We derive a microscopic expression for the instantaneous diagonal elements of the density matrix $ρ_{nn}(t)$ in the adiabatic basis for an arbitrary time dependent process in a closed Hamiltonian system. If the initial density matrix is stationary (diagonal) then this expression contains only squares of absolute values of matrix elements of the evolution operator, which can be interpreted as transition probabilities. We then derive the microscopic expression for the heat defined as the energy generated due to transitions between instantaneous energy levels. If the initial density matrix is passive (diagonal with $ρ_{nn}(0)$ monotonically decreasing with energy) then the heat is non-negative in agreement with basic expectations of thermodynamics. Our findings also can be used for systematic expansion of various observables around the adiabatic limit.
We construct a microscopic model of low-dissipation engines by driving a Brownian particle in a time-dependent harmonic potential. Shortcuts to adiabaticity and shortcuts to isothermality are introduced to realize the adiabatic and isothermal branches in a thermodynamic cycle, respectively. We derive an analytical expression of the efficiency at maximum power for this kind of engines. This expression satisfies the universal law of efficiency at maximum power up to the second order of the Carnot efficiency. We also analyze the issue of power at any given efficiency for general low-dissipation e…
We study the efficiency at maximum power of non-adiabatic dissipative (internally dissipative friction in finite time adiabatic processes) Carnot-like heat engines operate in finite time under the power law dissipation regime. We find that the non-adiabatic dissipation does not influence the universal minimum and maximum bounds on the efficiency at maximum power obtained in the generalized dissipative Carnot-like heat engines which does not take in to account the non-adiabatic dissipation.
Specific heat and entropy are relevant thermodynamic properties, which may be used as macroscopic probes to microscopic properties of materials under ambient conditions and under high applied fields. However, the measurement of specific heat under intense external fields can be a challenging task, as well as to obtain the entropy in the same conditions. Here, we describe a method to obtain high-field specific heat and entropy from measurements of specific heat under ambient conditions and direct temperature change induced by adiabatic field changes. We derive straightforward thermodynamic equa…
We investigate a heat engine under an adiabatic (Thouless) pumping process. In this process, the extracted work and lower bound on dissipated availability are characterized by a vector potential and a Riemannian metric tensor, respectively. We derive a trade-off relation between the power and effective efficiency. We also explicitly calculate the trade-off relation as well as the power and effective efficiency for a spin-boson model coupled to two reservoirs.
The impossibility of separating into work and heat the energy transmitted between two subsystems through a movable piston is analyzed in this article. The process here described, although "quasi-static", is not reversible. It is shown that the First Principle, dU=dW+dQ, introduced by Clausius, does not generally allow a physical identification of dW and dQ, although dU=-pdV+TdS is verified along the equilibrium points of each subsystem.
The energetics of the stochastic process has shown the balance of energy on the mesoscopic level. The heat and the energy defined there are, however, generally different from their macroscopic counterpart. We show that this discrepancy can be removed by adding to these quantities the reversible heat associated with the mesoscopic free energy.
Starting from microscopic mechanics, we derive thermodynamic relations for heat conducting nonequilibrium steady states. The extended Clausius relation enables one to experimentally determine nonequilibrium entropy to the second order in the heat current. The associated Shannon-like microscopic expression of the entropy is suggestive. When the heat current is fixed, the extended Gibbs relation provides a unified treatment of thermodynamic forces in linear nonequilibrium regime.
We present a quantum Otto engine model consisting of two isochoric and two adiabatic strokes, where the adiabatic expansion or compression is realized by adiabatically changing the shape of the potential. Here we show that such an adiabatic deformation may alter operation mode and enhance machine performance by increasing output work and efficiency, even with the advantage of decreasing work fluctuations. If the heat engine operates under maximal power by optimizing the control parameter, the efficiency shows certain universal behavior.
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…
We consider a finite-time quantum Otto cycle with single and two-spin-$1/2$ systems as its working medium. In order to mimic adiabatic dynamics at a finite-time, we employ a shortcut-to-adiabaticity technique and evaluate the performance of the engine including the cost of the shortcut. We compare our results with the true adiabatic and non-adiabatic performances of the same cycle. Our findings clearly indicate that the use of shortcut-to-adiabaticity scheme significantly enhances the performance of the quantum Otto engine as compared to its adiabatic and non-adiabatic counterparts for differe…
We obtain macroscopic adiabatic thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics subject to random collisions with the environment. The microscopic dynamics is given by a chain of oscillators subject to a varying tension (external force) and to collisions with external independent particles of "infinite mass". The effect of each collision is to change the sign of the velocity without changing the modulus. This way the energy is conserved by the resulting dynamics. After a diffusive space-time scaling and cross-graining, the profiles of volume and energ…
We model a microscopic heat engine as a particle hopping on a one-dimensional lattice in a periodic sawtooth potential, with or without load, assisted by the thermal kicks it gets from alternately placed hot and cold thermal baths. We find analytic expressions for current and rate of heat flow when the engine operates at steady state. Three regions are identified where the model acts either as a heat engine or as a refrigerator or as neither of the two. At quasistatic limit both efficiency of the engine and coefficient of performance of the refrigerator go to that for Carnot engine and Carnot …
Abstract The Curzon-Ahlborn (CA) efficiency, as the efficiency at the maximum power (EMP) of the endoreversible Carnot engine, has a significant impact on finite-time thermodynamics. However, the CA engine model is based on many assumptions. In the past few decades, although a lot of efforts have been made, a microscopic theory of the CA engine is still lacking. By adopting the method of the stochastic differential equation of energy, we formulate a microscopic theory of the CA engine realized with an underdamped Brownian particle in a class of non-harmonic potentials. This theory gives micros…
We give a microscopic explanation of both Debye and non-Debye thermalization processes that have been recently reported by Gall and Kutner (Physica A 352, 347 (2005)). Due to reduction of the problem to first passage phenomena we argue that relaxation functions f(t) introduced by the authors directly correspond to survival probabilities S(t) of particles in the considered systems. We show that in the case of broken ergodicity (i.e. in the case of mirror collisions) the survival probability decays as a power law S(t)=τ/t.
A unified framework to describe the adiabatic class of ensembles in the generalized statistical mechanics based on Schwammle-Tsallis two parameter (q, q') entropy is proposed. The generalized form of the equipartition theorem, virial theorem and the adiabatic theorem are derived. Each member of the class of ensembles is illustrated using the classical nonrelativistic ideal gas and we observe that the heat functions could be written in terms of the Lambert's W-function in the large N limit. In the microcanonical ensemble we study the effect of gravitational field on classical nonrelativistic id…
In the present work, a power law dissipative Carnot like heat engine cycle of two irreversible isothermal and two irreversible adiabatic processes with finite time non-adiabatic dissipation is considered and the efficiency under two optimization criteria $\dotΩ$ figure of merit and efficient power, $χ_{ep}$ is studied. The generalized extreme bounds of the optimized efficiency under the above said optimization criteria are obtained. The lower and upper bounds of the efficiency for the low dissipation Carnot-like heat engine under these optimization criteria are obtained with dissipation level …
Under a general framework, shortcuts to adiabatic processes are shown to be possible in classical systems. We then study the distribution function of the work done on a small system initially prepared at thermal equilibrium. It is found that the work fluctuations can be significantly reduced via shortcuts to adiabatic processes. For example, in the classical case probabilities of having very large or almost zero work values are suppressed. In the quantum case negative work may be totally removed from the otherwise non-positive-definite work values. We also apply our findings to a micro Otto-cy…
We investigate a microscopic motor based on an externally controlled two-level system. One cycle of the motor operation consists of two strokes. Within each stroke, the two-level system is in contact with a given thermal bath and its energy levels are driven with a constant rate. The time evolution of the occupation probabilities of the two states are controlled by one rate equation and represent the system's response with respect to the external driving. We give the exact solution of the rate equation for the limit cycle and discuss the emerging thermodynamics: the work done on the environmen…
We present a general approach to describe slowly driven quantum systems both in real and imaginary time. We highlight many similarities, qualitative and quantitative, between real and imaginary time evolution. We discuss how the metric tensor and the Berry curvature can be extracted from both real and imaginary time simulations as a response of physical observables. For quenches ending at or near the quantum critical point, we show the utility of the scaling theory for detecting the location of the quantum critical point by comparing sweeps at different velocities. We briefly discuss the unive…
Microscopic heat engines are microscale systems that convert energy flows between heat reservoirs into work or systematic motion. We have experimentally realized a minimal microscopic heat engine. It consists of a colloidal Brownian particle optically trapped in an elliptical potential well and simultaneously coupled to two heat baths at different temperatures acting along perpendicular directions. For a generic arrangement of the principal directions of the baths and the potential, the symmetry of the system is broken, such that the heat flow drives a systematic gyrating motion of the particl…
The efficiency of microscopic heat engines in a thermally heterogenous environment is considered. We show that, as a consequence of the recently discovered entropic anomaly, quasi-static engines, whose efficiency is maximal in a fluid at uniform temperature, have in fact vanishing efficiency in presence of temperature gradients. For slow cycles the efficiency falls off as the inverse of the period. The maximum efficiency is reached at a finite value of the cycle period that is inversely proportional to the square root of the gradient intensity. The relative loss in maximal efficiency with resp…
Heat engines are fundamental physical objects to develop nonequilibrium thermodynamics. The thermodynamic performance of the heat engine is determined by the choice of cycle and time-dependence of parameters. Here, we propose a systematic numerical method to find a heat engine cycle to optimize some target functions. We apply the method to heat engines with slowly varying parameters and show that the method works well. Our numerical method is based on the genetic algorithm which is widely applied to various optimization problems.
In this paper, we make use of the exact hydrodynamic solution for the Stokes equation for the velocity of a binary ionic solution that we have recently obtained, and show that the nonequilibrium pressure in an electrolyte solution subjected to an external electric field can be not only compressive, but also divergent in the region containing the coordinate origin at which the center ion of its ion atmosphere is located. This divergent compressive pressure implies that it would be theoretically possible to locally confine the ion and also to adiabatically generate heat in the local by means of …
We demonstrate that the Gibbs-Shannon entropy is applicable to non-equilibrium systems of any size and boundary conditions. The change in microscopic entropy can be attributed to the stochastic nature of dynamic processes and to the inherent uncertainties of thermodynamic systems. The latter predicts that entropy production is nonnegative on average and varies with different trajectories according to the fluctuation theorem. By contrast, heat is affiliated with stochastic processes underlying particle motions and the ensemble average over all possible trajectories leads to the Clausius inequal…
The efficiency of small thermal machines is typically a fluctuating quantity. We here study the efficiency large deviation function of two exemplary quantum heat engines, the harmonic oscillator and the two-level Otto cycles. While the efficiency statistics follows the 'universal' theory of Verley et al. [Nature Commun. 5, 4721 (2014)] for nonadiabatic driving, we find that the latter framework does not apply in the adiabatic regime. We relate this unusual property to the perfect anticorrelation between work output and heat input that generically occurs in the broad class of scale-invariant ad…
We develop a geometric framework to describe the thermodynamics of microscopic heat engines driven by slow periodic temperature variations and modulations of a mechanical control parameter. Covering both the classical and the quantum regime, our approach reveals a universal trade-off relation between efficiency and power that follows solely from geometric arguments and holds for any thermodynamically consistent microdynamics. Focusing on Lindblad dynamics, we derive a second bound showing that coherence as a genuine quantum effect inevitably reduces the performance of slow engine cycles regard…
For cyclic heat engines operating in a finite cycle period, thermodynamic quantities have intercycle and intracycle correlations. By tuning the driving protocol appropriately, we can get the negative intercycle correlation to reduce the fluctuation of work through multiple cycles, which leads to the enhanced stability compared to the single-cycle operation. Taking the Otto engine with an overdamped Brownian particle as a working substance, we identify a scenario to get such enhanced stability by the intercycle correlation. Furthermore, we demonstrate that the enhancement can be readily realize…
Thermodynamics allows the application of Statistical Mechanics to finite and even small systems. As surface effects cannot be scaled away, one has to be careful with the standard arguments of splitting a system into two or bringing two systems into thermal contact with energy or particle exchange: Not only the volume part of the entropy must be considered. The addition of any other macroscopic constraint like a dividing surface, or the enforcement of gradients of the energy/particle reduce the entropy. As will be shown here, when removing such constraint in regions of a negative heat capacity,…
The heat engine, a machine that extracts useful work from thermal sources, is one of the basic theoretical constructs and fundamental applications of classical thermodynamics. The classical description of a heat engine does not include coherence in its microscopic degrees of freedom. By contrast, a quantum heat engine might possess coherence between its internal states. Although the Carnot efficiency cannot be surpassed, and coherence can be performance degrading in certain conditions, it was recently predicted that even when using only thermal resources, internal coherence can enable a quantu…
Fluctuations of thermodynamic quantities become non-negligible and play an important role when the system size is small. We develop finite-time thermodynamics of fluctuations in microscopic heat engines whose environmental temperature and mechanical parameter are driven periodically in time. Within the slow-driving regime, this formalism universally holds in a coarse-grained time scale whose resolution is much longer than the correlation time of the fluctuations, and is shown to be consistent with the relation analogous to the fluctuation-dissipation relation. Employing a geometric argument, a…
Modern technologies could soon make it possible to investigate the operation cycles of quantum heat engines by counting the photons that are emitted and absorbed by their working systems. Using the quantum jump approach to open-system dynamics, we show that such experiments would give access to a set of observables that determine the trade-off between power and efficiency in finite-time engine cycles. By analyzing the single-jump statistics of thermodynamic fluxes such as heat and entropy production, we obtain a family of general bounds on the power of microscopic heat engines. Our new bounds …
We consider two spin-1/2 particles with isotropic Heisenberg interaction, as the working substance of a quantum heat engine. We observe a frictional effect on the adiabatic branches of the heat cycle, which arises due to an inhomogeneous driving at a finite rate of the external magnetic field. The frictional effect is characterized by entropy production in the system and reduction in the work extracted. Corresponding to a sudden and a very slow driving, we find expressions for the lower and upper bounds of work that can be extracted on the adiabatic branches. These bounds are also confirmed wi…
In this letter, we analyze a conceptual design for the operation of an Otto cycle heat machine driven by adiabatic modifications on the electronic effective mass. Such tailoring of it can be implemented, for instance, via the application of external electron fields in some materials, as in Gallium Nitride (GaN). We show that due both the energy quantization on this structure and the adiabatic transformation of the effective mass, the machine performance can be improved. The realization of classically inconceivable Otto machines, with an incompressible working substance, can be realized as well…
The status of heat and work in nonequilibrium thermodynamics is quite confusing and non-unique at present with conflicting interpretations even after a long history of the first law in terms of exchange heat and work, and is far from settled. Moreover, the exchange quantities lack certain symmetry. By generalizing the traditional concept to also include their time-dependent irreversible components allows us to express the first law in a symmetric form dE(t)= dQ(t)-dW(t) in which dQ(t) and work dW(t) appear on an equal footing and possess the symmetry. We prove that irreversible work turns into…
We obtain macroscopic isothermal thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics in contact with a heat bath. The microscopic dynamics is given by a chain of anharmonic oscillators subject to a varying tension (external force) and the contact with the heat bath is modeled by independent Langevin dynamics acting on each particle. After a diffusive space-time scaling and cross-graining, the profile of volume converges to the solution of a deterministic diffusive equation with boundary conditions given by the applied tension. This defines an irreversible…
We analyze the performance of slowly driven meso- and micro-scale refrigerators and heat engines that operate between two thermal baths with small temperature difference. Using a general scaling argument, we show that such devices can work arbitrarily close to their Carnot limit only if heat-leaks between the baths are fully suppressed. Their power output is then subject to a universal geometric bound that decays quadratically to zero at the Carnot limit. This bound can be asymptotically saturated in the quasi-static limit if the driving protocols are suitably optimized and the temperature dif…
Microcanonical Thermodynamics allows the application of Statistical Mechanics on one hand to closed finite and even small systems and on the other to the largest,self-gravitating ones. However, one has to reconsider the fundamental principles of Statistical Mechanics especially its key quantity, entropy. Whereas in conventional Thermostatistics the homogeneity and extensivity of the system and the concavity of its entropy S(E) are central conditions, these fail for the systems considered here. E.g. at phase separation the entropy S(E) is necessarily convex to make e^{S(E)-E/T} bimodal in E (th…
Microcanonical thermodynamics allows the application of statistical mechanics both to finite and even small systems and also to the largest, self-gravitating ones. However, one must reconsider the fundamental principles of statistical mechanics especially its key quantity, entropy. Whereas in conventional thermostatistics, the homogeneity and extensivity of the system and the concavity of its entropy are central conditions, these fail for the systems considered here. For example, at phase separation, the entropy, S(E), is necessarily convex to make exp[S(E)-E/T] bimodal in E. Particularly, as …
