We present a complete-quantum description of multi-particle Szilard engine which consists of a working substance and a Maxwell's demon. The demon is modeled as a multi-level quantum system with specific quantum control and the working substance consists of identical particles obeying Bose-Einstein or Fermi-Dirac statistics. In this description, a reversible scheme to erase the demon's memory by a lower temperature heat bath is used. We demonstrate that (1) the quantum control of the demon can be optimized for single-particle Szilard engine so that the efficiency of the demon-assisted thermodynamic cycle could reach the Carnot cycle's efficiency; (2) the low-temperature behavior of the working substance is very sensitive to the quantum statistics of the particles and the insertion position of the partition.
Maxwel introduced, in 1871, a notorious being, known as Maxwell's demon nowadays, to discuss the "limitations of the second law of thermodynamics" [1]. Such a demon distinguishes the velocities of the gas particles and controls a tiny door on a partition of the gas container to create a temperature difference, which breaks the Clausius statement of the Second Law of Thermodynamics (SLoT). To reveal the essence of the Maxwell's demon, Leo Szilard proposed a single-particle heat engine [2], named as Szilard heat engine (SHE). The demon in SHE distinguishes the positions of the particles after the partition has been inserted. With the help of the demon, SHE can absorb heat from a single heat resource and convert it into work without apperant evoking other changes in the cycle. Szilard pointed out that the SLoT was no longer violated if one considered the entropy increase during the measurement. Brillouin generalized the Szilard's argument and identified the thermodynamic entropy with the informational entropy firstly [3].
However, the measurement could be carried out without any change in entropy [4]. Acutally, it was realized that a logically irreversible process must be “accompanied by dissipative effects” [5] in the physical realization of the information processing, which is known as Landauer’s erasure principle. Bennett used this point of view in the study of the Maxwell’s demon paradox and pointed out that the erasure of the demon’s memory instead of the measurement was logically irreversible and thus must be accompanied by dissipative effects [4]. With these observations, the conventional cycle presented by Szilard is indeed not a thermodynamic cycle because the demon’s memory has not been erased to complete the cycle. The SLoT will be saved if one considers the erasure process to finish the cycle of the demon. As people believe the essence of information should be discussed in the frame-work of quantum mechanics, various quantum versions of SHE have been proposed with different views about quantum measurements. One proposal is semi-classical [6,10]. The working substance in this proposal is quantum mechanic while the demon is considered as a classical controller whose role is to extract information through measurement and control the system. The paradox of Maxwell’s demon was solved by arguing Landauer’s erasure principle. However, the final solution should include the MD in the cycle and treat also the MD in a quantum fasion [7,11]. For SHE, it is also proved the existence of MD will not violate the SLoT in Ref. [8], where MD is modeled as a two-level system.
The focus of the study in both classical and quantum mechanical frameworks is the erasure process, which is crucial to solving the Maxwell’s demon paradox. In an odinary way, the demon and the working substance are in contact with the same heat bath. After erasing the demon by applying some work, one will find that SHE can not extract work in a cycle at all and the SLoT is not violated. However, a more general erasure should be done with a lower-temperature heat bath which is also called a heat sink. In this suituation, we turn the SHE into a thermal dynamic cycle, where the non-violation of SLoT can be proved by illustrating non-exceeding of Carnot’s efficiency. It is realized that the effective temperature of the MD’s initial state actually charasticrizes the error in the control of the heat engine [7,8]. Our previous work in Ref. [8] emphasized the functions of the demon with errors in the study of the single-particle SHE. But the erasure schemes in Refs. [7,8] are irreversible. Therefore, the efficiencies of the heat engines in these papers can not reach the Carnot cycle’s efficiency. One purpose of the present paper is to establish an optimal scheme of the thermodynamic cycle with a reversible erasure process, assisted by demon. It is shown that the partition-removing process is not always reversible, which leads to the lower efficiency. We aso find the existence of the optimal expansion position to improve the efficiency of the single particle SHE to the Carnot cycle’s efficiency. The other purpose of this paper is to reveal the role of the quantum statistical properties of the work-ing substance. We generalize our previous works about demon-assisted quantum heat engine by using a multiparticle working substance, which is the ideal Bose or Fermi gases, and find that the ratio of the work extracted to the working temperature has some discontinuous behavior, and that discontinuous behavior is closely related to the degenerate-ground-state phenomenon.
The paper is organized as follows: In Sec. II we describe the model of quantum multi-particle SHE and the working scheme briefly. In Sec. III, we study in details the five steps of the working scheme: insertion, measurement, controlled expansion, removing and erasure separately and calculate the work applied and heat transferred in each step. In Sec. IV, the efficiency of the engine is evaluat
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