Exact fluctuation theorem without ensemble quantities

Work distribution in thermal processes

We find the moment generating function (mgf) of the nonequilibrium work for open systems undergoing a thermal process, i.e., when the stochastic dynamics maps thermal states into time-dependent thermal states. The mgf is given in terms of a temperaturelike scalar satisfying a first-order ordinary differential equation. We apply the result to some paradigmatic situations: a levitated nanoparticle in a breathing optical trap, a Brownian particle in a box with a moving piston, and a two-state system driven by an external field, where the work mgfs are obtained for different timescales and compared with Monte Carlo simulations.

Quantum heat distribution in thermal relaxation processes

We analyze the heat exchange distribution of open quantum systems undergoing a thermal relaxation process with a time-dependent effective temperature. We show that such processes arise, for example, if the dynamics maximizes the entropy production. Using a two-point measurement scheme, we find an expression for the heat moment generating function that depends solely on the system's partition function and on the thermalization function (i.e., the law of cooling) describing the effective temperature. Applications include the relaxation of free bosonic and fermionic modes, for which closed-form expressions for the time-dependent heat distribution function are derived. Multiple free modes with arbitrary dispersion relations are also briefly discussed. In the semiclassical limit our formula agrees with previous results of the literature for the heat distribution of an optically trapped nanoscopic particle far from equilibrium.

Detailed Fluctuation Theorems: A Unifying Perspective

We present a general method to identify an arbitrary number of fluctuating quantities which satisfy a detailed fluctuation theorem for all times within the framework of time-inhomogeneous Markovian jump processes. In doing so, we provide a unified perspective on many fluctuation theorems derived in the literature. By complementing the stochastic dynamics with a thermodynamic structure (i.e., using stochastic thermodynamics), we also express these fluctuating quantities in terms of physical observables.

Effective Thermodynamics for a Marginal Observer

Thermodynamics is usually formulated on the presumption that the observer has complete information about the system he or she deals with: no parasitic current, exact evaluation of the forces that drive the system. For example, the acclaimed fluctuation relation (FR), relating the probability of time-forward and time-reversed trajectories, assumes that the measurable transitions suffice to characterize the process as Markovian (in our case, a continuous-time jump process). However, most often the observer only measures a marginal current. We show that he or she will nonetheless produce an effective description that does not dispense with the fundamentals of thermodynamics, including the FR and the 2nd law. Our results stand on the mathematical construction of a hidden time reversal of the dynamics, and on the physical requirement that the observed current only accounts for a single transition in the configuration space of the system. We employ a simple abstract example to illustrate our results and to discuss the feasibility of generalizations.

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.

Entropy production and irreversibility of dissipative trajectories in electric circuits

We experimentally examine the equivalence between the entropy production evaluated from irreversibility of trajectories and the physical dissipation in dissipative processes via electric resistor-capacitor (RC) circuits. The examinations are performed for two nonequilibrium steady states that are driven by an injected current and temperature difference, respectively. Such an equivalence demonstrates a parameter-free method to evaluate the entropy production of a system. The effects of configurational and temporal resolutions are also studied.

Theoretical description of effective heat transfer between two viscously coupled beads

We analytically study the role of nonconservative forces, namely viscous couplings, on the statistical properties of the energy flux between two Brownian particles kept at different temperatures. From the dynamical model describing the system, we identify an energy flow that satisfies a fluctuation theorem both in the stationary and in transient states. In particular, for the specific case of a linear nonconservative interaction, we derive an exact fluctuation theorem that holds for any measurement time in the transient regime, and which involves the energy flux alone. Moreover, in this regime the system presents an interesting asymmetry between the hot and cold particles. The theoretical predictions are in good agreement with the experimental results already presented in our previous article [Imparato et al., Phys. Rev. Lett. 116, 068301 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.068301], where we investigated the thermodynamic properties of two Brownian particles, trapped with optical tweezers, interacting through a dissipative hydrodynamic coupling.

Dissipation in small systems: Landau-Zener approach

We establish a stochastic thermodynamics for a Fermionic level driven by a time-dependent force and interacting with initially thermalized levels playing the role of a reservoir. The driving induces consecutive avoided crossings between system and reservoir levels described within Landau-Zener theory. We derive the resulting system dynamics and thermodynamics and identify energy, work, heat, entropy, and dissipation. Our theory perfectly reproduces the numerically exact quantum work statistics obtained using a two point measurements approach of the total energy and provides an explicit expression for the dissipation in terms of diabatic transitions.

Efficiency fluctuations of small machines with unknown losses

The efficiency statistics of a small thermodynamic machine has been recently investigated assuming that the total dissipation is a linear combination of two currents: the input and output currents. Here, we relax this standard assumption and consider the question of the efficiency fluctuations for a machine involving three different currents, first in full generality and second for two different examples. Since the third current may not be measurable and/or may decrease the machine efficiency, our motivation is to study the effect of unknown losses in small machines.

Nonequilibrium thermodynamic potentials for continuous-time Markov chains

We connect the rare fluctuations of an equilibrium (EQ) process and the typical fluctuations of a nonequilibrium (NE) stationary process. In the framework of large deviation theory, this observation allows us to introduce NE thermodynamic potentials. For continuous-time Markov chains, we identify the relevant pairs of conjugated variables and propose two NE ensembles: one with fixed dynamics and fluctuating time-averaged variables, and another with fixed time-averaged variables, but a fluctuating dynamics. Accordingly, we show that NE processes are equivalent to conditioned EQ processes ensuring that NE potentials are Legendre dual. We find a variational principle satisfied by the NE potentials that reach their maximum in the NE stationary state and whose first derivatives produce the NE equations of state and second derivatives produce the NE Maxwell relations generalizing the Onsager reciprocity relations.

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.

Fluctuations of work in nearly adiabatically driven open quantum systems

We extend the quantum jump method to nearly adiabatically driven open quantum systems in a way that allows for an accurate account of the external driving in the system-environment interaction. Using this framework, we construct the corresponding trajectory-dependent work performed on the system and derive the integral fluctuation theorem and the Jarzynski equality for nearly adiabatic driving. We show that such identities hold as long as the stochastic dynamics and work variable are consistently defined. We numerically study the emerging work statistics for a two-level quantum system and find that the conventional diabatic approximation is unable to capture some prominent features arising from driving, such as the continuity of the probability density of work. Our results reveal the necessity of using accurate expressions for the drive-dressed heat exchange in future experiments probing jump time distributions.

Echo states for detailed fluctuation theorems

Detailed fluctuation theorems are statements about the probability distribution for the stochastic entropy production along a trajectory. It involves the consideration of a suitably transformed dynamics, such as the time reversed, the adjoint, or a combination of these. We identify specific, typically unique, initial conditions, called echo states, for which the final probability distribution of the transformed dynamics reproduces the initial distribution. In this case the detailed fluctuation theorems relate the stochastic entropy production of the direct process to that of the transformed one. We illustrate our results by an explicit analytical calculation and numerical simulations for a modulated two-state quantum dot.

Experimental observation of the role of mutual information in the nonequilibrium dynamics of a Maxwell demon

We validate experimentally a fluctuation relation known as generalized Jarzynski equality governing the work distribution in a feedback-controlled system. The feedback control is performed on a single electron box analogously to the original Szilard engine. In the generalized Jarzynski equality, mutual information is treated on an equal footing with the thermodynamic work. Our measurements provide the first evidence of the role of mutual information in the fluctuation theorem and thermodynamics of irreversible processes.