Chapter 2 Review of Maxwell ’ s Demon

The paradox of Maxwell’s demon was proposed in a letter from James C. Maxwell to Peter G. Tait for the first time. In the letter, Maxwell mentioned his gedankenexperiment of “a being whose faculties are so sharpened that he can follow every molecule” [1]. The being may be like a tiny fairy, and may violate the second law of thermodynamics. In 1874, William Thomson, who is also well-known as Lord Kelvin, gave it an impressive but opprobrious name—“demon.” Later, Leo Szilard proposed an important model of the demon, which quantitatively connects the thermodynamic work to information [2]. Since then, numerous researchers have been discussed the foundation of the second law of thermodynamics in terms of Maxwell’s demon [3–16]. In this chapter, we review the historical arguments and the basic ideas related to the problem of the demon. The modern aspects of the demon [5, 6, 17–25] will be discussed in the following chapters. 2.1 Original Maxwell’s Demon First of all, we consider the original version of the demon proposed by Maxwell (see also Fig. 2.1) [1]. A classical ideal gas is in a box that is adiabatically separated from the environment. In the initial state, the gas is in thermal equilibrium at temperature T . Suppose that a barrier is inserted at the center of the box, and a small door is attached to the barrier. A small being, which is named as a “demon” by Kelvin, is in the front of the door. It has the capability of measuring the velocity of each molecule in the gas, and it opens or closes the door depending on the measurement outcomes. If a molecule whose velocity is higher than the averaged one comes from the left box, then the demon opens the door. If a molecule whose velocity is slower than the average one comes from the right box, then the demon also opens the door. Otherwise the door is closed. By repeating this operation again and again, the gas in the left box gradually becomes cooler than the initial temperature, and the gas in the right box becomes hotter. After all, the demon is able to adiabatically create the temperature T. Sagawa, Thermodynamics of Information Processing in Small Systems, 9 Springer Theses, DOI: 10.1007/978-4-431-54168-4_2, © Springer Japan 2013 10 2 Review of Maxwell’s Demon Fig. 2.1 The original gedankenexperiment of Maxwell’s demon (reproduced from Ref. [25] with permission). A white (black) particle indicates a molecule whose velocity is slower (faster) than the average. The demon adiabatically realizes a temperature difference by measuring the velocities of molecules and controlling the door based on the measurement outcomes difference starting from the initial uniform temperature. In other words, the entropy of the gas is more and more decreased by the action of the demon, though the box is adiabatically separated from the outside. This apparent contradiction to the second law has been known as the paradox of Maxwell’s demon. The important point of this gedankenexperiment is that the demon can perform the measurement at the single-molecule level, and can control the door based on the measurement outcomes (i.e., the molecule’s velocity is faster or slower than the average), which implies the demon can perform feedback control of the thermal fluctuation.