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arXiv:2607.09615

Fluctuation theorems for thermally isolated driven quantum systems: nonadiabaticity, excess work and strong inequalities

J. V. M. Steimetz, M. Campisi, M. V. S. Bonança

cond-mat.stat-mechquant-phSubmitted Jul 10, 2026• Announced Jul 13, 2026• 0 citations
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Abstract

We expand on the ideas developed by C. Jarzynski in Physica A 552, 122077 (2020), where an integral fluctuation theorem was derived with the aim of obtaining thermodynamic inequalities stronger than those implied by the Jarzynski equality. Restricting ourselves to the quantum setting, we derive the corresponding detailed fluctuation theorem and additional detailed and integral fluctuation theorems; we also provide a clear physical interpretation of the stochastic quantities defined in the previous reference. Furthermore, we show that their averages are given by the nonadiabaticity parameter (i.e., the relative entropy between the final state after a finite-time driving protocol and the corresponding adiabatically evolved state) and the excess work (also known as inner friction). We elaborate on the inequalities derived from the fluctuation theorems and discuss their connection to irreversibility and formulations of the Second Law.

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19 pages, 6 figures

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We expand on the ideas developed by C. Jarzynski in Physica A 552, 122077 (2020), where an integral fluctuation theorem was derived with the aim of obtaining thermodynamic inequalities stronger than those implied by the Jarzynski equality. Restricting ourselves to the quantum setting, we derive the corresponding detailed fluctuation theorem and additional detailed and integral fluctuation theorems; we also provide a clear physical interpretation of the stochastic quantities defined in the previous reference. Furthermore, we show that their averages are given by the nonadiabaticity parameter (i.e., the relative entropy between the final state after a finite-time driving protocol and the corresponding adiabatically evolved state) and the excess work (also known as inner friction). We elaborate on the inequalities derived from the fluctuation theorems and discuss their connection to irreversibility and formulations of the Second Law.
The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.
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.
We discuss the thermodynamics of closed quantum systems driven out of equilibrium by a change in a control parameter and undergoing a unitary process. We compare the work actually done on the system with the one that would be performed along ideal adiabatic and isothermal transformations. The comparison with the latter leads to the introduction of irreversible work, while that with the former leads to the introduction of inner friction. We show that these two quantities can be treated on an equal footing, as both can be linked with the heat exchanged in thermalization processes and both can be expressed as relative entropies. Furthermore, we show that a specific fluctuation relation for the entropy production associated with the inner friction exists, which allows the inner friction to be written in terms of its cumulants.
Firstly the fluctuation theorems (FT) for expended work in a driven nonequilibrium system, isolated or thermostatted, together with the ensuing Jarzynski work-energy (W-E) relationships, will be discussed and reobtained. Secondly, the fluctuation theorems for entropy flow will be reconsidered. Our treatment will be fully quantum-statistical, being an ex-tension of our previous research reported in Phys. Rev. E (2012), and will avoid the deficiencies that afflicted previous works such as: arguments based on classical trajectories in phase space, a reliance on the 'pure' von Neumann equation or 'non-reduced' Heisenberg operators, or other departures from the general tenets spelled out by Lindblad and others (e.g. Breuer and Petruccione) such as stochastic 'jump-induced' random trajectories. While a number of relationships from such previous works will still be employed, our Markov probability P(σf,tf∣σ0,t0)P(σ_f,t_f|σ_0,t_0)P(σf​,tf​∣σ0​,t0​) shall only denote the two state-points, with no reference whatsoever to stochastic trajectories, these being meaningless in a quantum description.
Firstly the fluctuation theorems (FT) for expended work in a driven nonequilibrium system, isolated or thermostatted, together with the ensuing Jarzynski work-energy (W-E) relationships, will be discussed and reobtained. Secondly, the fluctuation theorems for entropy flow will be reconsidered. Our treatment will be fully quantum-statistical, being an ex-tension of our previous research reported in Phys. Rev. E (2012), and will avoid the deficiencies that afflicted previous works such as: arguments based on classical trajectories in phase space, a reliance on the 'pure' von Neumann equation or 'non-reduced' Heisenberg operators, or other departures from the general tenets spelled out by Lindblad and others (e.g. Breuer and Petruccione) such as stochastic 'jump-induced' random trajectories. While a number of relationships from such previous works will still be employed, our Markov probability P(σf,tf∣σ0,t0)P(\sigma_f,t_f|\sigma_0,t_0)P(σf​,tf​∣σ0​,t0​) shall only denote the two state-points, with no reference whatsoever to stochastic trajectories, these being meaningless in a quantum description.
This article sets up a formalism to describe stochastic thermodynamics for driven out-of-equilibrium open quantum systems. A stochastic Schrödinger equation allows to construct quantum trajectories describing the dynamics of the system state vector in presence of an eventually monitored environment. Thermodynamic quantities are defined at the single quantum trajectory level, independently of any energy measurement, at any time of the protocol. We thereby identify coherent contributions, without classical counterparts, leading to quantum fluctuations of thermodynamic quantities. This formalism eventually leads to central fluctuation theorems for entropy, extending in the quantum regime results obtained in classical stochastic thermodynamics. The thermal imprint of coherences on a simple implementation of Jarzynski equality is investigated, opening avenues for a thermodynamic approach to decoherence.