Fluctuation theorem in quantum heat conduction

Generalized Quantum Fluctuation Theorem for Energy Exchange

The nonequilibrium fluctuation relation is a cornerstone of quantum thermodynamics. It is widely believed that the system-bath heat exchange obeys the famous Jarzynski-Wójcik fluctuation theorem. However, this theorem is established in the Born-Markovian approximation under the weak-coupling condition. Via studying the energy exchange between a harmonic oscillator and its coupled bath in the non-Markovian dynamics, we establish a generalized quantum fluctuation theorem for energy exchange being valid for arbitrary coupling strength. The Jarzynski-Wójcik fluctuation theorem is recovered in the weak-coupling limit. We also find the average energy exchange exhibits rich nonequilibrium characteristics when different numbers of system-bath bound states are formed, which suggests a useful way to control the quantum heat. Deepening our understanding of the fluctuation relation in quantum thermodynamics, our result lays the foundation to design high-efficiency quantum heat engines.

Challenges in molecular dynamics simulations of heat exchange statistics

We study heat exchange in temperature-biased metal-molecule-metal molecular junctions by employing the molecular dynamics simulator LAMMPS. Generating the nonequilibrium steady state with Langevin thermostats at the boundaries of the junction, we show that the average heat current across a gold-alkanedithiol-gold nanojunction behaves physically, with the thermal conductance value matching the literature. In contrast, the full probability distribution function for heat exchange, as generated by the simulator, violates the fundamental fluctuation symmetry for entropy production. We trace this failure back to the implementation of the thermostats and the expression used to calculate the heat exchange. To rectify this issue and produce the correct statistics, we introduce single-atom thermostats as an alternative to conventional many-atom thermostats. Once averaging heat exchange over the hot and cold thermostats, this approach successfully generates the correct probability distribution function, which we use to study the behavior of both the average heat current and its noise. We further examine the thermodynamic uncertainty relation in the molecular junction and show that it holds, albeit demonstrating nontrivial trends. Our study points to the need to carefully implement nonequilibrium molecular dynamics solvers in atomistic simulation software tools for future investigations of noise phenomena in thermal transport.

Arbitrary-time thermodynamic uncertainty relation from fluctuation theorem

The thermodynamic uncertainty relation (TUR) provides a universal entropic bound for the precision of the fluctuation of the charge transfer, for example, for a class of continuous-time stochastic processes. However, its extension to general nonequilibrium dynamics is still an unsolved problem. We derive TUR for an arbitrary finite time from exchange fluctuation theorem under a geometric necessary and sufficient condition. We also generally show a necessary and sufficient condition of multidimensional TUR in a unified manner. As a nontrivial practical consequence, we obtain universal scaling relations among the mean and variance of the charge transfer in short time regime. In this manner, we can deepen our understanding of a link between two important rigorous relations, i.e., the fluctuation theorem and the thermodynamic uncertainty relation.

Heat distribution in quantum Brownian motion

We investigate the heat statistics in a relaxation process of quantum Brownian motion described by the Caldeira-Leggett model. By employing the normal mode transformation and the phase-space formulation approach, we can analyze the quantum heat distribution within an exactly dynamical framework beyond the traditional paradigm of Born-Markovian and weak-coupling approximations. It is revealed that the exchange fluctuation theorem for quantum heat generally breaks down in the strongly non-Markovian regime. Our results may improve the understanding about the nonequilibrium thermodynamics of open quantum systems when the usual Markovian treatment is no longer appropriate.

Performance of Quantum Heat Engines Enhanced by Adiabatic Deformation of Trapping Potential

We present a quantum Otto engine model alternatively driven by a hot and a cold heat reservoir and 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 in the sudden limit operates under maximal power by optimizing the control parameter, the efficiency shows certain universal behavior, η*=ηC/2+ηC2/8+O(ηC3), where ηC=1-βhr/βcr is the Carnot efficiency, with βhr(βcr) being the inverse temperature of the hot (cold) reservoir. However, such efficiency under maximal power can be produced by our machine model in the regimes where the machine without adiabatic deformation can only operate as a heater or a refrigerator.

Hierarchical structure of fluctuation theorems for a driven system in contact with multiple heat reservoirs

For driven open systems in contact with multiple heat reservoirs, we find the marginal distributions of work or heat do not satisfy any fluctuation theorem, but only the joint distribution of work and heat satisfies a family of fluctuation theorems. A hierarchical structure of these fluctuation theorems is discovered from microreversibility of the dynamics by adopting a step-by-step coarse-graining procedure in both classical and quantum regimes. Thus, we put all fluctuation theorems concerning work and heat into a unified framework. We also propose a general method to calculate the joint statistics of work and heat in the situation of multiple heat reservoirs via the Feynman-Kac equation. For a classical Brownian particle in contact with multiple heat reservoirs, we verify the validity of the fluctuation theorems for the joint distribution of work and heat.

Geometric heat pump and no-go restrictions of nonreciprocity in modulated thermal diffusion

Thermodynamics strongly restricts the direction of heat flow in static macroscopic thermal diffusive systems. To overcome this constraint, spatiotemporal modulated systems are used instead. Here, we unveil the underlying geometric heat pump effect in macroscopic driven thermal diffusion, which is crucial for achieving thermal nonreciprocity. We obtain a geometric expression to formulate the nontrivial current in a driven system, manifesting as an extra pumped heat ably diffusing from cold to hot that has no analogy in static setups. Moreover, we analyze the underlying geometric curvature of driven diffusive systems and derive no-pumping restriction theorems that constrain the thermal action under modulations and guide the optimization of driving protocols. Following the restrictions from geometry, we finally implement a minimum experiment and observe the predicted pumped heat in the absence of thermal bias at every instant, which is independent of the driving speed in the adiabatic limit, clearly validating the geometric theory. An extension of the geometric pump effect and no-pumping restrictions to macroscopic mass diffusion governed by Fick's law is also discussed. These results pave the way for designing and implementing nonreciprocal and topological diffusive systems under spatiotemporal modulations.

Bounds on skewness and kurtosis of steady-state currents

Current fluctuations are a powerful tool to unravel the underlying physics of the observed transport process. This work discusses some general properties of the third and the fourth current cumulant (skewness and kurtosis) related to dynamics and thermodynamics of a transport setup. Specifically, several distinct bounds on these quantities are either analytically derived or numerically conjectured, which are applicable to (1) noninteracting fermionic systems, (2) noninteracting bosonic systems, (3) thermally driven classical Markovian systems, and (4) unicyclic Markovian networks. Finally, it is demonstrated that violation of the obtained inequalities can provide a broad spectrum of information about the physics of the analyzed system; e.g., it can enable one to infer the presence of interactions or unitary dynamics, unravel the topology of the Markovian network, or characterize the nature of thermodynamic forces driving the system. In particular, relevant information about the microscopic dynamics can be gained even at equilibrium when the current variance-a standard measure of current fluctuations-is determined mostly by the thermal noise.

Quantum Thermodynamic Uncertainty Relations, Generalized Current Fluctuations and Nonequilibrium Fluctuation–Dissipation Inequalities

Thermodynamic uncertainty relations (TURs) represent one of the few broad-based and fundamental relations in our toolbox for tackling the thermodynamics of nonequilibrium systems. One form of TUR quantifies the minimal energetic cost of achieving a certain precision in determining a nonequilibrium current. In this initial stage of our research program, our goal is to provide the quantum theoretical basis of TURs using microphysics models of linear open quantum systems where it is possible to obtain exact solutions. In paper [Dong et al., Entropy 2022, 24, 870], we show how TURs are rooted in the quantum uncertainty principles and the fluctuation–dissipation inequalities (FDI) under fully nonequilibrium conditions. In this paper, we shift our attention from the quantum basis to the thermal manifests. Using a microscopic model for the bath’s spectral density in quantum Brownian motion studies, we formulate a “thermal” FDI in the quantum nonequilibrium dynamics which is valid at high temperatures. This brings the quantum TURs we derive here to the classical domain and can thus be compared with some popular forms of TURs. In the thermal-energy-dominated regimes, our FDIs provide better estimates on the uncertainty of thermodynamic quantities. Our treatment includes full back-action from the environment onto the system. As a concrete example of the generalized current, we examine the energy flux or power entering the Brownian particle and find an exact expression of the corresponding current–current correlations. In so doing, we show that the statistical properties of the bath and the causality of the system+bath interaction both enter into the TURs obeyed by the thermodynamic quantities.

Thermodynamic uncertainty relation for quantum work distribution: Exact case study for a perturbed oscillator

Recently, some general relations have been intensively investigated in nonequilibrium mesoscopic systems. In particular, the thermodynamic uncertainty relation (TUR) provides a general bound of the precision for the fluctuation of some currents in terms of the corresponding entropy production. On the other hand, the fluctuation of the work performed is also a significant quantity, which is supposed to satisfy TUR under some conditions, such as symmetric driving protocol. In this paper, we analytically show that the TUR holds for the work performed on an externally perturbed quantum harmonic oscillator interacting with multiple reservoirs in full quantum regime. In this manner, we evaluate how the noncommutativity affects the thermodynamic precision. We also explore its experimental accessibility.

Quantum-Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion

Quantum Brownian motion, described by the Caldeira-Leggett model, brings insights to the understanding of 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 classical Brownian motion described by the Langevin equation, indicating the quantum-classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.

Fluctuation theorem convergence in a viscoelastic medium demonstrated experimentally using a dusty plasma

The convergence of the steady-state fluctuation theorem (SSFT) is investigated in a shear-flow experiment performed in a dusty plasma. This medium has a viscoelastic property characterized by the Maxwell relaxation time τ_{M}. Using measurements of the time series of the entropy production rate, for subsystems of various sizes, it is discovered that the SSFT convergence time decreases with the increasing system size until it eventually reaches a minimum value of τ_{M}, no matter the size of the subsystem. This result indicates that the convergence of the SSFT is limited by the energy-storage property of the viscoelastic medium.

Full counting statistics of the particle currents through a Kitaev chain and the exchange fluctuation theorem

Exchange fluctuation theorems (XFTs) describe a fundamental symmetry relation for particle and energy exchange between several systems. Here we study the XFTs of a Kitaev chain connected to two reservoirs at the same temperature but different bias. By varying the parameters in the Kitaev chain model, we calculate analytically the full counting statistics of the transport current and formulate the corresponding XFTs for multiple current components. We also demonstrate the XFTs with numerical results. We find that due to the presence of the U(1) symmetry breaking terms in the Hamiltonian of the Kitaev chain, various forms of the XFTs emerge, and they can be interpreted in terms of various well-known transport processes.

Thermodynamic uncertainty relation for energy transport in a transient regime: A model study

We investigate a transient version of the recently discovered thermodynamic uncertainty relation (TUR) which provides a precision-cost trade-off relation for certain out-of-equilibrium thermodynamic observables in terms of net entropy production. We explore this relation in the context of energy transport in a bipartite setting for three exactly solvable toy model systems (two coupled harmonic oscillators, two coupled qubits, and a hybrid coupled oscillator-qubit system) and analyze the role played by the underlying statistics of the transport carriers in the TUR. Interestingly, for all these models, depending on the statistics, the TUR ratio can be expressed as a sum or a difference of a universal term which is always greater than or equal to 2 and a corresponding entropy production term. We find that the generalized version of the TUR, originating from the universal fluctuation symmetry, is always satisfied. However, interestingly, the specialized TUR, a tighter bound, is always satisfied for the coupled harmonic oscillator system obeying Bose-Einstein statistics. Whereas, for both the coupled qubit, obeying Fermi-like statistics, and the hybrid qubit-oscillator system with mixed Fermi-Bose statistics, violation of the tighter bound is observed in certain parameter regimes. We have provided conditions for such violations. We also provide a rigorous proof following the nonequilibrium Green's function approach that the tighter bound is always satisfied in the weak-coupling regime for generic bipartite systems.

Dynamics of large deviations in the hydrodynamic limit: Noninteracting systems

We study the dynamics of the statistics of the energy transferred across a point along a quantum chain which is prepared in the inhomogeneous initial state obtained by joining two identical semi-infinite parts thermalized at two different temperatures. In particular, we consider the transverse field Ising and harmonic chains as prototypical models of noninteracting fermionic and bosonic excitations, respectively. Within the so-called hydrodynamic limit of large space-time scales we first discuss the mean values of the energy density and current, and then, aiming at the statistics of fluctuations, we calculate exactly the scaled cumulant generating function of the transferred energy. From the latter, the evolution of the associated large deviation function is obtained. A natural interpretation of our results is provided in terms of a semiclassical picture of quasiparticles moving ballistically along classical trajectories. Similarities and differences between the transferred energy scaled cumulant and the large deviation functions in the cases of noninteracting fermions and bosons are discussed.

Universal scaling for recovery of Fourier's law in low-dimensional solids under momentum conservation

Dynamic renormalization group (RG) of fluctuating viscoelastic equations is investigated to clarify the cause for numerically reported disappearance of anomalous heat conduction (recovery of Fourier's law) in low-dimensional momentum-conserving systems. RG flow is obtained explicitly for simplified two model cases: a one-dimensional continuous medium under low pressure and incompressible viscoelastic medium of arbitrary dimensions. Analyses of these clarify that the inviscid fixed point of contributing the anomalous heat conduction becomes unstable under the RG flow of nonzero elastic-wave speeds. The dynamic RG analysis further predicts a universal scaling of describing the crossover between the growth and saturation of observed heat conductivity, which is confirmed through the numerical experiments of Fermi-Pasta-Ulam β (FPU-β) lattices.

Fluctuation relations for adiabatic pumping

We derive an extended fluctuation relation for an open system coupled with two reservoirs under adiabatic one-cycle modulation. We confirm that the geometrical phase caused by the Berry-Sinitsyn-Nemenman curvature in the parameter space generates non-Gaussian fluctuations. This non-Gaussianity is enhanced for the instantaneous fluctuation relation when the bias between the two reservoirs disappears.

Nonequilibrium Green's Function's Approach to the Calculation of Work Statistics

The calculation of work distributions in a quantum many-body system is of significant importance and also of formidable difficulty in the field of nonequilibrium quantum statistical mechanics. To solve this problem, inspired by the Schwinger-Keldysh formalism, we propose the contour-integral formulation for work statistics. Based on this contour integral, we show how to do the perturbation expansion of the characteristic function of work (CFW) and obtain the approximate expression of the CFW to the second order of the work parameter for an arbitrary system under a perturbative protocol. We also demonstrate the validity of fluctuation theorems by utilizing the Kubo-Martin-Schwinger condition. Finally, we use noninteracting identical particles in a forced harmonic potential as an example to demonstrate the powerfulness of our approach.

Shortcuts to Adiabatic Pumping in Classical Stochastic Systems

Adiabatic pumping is characterized by a geometric contribution to the pumped charge, which can be nonzero even in the absence of a bias. However, as the driving speed is increased, nonadiabatic excitations gradually reduce the pumped charge, thereby limiting the maximal applicable driving frequencies. To circumvent this problem, we here extend the concept of shortcuts to adiabaticity to construct a control protocol which enables geometric pumping well beyond the adiabatic regime. Our protocol allows for an increase, by more than an order of magnitude, in the driving frequencies, and the method is also robust against moderate fluctuations of the control field. We provide a geometric interpretation of the control protocol and analyze the thermodynamic cost of implementing it. Our findings can be realized using current technology and potentially enable fast pumping of charge or heat in quantum dots, as well as in other stochastic systems from physics, chemistry, and biology.

Work Fluctuations and Jarzynski Equality in Stochastic Resetting

We consider the paradigm of an overdamped Brownian particle in a potential well, which is modulated through an external protocol, in the presence of stochastic resetting. Thus, in addition to the short range diffusive motion, the particle also experiences intermittent long jumps that reset the particle back at a preferred location. Due to the modulation of the trap, work is done on the system and we investigate the statistical properties of the work fluctuations. We find that the distribution function of the work typically, in asymptotic times, converges to a universal Gaussian form for any protocol as long as that is also renewed after each resetting event. When observed for a finite time, we show that the system does not generically obey the Jarzynski equality that connects the finite time work fluctuations to the difference in free energy. Nonetheless, we identify herein a restricted set of protocols which embraces the relation. In stark contrast, the Jarzynski equality is always fulfilled when the protocols continue to evolve without being reset. We present a set of exactly solvable models, demonstrate the validation of our theory and carry out numerical simulations to illustrate these findings. Finally, we have pointed out possible realistic implementations for resetting in experiments using the so-called engineered swift equilibration.

Thermodynamic uncertainty relation in thermal transport

We use the fundamental nonequilibrium steady-state fluctuation symmetry and derive a condition on the validity of the thermodynamic uncertainty relation (TUR) in thermal transport problems, classical and quantum alike. We test this condition and study the breakdown of the TUR in different thermal transport junctions of bosonic and electronic degrees of freedom. We prove that the TUR is valid in harmonic oscillator junctions. In contrast, in the nonequilibrium spin-boson model, which realizes many-body effects, it is satisfied in the Markovian limit, but violations arise as we tune (reduce) the cutoff frequency of the thermal baths, thus observing non-Markovian dynamics. We consider heat transport by noninteracting electrons in a tight-binding chain model. We show that the TUR is feasibly violated by tuning, e.g., the hybridization energy of the chain to the metal leads. These results manifest that the validity of the TUR relies on the statistics of the participating carriers, their interaction, and the nature of their couplings to the macroscopic contacts (metal electrodes and phonon baths).

Path-integral methodology and simulations of quantum thermal transport: Full counting statistics approach

We develop and test a computational framework to study heat exchange in interacting, nonequilibrium open quantum systems. Our iterative full counting statistics path integral (iFCSPI) approach extends a previously well-established influence functional path integral method, by going beyond reduced system dynamics to provide the cumulant generating function of heat exchange. The method is straightforward; we implement it for the nonequilibrium spin boson model to calculate transient and long-time observables, focusing on the steady-state heat current flowing through the system under a temperature difference. Results are compared to perturbative treatments and demonstrate good agreement in the appropriate limits. The challenge of converging nonequilibrium quantities, currents and high order cumulants, is discussed in detail. The iFCSPI, a numerically exact technique, naturally captures strong system-bath coupling and non-Markovian effects of the environment. As such, it is a promising tool for probing fundamental questions in quantum transport and quantum thermodynamics.

Path integral approach to heat in quantum thermodynamics

We study the heat statistics of a quantum Brownian motion described by the Caldeira-Leggett model. By using the path integral approach, we introduce a concept of the quantum heat functional along every pair of Feynman paths. This approach has the advantage of improving our understanding about heat in quantum systems. First, we demonstrate the microscopic reversibility of the system by connecting the heat functional to the forward and time-reversed probabilities. Second, we analytically prove the quantum-classical correspondence of the heat functional and their statistics, which allows us to obtain better intuitions about the difference between classical and quantum heat.

Path Integral Approach to Quantum Thermodynamics

Work belongs to the most basic notions in thermodynamics but it is not well understood in quantum systems, especially in open quantum systems. By introducing a novel concept of the work functional along an individual Feynman path, we invent a new approach to study thermodynamics in the quantum regime. Using the work functional, we derive a path integral expression for the work statistics. By performing the ℏ expansion, we analytically prove the quantum-classical correspondence of the work statistics. In addition, we obtain the quantum correction to the classical fluctuating work. We can also apply this approach to an open quantum system in the strong coupling regime described by the quantum Brownian motion model. This approach provides an effective way to calculate the work in open quantum systems by utilizing various path integral techniques. As an example, we calculate the work statistics for a dragged harmonic oscillator in both isolated and open quantum systems.

Frequency-dependent current noise in quantum heat transfer: A unified polaron calculation

To investigate frequency-dependent current noise (FDCN) in open quantum systems at steady states, we present a theory which combines Markovian quantum master equations with a finite time full counting statistics. Our formulation of the FDCN generalizes previous zero-frequency expressions and can be viewed as an application of MacDonald's formula for electron transport to heat transfer. As a demonstration, we consider the paradigmatic example of quantum heat transfer in the context of a non-equilibrium spin-boson model. We adopt a recently developed polaron-transformed Redfield equation which allows us to accurately investigate heat transfer with arbitrary system-reservoir coupling strength, arbitrary values of spin bias, and temperature differences. We observe a turn-over of FDCN in the intermediate coupling regimes, similar to the zero-frequency case. We find that the FDCN with varying coupling strengths or bias displays a universal Lorentzian-shape scaling form in the weak coupling regime, and a white noise spectrum emerges with zero bias in the strong coupling regime due to distinctive spin dynamics. We also find that the bias can suppress the FDCN in the strong coupling regime, in contrast to its zero-frequency counterpart which is insensitive to bias changes. Furthermore, we utilize the Saito-Utsumi relation as a benchmark to validate our theory and study the impact of temperature differences at finite frequencies. Together, our results provide detailed dissections of the finite time fluctuation of heat current in open quantum systems.

Energy Current Cumulants in One-Dimensional Systems in Equilibrium

A recent theory based on fluctuating hydrodynamics predicts that one-dimensional interacting systems with particle, momentum, and energy conservation exhibit anomalous transport that falls into two main universality classes. The classification is based on behavior of equilibrium dynamical correlations of the conserved quantities. One class is characterized by sound modes with Kardar-Parisi-Zhang scaling, while the second class has diffusive sound modes. The heat mode follows Lévy statistics, with different exponents for the two classes. Here we consider heat current fluctuations in two specific systems, which are expected to be in the above two universality classes, namely, a hard particle gas with Hamiltonian dynamics and a harmonic chain with momentum conserving stochastic dynamics. Numerical simulations show completely different system-size dependence of current cumulants in these two systems. We explain this numerical observation using a phenomenological model of Lévy walkers with inputs from fluctuating hydrodynamics. This consistently explains the system-size dependence of heat current fluctuations. For the latter system, we derive the cumulant-generating function from a more microscopic theory, which also gives the same system-size dependence of cumulants.

Geometric phaselike effects in a quantum heat engine

By periodically driving the temperature of reservoirs in a quantum heat engine, geometric or Pancharatnam-Berry phaselike (PBp) effects in the thermodynamics can be observed. The PBp can be identified from a generating function (GF) method within an adiabatic quantum Markovian master equation formalism. The GF is shown not to lead to a standard open quantum system's fluctuation theorem in the presence of phase-different modulations with an inapplicability in the use of large deviation theory. Effect of quantum coherences in optimizing the flux is nullified due to PBp contributions. The linear coefficient, 1/2, which is universal in the expansion of the efficiency at maximum power in terms of Carnot efficiency no longer holds true in the presence of PBp effects.

Geometric fluctuation theorem for a spin-boson system

We derive an extended fluctuation theorem for geometric pumping of a spin-boson system under periodic control of environmental temperatures by using a Markovian quantum master equation. We obtain the current distribution, the average current, and the fluctuation in terms of the Monte Carlo simulation. To explain the results of our simulation we derive an extended fluctuation theorem. This fluctuation theorem leads to the fluctuation dissipation relations but the absence of the conventional reciprocal relation.

Nonequilibrium quantum chains under multisite Lindblad baths

We study a quantum XX chain coupled to two heat reservoirs that act on multiple sites and are kept at different temperatures and chemical potentials. The baths are described by Lindblad dissipators, which are constructed by direct coupling to the fermionic normal modes of the chain. Using a perturbative method, we are able to find analytical formulas for all steady-state properties of the system. We compute both the particle or magnetization current and the energy current, both of which are found to have the structure of Landauer's formula. We also obtain exact formulas for the Onsager coefficients. All properties are found to differ substantially from those of a single-site bath. In particular, we find a strong dependence on the intensity of the bath couplings. In the weak-coupling regime, we show that the Onsager reciprocal relations are satisfied.

Typical pure nonequilibrium steady states and irreversibility for quantum transport

It is known that each single typical pure state in an energy shell of a large isolated quantum system well represents a thermal equilibrium state of the system. We show that such typicality holds also for nonequilibrium steady states (NESS's). We consider a small quantum system coupled to multiple infinite reservoirs. In the long run, the total system reaches a unique NESS. We identify a large Hilbert space from which pure states of the system are to be sampled randomly and show that the typical pure states well describe the NESS. We also point out that the irreversible relaxation to the unique NESS is important to the typicality of the pure NESS's.

Transitional steady states of exchange dynamics between finite quantum systems

We examine energy and particle exchange between finite-sized quantum systems and find a new form of nonequilibrium state. The exchange rate undergoes stepwise evolution in time, and its magnitude and sign dramatically change according to system size differences. The origin lies in interference effects contributed by multiply scattered waves at system boundaries. Although such characteristics are utterly different from those of true steady state for infinite systems, Onsager's reciprocal relation remains universally valid.

Le Chatelier Principle for Out-of-Equilibrium and Boundary-Driven Systems: Application to Dynamical Phase Transitions

A stability analysis is presented for boundary-driven and out-of-equilibrium systems in the framework of the hydrodynamic macroscopic fluctuation theory. A Hamiltonian description is proposed which allows us to thermodynamically interpret the additivity principle. A necessary and sufficient condition for the validity of the additivity principle is obtained as an extension of the Le Chatelier principle. These stability conditions result from a diagonal quadratic form obtained using the cumulant generating function. This approach allows us to provide a proof for the stability of the weakly asymmetric exclusion process and to reduce the search for stability to the solution of two coupled linear ordinary differential equations instead of nonlinear partial differential equations. Additional potential applications of these results are discussed in the realm of classical and quantum systems.

Vibrational Heat Transport in Molecular Junctions

We review studies of vibrational energy transfer in a molecular junction geometry, consisting of a molecule bridging two heat reservoirs, solids or large chemical compounds. This setup is of interest for applications in molecular electronics, thermoelectrics, and nanophononics, and for addressing basic questions in the theory of classical and quantum transport. Calculations show that system size, disorder, structure, dimensionality, internal anharmonicities, contact interaction, and quantum coherent effects are factors that combine to determine the predominant mechanism (ballistic/diffusive), effectiveness (poor/good), and functionality (linear/nonlinear) of thermal conduction at the nanoscale. We review recent experiments and relevant calculations of quantum heat transfer in molecular junctions. We recount the Landauer approach, appropriate for the study of elastic (harmonic) phononic transport, and outline techniques that incorporate molecular anharmonicities. Theoretical methods are described along with examples illustrating the challenge of reaching control over vibrational heat conduction in molecules.

Fluctuations of the heat exchanged between two quantum spin chains

The statistics of the heat exchanged between two quantum XX spin chains prepared at different temperatures is studied within the assumption of weak coupling. This provides simple formulas for the average heat and its corresponding characteristic function, from which the probability distribution may be computed numerically. These formulas are valid for arbitrary sizes and therefore allow us to analyze the role of the thermodynamic limit in this nonequilibrium setting. It is found that all thermodynamic quantities are extremely sensitive to the quantum phase transition of the XX chain.

Stationary and Transient Fluctuation Theorems for Effective Heat Fluxes between Hydrodynamically Coupled Particles in Optical Traps

We experimentally study the statistical properties of the energy fluxes between two trapped Brownian particles, interacting through dissipative hydrodynamic coupling, and submitted to an effective temperature difference ΔT, obtained by random forcing the position of one trap. We identify effective heat fluxes between the two particles and show that they satisfy an exchange fluctuation theorem in the stationary state. We also show that after the sudden application of a temperature gradient ΔT, the total hot-cold flux satisfies a transient exchange fluctuation theorem for any integration time, whereas the total cold-hot flux only does it asymptotically for long times.

Negative Full Counting Statistics Arise from Interference Effects

The Keldysh-ordered full counting statistics is a quasiprobability distribution describing the fluctuations of a time-integrated quantum observable. While it is well known that this distribution can fail to be positive, the interpretation and origin of this negativity has been somewhat unclear. Here, we show how the full counting statistics can be tied to trajectories through Hilbert space, and how this directly connects negative quasiprobabilities to an unusual interference effect. Our findings are illustrated with the example of energy fluctuations in a driven bosonic resonator; we discuss how negative quasiprobability here could be detected experimentally using superconducting microwave circuits.

Stochastic thermodynamics for linear kinetic equations

Stochastic thermodynamics is formulated for variables that are odd under time reversal. The invariance under spatial rotation of the collision rates due to the isotropy of the heat bath is shown to be a crucial ingredient. An alternative detailed fluctuation theorem is derived, expressed solely in terms of forward statistics. It is illustrated for a linear kinetic equation with kangaroo rates.

Thermal transport in phononic Cayley-tree networks

We analytically investigate the heat current I and its thermal fluctuations Δ in a branching network without loops (Cayley tree). The network consists of two types of harmonic masses: vertex masses M placed at the branching points where phononic scattering occurs and masses m at the bonds between branching points where phonon propagation takes place. The network is coupled to thermal reservoirs consisting of one-dimensional harmonic chains of coupled masses m. Due to impedance mismatch phenomena, both I and Δ are non-monotonic functions of the mass ratio μ=M/m. Furthermore, we find that in the low-temperature limit the thermal conductance approaches zero faster than linearly due to the small transmittance of the long-wavelength modes.

Work fluctuations for a Brownian particle driven by a correlated external random force

We have considered the underdamped motion of a Brownian particle in the presence of a correlated external random force. The force is modeled by an Ornstein-Uhlenbeck process. We investigate the fluctuations of the work done by the external force on the Brownian particle in a given time interval in the steady state. We calculate the large deviation functions as well as the complete asymptotic form of the probability density function of the performed work. We also discuss the symmetry properties of the large deviation functions for this system. Finally we perform numerical simulations and they are in a very good agreement with the analytic results.

Heat fluctuations and initial ensembles

Time-integrated quantities such as work and heat increase incessantly in time during nonequilibrium processes near steady states. In the long-time limit, the average values of work and heat become asymptotically equivalent to each other, since they only differ by a finite energy change in average. However, the fluctuation theorem (FT) for the heat is found not to hold with the equilibrium initial ensemble, while the FT for the work holds. This reveals an intriguing effect of everlasting initial memory stored in rare events. We revisit the problem of a Brownian particle in a harmonic potential dragged with a constant velocity, which is in contact with a thermal reservoir. The heat and work fluctuations are investigated with initial Boltzmann ensembles at temperatures generally different from the reservoir temperature. We find that, in the infinite-time limit, the FT for the work is fully recovered for arbitrary initial temperatures, while the heat fluctuations significantly deviate from the FT characteristics except for the infinite initial-temperature limit (a uniform initial ensemble). Furthermore, we succeed in calculating finite-time corrections to the heat and work distributions analytically, using the modified saddle point integral method recently developed by us. Interestingly, we find noncommutativity between the infinite-time limit and the infinite-initial-temperature limit for the probability distribution function (PDF) of the heat.

Transport as a sensitive indicator of quantum criticality

We consider bosonic transport through one-dimensional spin systems. Transport is induced by coupling the spin systems to bosonic reservoirs kept at different temperatures. In the limit of weak-coupling between spins and bosons we apply the quantum-optical master equation to calculate the energy transmitted from source to drain reservoirs. At large thermal bias, we find that the current for longitudinal transport becomes independent of the chain length and is also not drastically affected by the presence of disorder. In contrast, at small temperatures, the current scales inversely with the chain length and is further suppressed in the presence of disorder. We also find that the critical behaviour of the ground state is mapped to critical behaviour of the current--even in configurations with infinite thermal bias.

Exchange fluctuation theorem for heat transport between multiterminal harmonic systems

We study full counting statistics for transferred heat and entropy production between multiterminal systems in absence of a finite junction. The systems are modeled as collections of coupled harmonic oscillators, which are kept at different equilibrium temperatures and are connected via arbitrary time-dependent couplings. Following consistent quantum framework and two-time measurement concept we obtain analytical expressions for the generalized cumulant generating function. We discuss transient and steady-state fluctuation theorems for the transferred quantities. We also address the effect of coupling strength on the exchange fluctuation theorem.

Nonequilibrium fluctuation-dissipation relations for one- and two-particle correlation functions in steady-state quantum transport

We study the non-equilibrium (NE) fluctuation-dissipation (FD) relations in the context of quantum thermoelectric transport through a two-terminal nanodevice in the steady-state. The FD relations for the one- and two-particle correlation functions are derived for a model of the central region consisting of a single electron level. Explicit expressions for the FD relations of the Green's functions (one-particle correlations) are provided. The FD relations for the current-current and charge-charge (two-particle) correlations are calculated numerically. We use self-consistent NE Green's functions calculations to treat the system in the absence and in the presence of interaction (electron-phonon) in the central region. We show that, for this model, there is no single universal FD theorem for the NE steady state. There are different FD relations for each different class of problems. We find that the FD relations for the one-particle correlation function are strongly dependent on both the NE conditions and the interactions, while the FD relations of the current-current correlation function are much less dependent on the interaction. The latter property suggests interesting applications for single-molecule and other nanoscale transport experiments.

Large-deviation statistics of a diffusive quantum spin chain and the additivity principle

Using the large-deviation formalism, we study the statistics of current fluctuations in a diffusive nonequilibrium quantum spin chain. The boundary-driven XX chain with dephasing consists of a coherent bulk hopping and a local dissipative dephasing. We analytically calculate the exact expression for the second current moment in a system of any length and then numerically demonstrate that in the thermodynamic limit, higher-order cumulants and the large-deviation function can be calculated using the additivity principle or macroscopic hydrodynamic theory. This shows that the additivity principle can also hold in systems that are not purely stochastic, and can in particular be valid in quantum systems. We also show that in large systems, the current fluctuations are the same as in the classical symmetric simple exclusion process.

Exactly solvable counting statistics in open weakly coupled interacting spin systems

We study the full counting statistics for interacting quantum many-body spin systems weakly coupled to the environment. In the leading order in the system-bath coupling, we derive exact spin current statistics for a large class of parity symmetric spin-1/2 systems driven by a pair of Markovian baths with local coupling operators. Interestingly, in this class of systems the leading-order current statistics are universal and do not depend on details of the Hamiltonian. Furthermore, in the specific case of a symmetrically boundary driven anisotropic Heisenberg (XXZ) spin-1/2 chain, we explicitly derive the third-order nonlinear corrections to the current statistics.

Exact large-deviation statistics for a nonequilibrium quantum spin chain

We consider a one-dimensional XX spin chain in a nonequilibrium setting with a Lindblad-type boundary driving. By calculating large-deviation rate function in the thermodynamic limit, a generalization of free energy to a nonequilibrium setting, we obtain a complete distribution of current, including closed expressions for lower-order cumulants. We also identify two phase-transition-like behaviors in either the thermodynamic limit, at which the current probability distribution becomes discontinuous, or at maximal driving, when the range of possible current values changes discontinuously. In the thermodynamic limit the current has a finite upper and lower bound. We also explicitly confirm nonequilibrium fluctuation relation and show that the current distribution is the same under mapping of the coupling strength Γ→1/Γ.

Heat flux and entropy produced by thermal fluctuations

We report an experimental and theoretical analysis of the energy exchanged between two conductors kept at different temperature and coupled by the electric thermal noise. Experimentally we determine, as functions of the temperature difference, the heat flux, the out-of-equilibrium variance, and a conservation law for the fluctuating entropy, which we justify theoretically. The system is ruled by the same equations as two Brownian particles kept at different temperatures and coupled by an elastic force. Our results set strong constraints on the energy exchanged between coupled nanosystems held at different temperatures.

Power fluctuation theorem for a Brownian harmonic oscillator

In this paper we study the validity of the total power fluctuation theorem spent on a Brownian harmonic oscillator when the system is driven out of equilibrium through the drag of the potential minimum. The theorem is first proved for an ordinary harmonic oscillator in two cases: The first one considers the particle in a thermal bath under the action of Gaussian white noise, and in the second one the drift is provided by an additional external Gaussian colored noise satisfying the characteristics of an Ornstein-Uhlenbeck process. We go further, by considering a charged harmonic oscillator under the action of an electromagnetic field. The theorem is also proven as in the two cases given above. In both of those cases, we illustrate the theorem for a uniform motion of the trap potential minimum and show that in the presence of external colored noise, the theorem is only valid in the stationary state.

Work fluctuations for a Brownian particle in a harmonic trap with fluctuating locations

We consider a Brownian particle in a harmonic trap. The location of the trap is modulated according to an Ornstein-Uhlenbeck process. We investigate the fluctuation of the work done by the modulated trap on the Brownian particle in a given time interval in the steady state. We compute the large deviation as well as the complete asymptotic form of the probability density function of the work done. The theoretical asymptotic forms of the probability density function are in very good agreement with the numerics. We also discuss the validity of the fluctuation theorem for this system.

Fluctuation relation for heat

We present a fluctuation relation for heat dissipation in a nonequilibrium system. A nonequilibrium work is known to obey the fluctuation theorem in any time interval t. Heat, which differs from work by an energy change, is shown to satisfy a modified fluctuation relation. Modification is brought about by the correlation between heat and energy change during nonequilibrium processes whose effect may not be negligible even in the t→∞ limit. The fluctuation relation is derived for overdamped Langevin equation systems, and tested in a linear diffusion system.