Fluctuation relation for heat

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.

Convergence to the Asymptotic Large Deviation Limit

Large deviation theory offers a powerful and general statistical framework to study the asymptotic dynamical properties of rare events. The application of the formalism to concrete experimental situations is, however, often restricted by finite statistics. Data might not suffice to reach the asymptotic regime or judge whether large deviation estimators converge at all. We here experimentally study the large deviation properties of the stochastic work and heat of a levitated nanoparticle subjected to nonequilibrium feedback control. This setting allows us to determine for each quantity the convergence domain of the large deviation estimators using a criterion that does not require the knowledge of the probability distribution. By extracting both the asymptotic exponential decay and the subexponential prefactors, we demonstrate that singular prefactors significantly restrict the convergence characteristics close to the singularity. Our results provide unique insight into the approach to the asymptotic large deviation limit and underscore the pivotal role of singular prefactors.

Effects of the non-Markovianity and non-Gaussianity of active environmental noises on engine performance

An active environment is a reservoir containing active materials, such as bacteria and Janus particles. Given the self-propelled motion of these materials, powered by chemical energy, an active environment has unique, nonequilibrium environmental noise. Recently, studies on engines that harvest energy from active environments have attracted a great deal of attention because the theoretical and experimental findings indicate that these engines outperform conventional ones. Studies have explored the features of active environments essential for outperformance, such as the non-Gaussian or non-Markovian nature of the active noise. We systematically study the effects of the non-Gaussianity and non-Markovianity of active noise on engine performance. We show that non-Gaussianity is irrelevant to the performance of an engine driven by any linear force (including a harmonic trap) regardless of time dependency, whereas non-Markovianity is relevant. However, for a system driven by a general nonlinear force, both non-Gaussianity and non-Markovianity enhance engine performance. Also, the memory effect of an active reservoir should be considered when fabricating a cyclic engine.

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.

Quantum Fluctuation Theorems beyond Two-Point Measurements

We derive detailed and integral quantum fluctuation theorems for heat exchange in a quantum correlated bipartite thermal system using the framework of dynamic Bayesian networks. Contrary to the usual two-projective-measurement scheme that is known to destroy quantum features, these fluctuation relations fully capture quantum correlations and quantum coherence at arbitrary times. We further obtain individual integral fluctuation theorems for classical and quantum correlations, as well as for local and global quantum coherences.

Additivity of multiple heat reservoirs in the Langevin equation

The Langevin equation greatly simplifies the mathematical expression of the effects of thermal noise by using only two terms, a dissipation term, and a random-noise term. The Langevin description was originally applied to a system in contact with a single heat reservoir; however, many recent studies have also adopted a Langevin description for systems connected to multiple heat reservoirs. This is accomplished through the introduction of a simple summation for the dissipation and random-noise terms associated with each reservoir. However, the validity of this simple addition has been the focus of only limited discussion and has raised several criticisms. Moreover, this additive description has never been either experimentally or numerically verified, rendering its validity is still an open question. Here we perform molecular dynamics simulations for a Brownian system in simultaneous contact with multiple heat reservoirs to check the validity of this additive approach. Our simulation results confirm that the effect of multiple heat reservoirs is additive in general. A very small deviation in the total amount of dissipation and associated noise is found but seems not significant within statistical errors. We find that the steady-state properties satisfy the additivity perfectly and are not affected by this deviation.

Derivation of Markov processes that violate detailed balance

Time-reversal symmetry of the microscopic laws dictates that the equilibrium distribution of a stochastic process must obey the condition of detailed balance. However, cyclic Markov processes that do not admit equilibrium distributions with detailed balance are often used to model systems driven out of equilibrium by external agents. I show that for a Markov model without detailed balance, an extended Markov model can be constructed, which explicitly includes the degrees of freedom for the driving agent and satisfies the detailed balance condition. The original cyclic Markov model for the driven system is then recovered as an approximation at early times by summing over the degrees of freedom for the driving agent. I also show that the widely accepted expression for the entropy production in a cyclic Markov model is actually a time derivative of an entropy component in the extended model. Further, I present an analytic expression for the entropy component that is hidden in the cyclic Markov model.

Emergence of nonwhite noise in Langevin dynamics with magnetic Lorentz force

We investigate the low mass limit of Langevin dynamics for a charged Brownian particle driven by a magnetic Lorentz force. In the low mass limit, velocity variables relaxing quickly are coarse-grained out to yield effective dynamics for position variables. Without the Lorentz force, the low mass limit is equivalent to the high friction limit. Both cases share the same Langevin equation that is obtained by setting the mass to zero. The equivalence breaks down in the presence of the Lorentz force. The low mass limit cannot be achieved by setting the mass to zero. The limit is also distinct from the large friction limit. We derive the effective equations of motion in the low mass limit. The resulting stochastic differential equation involves a nonwhite noise whose correlation matrix has antisymmetric components. We demonstrate the importance of the nonwhite noise by investigating the heat dissipation by a driven Brownian particle, where the emergent nonwhite noise has a physically measurable effect.

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.

Heat fluctuations of Brownian oscillators in nonstationary processes: Fluctuation theorem and condensation transition

We study analytically the probability distribution of the heat released by an ensemble of harmonic oscillators to the thermal bath, in the nonequilibrium relaxation process following a temperature quench. We focus on the asymmetry properties of the heat distribution in the nonstationary dynamics, in order to study the forms taken by the fluctuation theorem as the number of degrees of freedom is varied. After analyzing in great detail the cases of one and two oscillators, we consider the limit of a large number of oscillators, where the behavior of fluctuations is enriched by a condensation transition with a nontrivial phase diagram, characterized by reentrant behavior. Numerical simulations confirm our analytical findings. We also discuss and highlight how concepts borrowed from the study of fluctuations in equilibrium under symmetry-breaking conditions [Gaspard, J. Stat. Mech. (2012) P0802110.1088/1742-5468/2012/08/P08021] turn out to be quite useful in understanding the deviations from the standard fluctuation theorem.

Transient exchange fluctuation theorems for heat using a Hamiltonian framework: Classical and quantum regimes

We investigate the statistics of heat exchange between a finite system coupled to reservoir(s). We have obtained analytical results for heat fluctuation theorems in the transient regime considering the Hamiltonian dynamics of the composite system consisting of the system of interest and the heat bath(s). The system of interest is driven by an external protocol. We first derive it in the context of a single heat bath. The result is in exact agreement with known result. We then generalize the treatment to two heat baths. We further extend the study to quantum systems and show that relations similar to the classical case hold in the quantum regime. For our study we invoke von Neumann two-point projective measurement in quantum mechanics in the transient regime. The study of quantum systems follows the same lines of argument as that of the classical system, and as a result the treatment used in the latter complements that used in the former. Our result is a generalization of Jarzynski-Wòjcik heat fluctuation theorem.

Stochastic thermodynamics of Langevin systems under time-delayed feedback control. II. Nonequilibrium steady-state fluctuations

This paper is the second in a series devoted to the study of Langevin systems subjected to a continuous time-delayed feedback control. The goal of our previous paper [Phys. Rev. E 91, 042114 (2015)PLEEE81539-375510.1103/PhysRevE.91.042114] was to derive second-law-like inequalities that provide bounds to the average extracted work. Here we study stochastic fluctuations of time-integrated observables such as the heat exchanged with the environment, the extracted work, or the (apparent) entropy production. We use a path-integral formalism and focus on the long-time behavior in the stationary cooling regime, stressing the role of rare events. This is illustrated by a detailed analytical and numerical study of a Langevin harmonic oscillator driven by a linear feedback.

Overdamped stochastic thermodynamics with multiple reservoirs

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

Efficiency at maximum power and efficiency fluctuations in a linear Brownian heat-engine model

We investigate the stochastic thermodynamics of a two-particle Langevin system. Each particle is in contact with a heat bath at different temperatures T_{1} and T_{2} (<T_{1}), respectively. Particles are trapped by a harmonic potential and driven by a linear external force. The system can act as an autonomous heat engine performing work against the external driving force. Linearity of the system enables us to examine thermodynamic properties of the engine analytically. We find that the efficiency of the engine at maximum power η_{MP} is given by η_{MP}=1-sqrt[T_{2}/T_{1}]. This universal form has been known as a characteristic of endoreversible heat engines. Our result extends the universal behavior of η_{MP} to nonendoreversible engines. We also obtain the large deviation function of the probability distribution for the stochastic efficiency in the overdamped limit. The large deviation function takes the minimum value at macroscopic efficiency η=η[over ¯] and increases monotonically until it reaches plateaus when η≤η_{L} and η≥η_{R} with model-dependent parameters η_{R} and η_{L}.

Hidden entropy production by fast variables

We investigate nonequilibrium underdamped Langevin dynamics of Brownian particles that interact through a harmonic potential with coupling constant K and are in thermal contact with two heat baths at different temperatures. The system is characterized by a net heat flow and an entropy production in the steady state. We compare the entropy production of the harmonic system with that of Brownian particles linked with a rigid rod. The harmonic system may be expected to reduce to the rigid rod system in the infinite K limit. However, we find that the harmonic system in the K→∞ limit produces more entropy than the rigid rod system. The harmonic system has the center-of-mass coordinate as a slow variable and the relative coordinate as a fast variable. By identifying the contributions of the degrees of freedom to the total entropy production, we show that the hidden entropy production by the fast variable is responsible for the extra entropy production. We discuss the K dependence of each contribution.

Work relations connecting nonequilibrium steady states without detailed balance

Bridging equilibrium and nonequilibrium statistical physics attracts sustained interest. Hallmarks of nonequilibrium systems include a breakdown of detailed balance, and an absence of a priori potential function corresponding to the Boltzmann-Gibbs distribution, without which classical equilibrium thermodynamical quantities could not be defined. Here, we construct dynamically the potential function through decomposing the system into a dissipative part and a conservative part, and develop a nonequilibrium theory by defining thermodynamical quantities based on the potential function. Concepts for equilibrium can thus be naturally extended to nonequilibrium steady state. We elucidate this procedure explicitly in a class of time-dependent linear diffusive systems without mathematical ambiguity. We further obtain the exact work distribution for an arbitrary control parameter, and work equalities connecting nonequilibrium steady states. Our results provide a direct generalization on Jarzynski equality and Crooks fluctuation theorem to systems without detailed balance.

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.

Energy pumping in electrical circuits under avalanche noise

We theoretically study energy pumping processes in an electrical circuit with avalanche diodes, where non-Gaussian athermal noise plays a crucial role. We show that a positive amount of energy (work) can be extracted by an external manipulation of the circuit in a cyclic way, even when the system is spatially symmetric. We discuss the properties of the energy pumping process for both quasistatic and finite-time cases, and analytically obtain formulas for the amounts of the work and the power. Our results demonstrate the significance of the non-Gaussianity in energetics of electrical circuits.

Multiple dynamic transitions in nonequilibrium work fluctuations

The time-dependent work probability distribution function P(W) is investigated analytically for a diffusing particle trapped by an anisotropic harmonic potential and driven by a nonconservative drift force in two dimensions. We find that the exponential tail shape of P(W) characterizing rare-event probabilities undergoes a sequence of dynamic transitions in time. These remarkable locking-unlocking type transitions result from an intricate interplay between a rotational mode induced by the nonconservative force and an anisotropic decaying mode due to the conservative attractive force. We expect that most of the high-dimensional dynamical systems should exhibit similar multiple dynamic transitions.

Heat conduction induced by non-Gaussian athermal fluctuations

We study the properties of heat conduction induced by non-Gaussian noises from athermal environments. We find that new terms should be added to the conventional Fourier law and the fluctuation theorem for the heat current, where its average and fluctuation are determined not only by the noise intensities but also by the non-Gaussian nature of the noises. Our results explicitly show the absence of the zeroth law of thermodynamics in athermal systems.

Everlasting initial memory threshold for rare events in equilibration processes

Conventional wisdom indicates that initial memory should decay away exponentially in time for general (noncritial) equilibration processes. In particular, time-integrated quantities such as heat are presumed to lose initial memory in a sufficiently long-time limit. However, we show that the large deviation function of time-integrated quantities may exhibit initial memory effect even in the infinite-time limit, if the system is initially prepared sufficiently far away from equilibrium. For a Brownian particle dynamics, as an example, we found a sharp finite threshold rigorously, beyond which the corresponding large deviation function contains everlasting initial memory. The physical origin for this phenomenon is explored with an intuitive argument and also from a toy model analysis. Our results can be applied to general nonequilibrium relaxation processes reaching (non)equilibrium steady states.

Zon-Cohen singularity and negative inverse temperature in a trapped-particle limit

We study a Brownian particle on a moving periodic potential. We focus on the statistical properties of the work done by the potential and the heat dissipated by the particle. When the period and the depth of the potential are both large, by using a boundary layer analysis, we calculate a cumulant generating function and a biased distribution function. The result allows us to understand a Zon-Cohen singularity for an extended fluctuation theorem from a viewpoint of rare trajectories characterized by a negative inverse temperature of the biased distribution function.