Energy requirement of control: Comments on Szilard’s engine and Maxwell’s demon

In mathematical physical analyses of Szilard’s engine and Maxwell’s demon, a general assumption (explicit or implicit) is that one can neglect the energy needed for relocating the piston in Szilard’s engine and for driving the trap door in Maxwell’s demon. If this basic assumption is wrong, then the conclusions of a vast literature on the implications of the second law of thermodynamics and of Landauer’s erasure theorem are incorrect, too. Our analyses of the fundamental information physical aspects of various types of control within Szilard’s engine and Maxwell’s demon indicate that the entropy production due to the necessary generation of information yield much greater energy dissipation than the energy Szilard’s engine is able to produce even if all sources of dissipation in the rest of these demons (due to measurement, decision, memory, etc.) are neglected. Copyright c © EPLA, 2012 Introduction: demons and the need of control. – Heat engines (e.g., [1]) use a temperature difference to produce work while heat demons (information demons), such as Szilard’s engine [2] and Maxwell’s demon [3], employ information on the instantaneous amplitude of thermal fluctuations and execute active control to produce a temperature difference or work. A useful collection of papers on this topic was presented recently [4]. In particular we want to draw attention to seminal work by Brillouin [5,6]. Our present paper raises the question of energy requirements for active and passive control, which to our knowledge has not been done before. The lack of such analyses makes it understandable that existing papers regard (implicitly or explicitly) the relocation of the piston of Szilard’s engine and the control of the trap door of Maxwell’s demon as operations that ideally do not demand energy, or that they need less energy than the demon is supposed to gain. In the opposite case —i.e., if an ultimate energy requirement exists which is beyond the energy that an ideal demon can produce— many earlier conclusions related to the second law of thermodynamics [4] and the claimed necessity of (a)E-mail: [email protected] Landauer’s erasure theorem [4,7–12] in the analysis of Szilard’s engine need reassessment. For example, if these control operations themselves demand more energy than Szilard’s engine can produce, then the second law of thermodynamics is not violated and Landauer’s erasure theorem is not needed in order to restore it. Perhaps, the main reason for the lack of studies of the kind referred to above is that these historical demons [2,3] basically are mechanical systems coupled to a heat reservoir where different types of energies coexist. In addition to the obvious heat and mechanical energies in these systems, energies (often of different type, such as electrical or photonic) are needed for the information collection (measurements, monitoring), decision (logic operations) and control. Information processing, decision, control, and system analysis typically are part of electronics, where these concepts have been mostly developed, and they do not belong to conventional physics; this may explain why they have been neglected. In the present paper we focus on the energy requirement for active and passive control and show that they have fundamental minimum values that are inherently related to the errors of the operation and that this energy, in the case of Szilard’s engine, is beyond the energy produced by the ideal engine. Thus, the energy