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arXiv:1504.05111 · Physical Review A

Fluctuations in single-shot ε -deterministic work extraction

Sina Salek, Karoline Wiesner

Published Nov 13, 2017• 14 citations
arXivHTMLPDF

Abstract

There has been an increasing interest in the quantification of nearly deterministic work extraction from a finite number of copies of microscopic particles in finite time. This paradigm, the so-called single-shot $\ensuremath{\epsilon}$-deterministic work extraction, considers processes with small failure probabilities. However, the resulting fluctuations in the extracted work entailed by this failure probability have yet to be studied. In the standard thermodynamics, paradigm fluctuation theorems are powerful tools to study fluctuating quantities. Given that standard fluctuation theorems are inadequate for a single-shot scenario, here we formulate and prove a fluctuation relation specific to the single-shot $\ensuremath{\epsilon}$-deterministic work extraction to bridge this gap. Our results are general in the sense that we allow the system to be in contact with the heat bath at all times. As a corollary of our theorem, we derive the known bounds on the $\ensuremath{\epsilon}$-deterministic work.

DOI
10.1103/physreva.96.052114

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There has been an increasing interest in the quantification of nearly deterministic work extraction from a finite number of copies of microscopic particles in finite time. This paradigm, the so-called single-shot \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-deterministic work extraction, considers processes with small failure probabilities. However, the resulting fluctuations in the extracted work entailed by this failure probability have yet to be studied. In the standard thermodynamics, paradigm fluctuation theorems are powerful tools to study fluctuating quantities. Given that standard fluctuation theorems are inadequate for a single-shot scenario, here we formulate and prove a fluctuation relation specific to the single-shot \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-deterministic work extraction to bridge this gap. Our results are general in the sense that we allow the system to be in contact with the heat bath at all times. As a corollary of our theorem, we derive the known bounds on the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-deterministic work.
Recent developments of the analysis of the second law of thermodynamics by using the information theoretic quantities has led to the discovery of the so-called nonequilibrium free energy and has allowed us to formulate thermodynamic relations for arbitrary nonequilibrium initial and final states.By using the nonequilibrium free energies, we can quantify thermodynamic costs of information encoding and erasure processes as well as to quantify the extractable work from information heat engines in a unified way.In those general situations, reducing dissipation allows us to increase the extractable work from the system (or to reduce the work cost), and reducing work fluctuation allows us to narrow the work probability distribution and to extract a fluctuation-free work from the system.Suppressing both work fluctuation and dissipation as much as possible is vital to control mesoscopic and nano systems that work at the level of thermal fluctuations.Previous studies have searched for a vanishing work fluctuation protocol by using techniques of quantum information theory, known as the single-shot statistical mechanics (and sometimes called the deterministic work extraction protocol) and the vanishing dissipation protocol, known as the thermodynamically reversible protocol by using the second law of thermodynamics.In this thesis, we address issues concerning a method to control nonequilibrium systems by using fluctuation theorems and trade-off relation between work fluctuation and fluctuation in dissipation.We derive quantum fluctuation theorems under measurement and feedback control, which allow us to study how to suppress dissipation during a general quantum measurement and a feedback control.We then consider a method to suppress work fluctuation and fluctuation in dissipation simultaneously as much as possible by showing the trade-off relation between work and dissipation in terms of their standard deviations and derive an explicit protocol that achieves the lower bound of the trade-off relation.We show that the explicit protocol reproduces the deterministic work extraction protocol in the limit of vanishing work fluctuation and the thermodynamically reversible protocol in the limit of vanishing fluctuation in dissipation.
According to the second law of thermodynamics, for every transformation performed on a system which is in contact with an environment of fixed temperature, the average extracted work is bounded by the decrease of the free energy of the system. However, in a single realization of a generic process, the extracted work is subject to statistical fluctuations which may allow for probabilistic violations of the previous bound. We are interested in enhancing this effect, i.e. we look for thermodynamic processes that maximize the probability of extracting work above a given arbitrary threshold. For any process obeying the Jarzynski identity, we determine an upper bound for the work extraction probability that depends also on the minimum amount of work that we are willing to extract in case of failure, or on the average work we wish to extract from the system. Then we show that this bound can be saturated within the thermodynamic formalism of quantum discrete processes composed by sequences of unitary quenches and complete thermalizations. We explicitly determine the optimal protocol which is given by two quasi-static isothermal transformations separated by a finite unitary quench.
Abstract We revisit the problem of work extraction from a system in contact with a heat bath to a work storage system, and the reverse problem of state formation from a thermal system state in single-shot quantum thermodynamics. A physically intuitive and mathematically simple approach using only elementary majorization theory and matrix analysis is developed, and a graphical interpretation of the maximum extractable work, minimum work cost of formation, and corresponding single-shot free energies is presented. This approach provides a bridge between two previous methods based respectively on the concept of thermomajorization and a comparison of subspace dimensions. In addition, a conceptual inconsistency with regard to general work extraction involving transitions between multiple energy levels of the work storage system is clarified and resolved. It is shown that an additional contribution to the maximum extractable work in those general cases should be interpreted not as work extracted from the system, but as heat transferred from the heat bath. Indeed, the additional contribution is an artifact of a work storage system (essentially a suspended ‘weight’ that can be raised or lowered) that does not truly distinguish work from heat. The result calls into question the common concept that a work storage system in quantum thermodynamics is simply the quantum version of a suspended weight in classical thermodynamics.
Abstract We consider work extraction from N copies of a quantum system. When the same work-extraction process is implemented on each copy, the relative size of fluctuations is expected to decay as <?CDATA 1/N1/\sqrt{N}1/N​?> 1 / N . Here, we consider protocols where the copies can be processed collectively, and show that in this case work fluctuations can disappear exponentially fast in N . As a consequence, a considerable proportion of the average extractable work <?CDATA W{ \mathcal W }W?>  can be obtained almost deterministically by globally processing a few copies of the state. This is derived in the two canonical scenarios for work extraction: (i) in thermally isolated systems, where <?CDATA W{ \mathcal W }W?>  corresponds to the energy difference between initial and passive states, known as the ergotropy, and (ii) in the presence of a thermal bath, where <?CDATA W{ \mathcal W }W?>  is given by the free energy difference between initial and thermal states.
Abstract We consider work extraction from N copies of a quantum system. When the same work-extraction process is implemented on each copy, the relative size of fluctuations is expected to decay as 1 / N . Here, we consider protocols where the copies can be processed collectively, and show that in this case work fluctuations can disappear exponentially fast in N. As a consequence, a considerable proportion of the average extractable work  can be obtained almost deterministically by globally processing a few copies of the state. This is derived in the two canonical scenarios for work extraction: (i) in thermally isolated systems, where  corresponds to the energy difference between initial and passive states, known as the ergotropy, and (ii) in the presence of a thermal bath, where  is given by the free energy difference between initial and thermal states.
We study the possibility of deterministic protocols for extracting work from quantum systems. Focusing on the two-point measurement work extraction scenario, we prove that, with enough copies of the system, such zero-fluctuation protocols always exist if the Hamiltonian has a rational spectrum. Leveraging this result, we show that for any Hamiltonian, it is possible to construct a unitary driving protocol on sufficiently many copies of the system with work fluctuations strictly bounded within an arbitrary interval $\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}$, albeit requiring exponentially many copies in $1/\ensuremath{\delta}$.