to constant value.…
The thermodynamics of quantum systems coupled to periodically modulated heat baths and work reservoirs is developed. By identifying affinities and fluxes, the first and second law are formulated consistently. In the linear response regime, entropy production becomes a quadratic form in the affinities. Specializing to Lindblad-dynamics, we identify the corresponding kinetic coefficients in terms of correlation functions of the unperturbed dynamics. Reciprocity relations follow from symmetries with respect to time reversal. The kinetic coefficients can be split into a classical and a quantum con…
Thermodynamics and information have intricate inter-relations. The justification of the fact that information is physical, is done by inter-linking information and thermodynamics - through Landauer's principle. This modern approach towards information recently has improved our understanding of thermodynamics, both in classical and quantum domains. Here we show thermodynamics as a consequence of information conservation. Our approach can be applied to most general situations, where systems and thermal-baths could be quantum, of arbitrary sizes and even could posses inter-system correlations. Th…
Information ratchets extract work from heat baths using low entropy information reservoirs. We find that the structure of the reservoirs affects both their performance and phase diagram. By deriving exact probabilities of recurrence in d-dimensional reservoirs we calculate the net extraction of work. Performance is characterised by two critical dimensions (i) For d < 3 driven exploration of the reservoir is essential to extract work (ii) For d > 4 purely diffusive exploration is optimal. This has important consequences for computation where sequential operations constitute a 1D path.
Thomson's formulation of the second law - no work can be extracted from a system coupled to a bath through a cyclic process - is believed to be a fundamental principle of nature. For the equilibrium situation a simple proof is presented, valid for macroscopic sources of work. Thomson's formulation gets limited when the source of work is mesoscopic, i.e. when its number of degrees of freedom is large but finite. Here work-extraction from a single equilibrium thermal bath is possible when its temperature is large enough. This result is illustrated by means of exactly solvable models. Finally we …
In information theory, one area of interest is gambling, where mutual information characterizes the maximal gain in wealth growth rate due to knowledge of side information; the betting strategy that achieves this maximum is named the Kelly strategy. In the field of physics, it was recently shown that mutual information can characterize the maximal amount of work that can be extracted from a single heat bath using measurement-based control protocols, i.e., using "information engines". However, to the best of our knowledge, no relation between gambling and information engines has been presented …
Recent years have seen an enormously revived interest in the study of thermodynamic notions in the quantum regime. This applies both to the study of notions of work extraction in thermal machines in the quantum regime, as well as to questions of equilibration and thermalisation of interacting quantum many-body systems as such. In this work we bring together these two lines of research by studying work extraction in a closed system that undergoes a sequence of quenches and equilibration steps concomitant with free evolutions. In this way, we incorporate an important insight from the study of th…
