We introduce and realize demons that follow a customary gambling strategy to stop a nonequilibrium process at stochastic times. We derive second-law-like inequalities for the average work done in the presence of gambling, and universal stopping-time fluctuation relations for classical and quantum non-stationary stochastic processes. We test experimentally our results in a single-electron box, where an electrostatic potential drives the dynamics of individual electrons tunneling into a metallic island. We also discuss the role of coherence in gambling demons measuring quantum jump trajectories.
The transfer entropy is a well-established measure of information flow, which quantifies directed influence between two stochastic time series and has been shown to be useful in a variety fields of science. Here we introduce the transfer entropy of the backward time series called the backward transfer entropy, and show that the backward transfer entropy quantifies how far it is from dynamics to a hidden Markov model. Furthermore, we discuss physical interpretations of the backward transfer entropy in completely different settings of thermodynamics for information processing and the gambling wi…
We study the physical mechanism of Maxwell's Demon (MD) helping to do extra work in thermodynamic cycles, by describing measurement of position, insertion of wall and information erasing of MD in a quantum mechanical fashion. The heat engine is exemplified with one molecule confined in an infinitely deep square potential inserted with a movable solid wall, while the MD is modeled as a two-level system (TLS) for measuring and controlling the motion of the molecule. It is discovered that the the MD with quantum coherence or on a lower temperature than that of the heat bath of the particle would …
We propose a new thermodynamic equality and several inequalities concerning the relationship between work and information for an isothermal process with Maxwell's demon. Our approach is based on the formulation a la Jarzynski of the thermodynamic engine and on the quantum information-theoretic characterization of the demon. The lower bound of each inequality, which is expressed in terms of the information gain by the demon and the accuracy of the demon's measurement, gives the minimum work that can be performed on a single heat bath in an isothermal process. These results are independent of th…
A system in equilibrium does not evolve -- time independence is its telltale characteristic. However, in Newtonian physics the microstate of an individual system (a point in its phase space) evolves incessantly in accord with its equations of motion. Ensembles were introduced in XIX century to bridge that chasm between continuous motion of phase space points in Newtonian dynamics and stasis of thermodynamics: While states of individual classical systems inevitably evolve, a phase space distribution of such states -- an ensemble -- can be time-independent. I show that entanglement (e.g., with t…
We review theory of information thermodynamics which incorporates effects of measurement and feedback into nonequilibrium thermodynamics of a small system, and discuss how the second law of thermodynamics should be extended for such situations. We address the issue of the maximum work that can be extracted from the system in the presence of a feedback controller (Maxwell's demon) and provide a few illustrative examples. We also review a recent experiment that realized a Maxwell's demon based on a feedback-controlled ratchet.
We propose a way to incorporate the effect of a specific class of feedback processes into stochastic thermodynamics. These "Maxwell demon" feedbacks do not affect the system energetics but only the energy barriers between the system states (in a way which depends on the system states). They are thus of a purely informational nature. We show that the resulting formalism can be applied to study the thermodynamic effect of a feedback process acting on electron transfers through a junction.
In his famous letter in 1870, Maxwell describes how Joule's law can be violated "only by the intelligent action of a mere guiding agent", later coined as Maxwell's demon by Lord Kelvin. In this letter we study thermodynamics of information using an experimentally feasible Maxwell's demon setup based a single electron transistor capacitively coupled to a single electron box, where both the system and the Demon can be clearly identified. Such an engineered on-chip Demon measures and performes feedback on the system, which can be observed as cooling whose efficiency can be adjusted. We present a …
We introduce a deterministic chaotic system---the Szilard Map---that encapsulates the measurement, control, and erasure protocol by which Maxwellian Demons extract work from a heat reservoir. Implementing the Demon's control function in a dynamical embodiment, our construction symmetrizes Demon and thermodynamic system, allowing one to explore their functionality and recover the fundamental trade-off between the thermodynamic costs of dissipation due to measurement and due to erasure. The map's degree of chaos---captured by the Kolmogorov-Sinai entropy---is the rate of energy extraction from t…
We present a physical implementation of a Maxwell demon which consists of a conventional single electron transistor (SET) capacitively coupled to another quantum dot detecting its state. Altogether, the system is described by stochastic thermodynamics. We identify the regime where the energetics of the SET is not affected by the detection, but where its coarse-grained entropy production is shown to contain a new contribution compared to the isolated SET. This additional contribution can be identified as the information flow generated by the "Maxwell demon" feedback in an idealized limit.
It is nearly 150 years since Maxwell challenged the validity of the second law of thermodynamics by imagining a tiny creature who could sort the molecules of a gas in such a way that would decrease entropy without exerting any work. The demon has been discussed largely using thought experiments, but it has recently become possible to exert control over nanoscale systems, just as Maxwell imagined, and the status of the second law has become a more practical matter, raising the issue of how measurements manage our ignorance in a way that can be exploited. The framework of stochastic thermodynami…
We consider the stationary state of a Markov process on a bipartite system from the perspective of stochastic thermodynamics. One subsystem is used to extract work from a heat bath while being affected by the second subsystem. We show that the latter allows for a transparent and thermodynamically consistent interpretation of a Maxwell's demon. Moreover, we obtain an integral fluctuation theorem involving the transfer entropy from one subsystem to the other. Comparing three different inequalities, we show that the entropy decrease of the first subsystem provides a tighter bound on the rate of e…
We expand the standard thermodynamic framework of a system coupled to a thermal reservoir by considering a stream of independently prepared units repeatedly put into contact with the system. These units can be in any nonequilibrium state and interact with the system with an arbitrary strength and duration. We show that this stream constitutes an effective resource of nonequilibrium free energy and identify the conditions under which it behaves as a heat, work or information reservoir. We also show that this setup provides a natural framework to analyze information erasure ("Landauer's principl…
In the past decade several theoretical Maxwell's demon models have been proposed exhibiting effects such as refrigerating, doing work at the cost of information, and some experiments have been done to realise these effects. Here we propose a model with a two level demon, information represented by a sequence of bits, and two heat reservoirs. Which reservoir the demon interact with depends on the bit. If information is pure, one reservoir will be refrigerated, on the other hand, information can be erased if temperature difference is large. Genuine examples of such a system are discussed.
We investigate a weak measurement described by a von Neumann type interaction $\hat{A}\otimes \hat{p}^2$, where $\hat{A}$ is a system observable and $\hat{p}^2$ is a measurement pointer observable. We consider the weak measurement in terms of thermodynamics by adopting a mixed Gaussian state as a quantum state of the measurement pointer. We show that Maxwell's demon appears as a measure who carries out post-selections. It is found that, even if the demon only knows the weak value, a difference in the von Neumann entropy between the initial and final system can be the QC mutual information cont…
We develop an exact quantum thermodynamic description for a noninteracting nanoscale steady state that couples strongly with multiple reservoirs. It is demonstrated that there exists a steady-state extension of the thermodynamic function that correctly accounts for the multiterminal Landauer-Büttiker formula of quantum transport of charge, energy or heat, via the nonequilibrium thermodynamic relations. Its explicit form is obtained for a single bosonic or fermionic level in the wide-band limit, and corresponding thermodynamic forces (affinities) are identified. Nonlinear generalization of the …
An equation of state for an ideal gas with a small number of particles is studied. The resulting equation is found to differ from that expected in conventional thermodynamics, which is strikingly illustrated when considering the traditional thermodynamic problem of Maxwell's demon. We clarify the mechanism of this different feature of thermodynamics arising in small systems.
The second law of thermodynamics dictates the fundamental limits to the amount of energy and information that can be exchanged between physical systems. In this work, we extend a thermodynamic formalism describing this flow of energy and information developed for a pair of bipartite systems to many multipartite systems. We identify a natural thermodynamic quantity that describes the information exchanged among these systems. We then introduce and discuss a refined version. Our results are illustrated with a model of two, competing Maxwell demons.
We develop a consistent stochastic thermodynamics for environments composed of thermodynamic reservoirs in an external conservative force field, that is environments described by the Generalized or Gibbs canonical ensemble. We demonstrate that small systems weakly coupled to such reservoirs exchange both heat and work by verifying a local detailed balance relation for the induced stochastic dynamics. Based on this analysis, we help to rationalize the observation that nonthermal reservoirs can increase the efficiency of thermodynamic heat engines.
We propose a setup based on two coupled quantum dots where thermodynamics of a measurement can be quantitatively characterized. The information obtained in the measurement can be utilized by performing feedback in a manner apparently breaking the second law of thermodynamics. In this way the setup can be operated as a Maxwell's Demon where both the measurement and feedback are performed separately by controlling an external parameter. This is analogous to the case of the original Szilard engine. Since the setup contains both the microscopic demon and the engine itself, the operation of the who…
We reformulate stochastic thermodynamics in terms of noise realizations for Langevin systems in contact with multiple reservoirs and investigated the structure of the second laws of thermodynamics. We derive a hierarchy of fluctuation theorems when one degree of freedom of the system is affected by multiple reservoirs simultaneously, that is, when noise mixing occurs. These theorems and the associated second laws of thermodynamics put stricter bounds on the thermodynamics of Langevin systems. We apply our results to a stochastic machine in noise-mixing environments and demonstrate that our new…
According to Landauer's principle, erasure of information is the only part of a computation process that unavoidably involves energy dissipation. If done reversibly, such an erasure generates the minimal heat of $k_BT\ln 2$ per erased bit of information. The goal of this work is to discuss the actual reversal of the optimal erasure which can serve as the basis for the Maxwell's demon operating with ultimate thermodynamic efficiency as dictated by the second law of thermodynamics. The demon extracts $k_BT\ln 2$ of heat from an equilibrium reservoir at temperature $T$ per one bit of information …
Signal transduction in living cells is vital to maintain life itself, where information transfer in noisy environment plays a significant role. In a rather different context, the recent intensive researches of "Maxwell's demon" - a feedback controller that utilizes information of individual molecules - has led to a unified theory of information and thermodynamics. Here we combine these two streams of researches, and show that the second law of thermodynamics with information reveals the fundamental limit of the robustness of signal transduction against environmental fluctuations. Especially, w…
We introduce a family of Maxwellian Demons for which correlations among information bearing degrees of freedom can be calculated exactly and in compact analytical form. This allows one to precisely determine Demon functional thermodynamic operating regimes, when previous methods either misclassify or simply fail due to approximations they invoke. This reveals that these Demons are more functional than previous candidates. They too behave either as engines, lifting a mass against gravity by extracting energy from a single heat reservoir, or as Landauer erasers, consuming external work to remove…
The common saying, that information is power, takes a rigorous form in stochastic thermodynamics, where a quantitative equivalence between the two helps explain the paradox of Maxwell's demon in its ability to reduce entropy. In the present paper, we build on earlier work on the interplay between the relative cost and benefits of information in producing work in cyclic operation of thermodynamic engines (by Sandberg etal. 2014). Specifically, we study the general case of overdamped particles in a time-varying potential (control action) in feedback that utilizes continuous measurements (nonline…
