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arXiv:2011.01180 · American Journal of Physics

The harmonic quantum Szilárd engine

Paul Davies, Logan Thomas, George Zahariade

Published Nov 22, 2021• 6 citations
arXivHTMLPDF

Abstract

The Szilárd engine is a mechanism (akin to Maxwell's demon) for converting information into energy, which seemingly violates the second law of thermodynamics. Originally a classical thought experiment, it was extended to a quantized treatment by Zurek. Here, we examine a new, elegant model of a quantum Szilárd engine by replacing the traditional rigid box with a harmonic potential, extending the scope of the model. Remarkably, almost all calculations are exact. This article is suitable for students, researchers, and educators interested in the conceptual links among information, entropy, and quantum measurement.

DOI
10.1119/10.0005946

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The Szilárd engine is a mechanism (akin to Maxwell's demon) for converting information into energy, which seemingly violates the second law of thermodynamics. Originally a classical thought experiment, it was extended to a quantized treatment by Zurek. Here, we examine a new, elegant model of a quantum Szilárd engine by replacing the traditional rigid box with a harmonic potential, extending the scope of the model. Remarkably, almost all calculations are exact. This article is suitable for students, researchers, and educators interested in the conceptual links among information, entropy, and quantum measurement.
Using an isolated measurement process, I've calculated the effect measurement has on entropy for the multi-cylinder Szilard engine. This calculation shows that the system of cylinders possesses an entropy associated with cylinder total energy states, and that it records information transferred at measurement. Contrary to other's results, I've found that the apparatus loses entropy due to measurement. The Second Law of Thermodynamics may be preserved if Maxwell's demon gains entropy moving the engine partition.
We analyze an engine whose working fluid consists of a single quantum particle, paralleling Szilard's construction of a classical single-particle engine. Following his resolution of Maxwell's Second Law paradox using the latter, which turned on physically instantiating the demon (control subsystem), the quantum engine's design mirrors the classically-chaotic Szilard Map that operates a thermodynamic cycle of measurement, thermal-energy extraction, and memory reset. Focusing on the thermodynamic costs to observe and control the particle and comparing these in the quantum and classical limits, we detail the thermodynamic tradeoffs behind Landauer's Principle for information-processing-induced thermodynamic dissipation in the quantum and classical regimes. In particular, and as found with the classical engine, we show that the sum of the thermodynamic costs over a cycle obeys a generalized Landauer Principle, exactly balancing energy extraction from the heat bath. Thus, the quantum engine obeys the Second Law. However, the quantum engine does so via substantially different mechanisms: classically measurement and erasure determine the thermodynamics, while in the quantum implementation the cost of partition insertion is key.
Abstract A reinvention of the classical Maxwel demon was proposed by Szilard around the time quantum mechanics was developed. His model continues to attract great interest, especially quantum versions of it. A quantum formulation of the Szilard engine is introduced and investigated here. It is made to operate through specified cycles in such a way that all thermodynamic quantities which pertain to the system can be evaluated exactly in closed form along each sequence of steps through a cycle. It is shown that as a result of the structure of the model, it is possible to calculate and compare various thermodynamic quantities as the engine proceeds around a well defined specific cycle.
We study the physical mechanism of Maxwell's demon (MD), which helps do extra work in thermodynamic cycles with the heat engine. This is exemplified with one molecule confined in an infinitely deep square potential with a movable solid wall. The MD is modeled as a two-level system (TLS) for measuring and controlling the motion of the molecule. The processes in the cycle are described in a quantum fashion. It is discovered that a MD with quantum coherence or one at a temperature lower than the molecule's heat bath can enhance the ability of the whole working substance, formed by the heat engine plus the MD, to do work outside. This observation reveals that the essential role of the MD is to drive the whole working substance off equilibrium, or equivalently, to work between two heat baths with different effective temperatures. The elaborate studies with this model explicitly reveal the effect of finite size off the classical limit or thermodynamic limit, which contradicts common sense on a Szilard heat engine (SHE). The quantum SHE's efficiency is evaluated in detail to prove the validity of the second law of thermodynamics.
With a class of quantum heat engines which consists of two-energy-eigenstate systems undergoing, respectively, quantum adiabatic processes and energy exchanges with heat baths at different stages of a cycle, we are able to clarify some important aspects of the second law of thermodynamics. The quantum heat engines also offer a practical way, as an alternative to Szilard's engine, to physically realize Maxwell's demon. While respecting the second law on the average, they are also capable of extracting more work from the heat baths than is otherwise possible in thermal equilibrium.