Maxwell's demon. (I) A thermodynamic exorcism

Abstract

It is shown that Maxwell's demon is unable to accomplish his task not because of considerations related to irreversibility, acquisition of information, and computers and erasure of information but because of limitations imposed by the properties of the system on which he is asked to perform his demonic manipulations. The limitations emerge from two recent but related developments of which Maxwell was completely unaware. One is an exposition of thermodynamics as a nonstatistical theory, valid for all systems, both large and small, including a system with only one degree of (translational) freedom, and for all states, both thermodynamic or stable equilibrium states and states that are not thermodynamic equilibrium, including states encountered in mechanics. In this theory, entropy is proven to be a nondestructible, nonstatistical property of any state in the same sense that inertial mass is a nonstatistical property of any state. In Part I, the demon is shown to be incapable of accomplishing his task because this would be equivalent either to reducing the nondestructible and nonstatistical entropy of air in a container without compensation by any other system, including himself, or to extracting only energy from the air under conditions that require the extraction of both energy and entropy. The second development is a unified, quantum-theoretic interpretation of mechanics and the thermodynamics just cited. In this theory: (a) the quantum-theoretic probabilities of measurement results are represented by a density operator ρ that corresponds to a homogeneous ensemble of identical systems, identically prepared; homogeneous is an ensemble in which every member is described by the same density operator ρ as any other member, that is, the ensemble is not a statistical mixture of projectors (wave functions); said differently, experimentally (as opposed to algebraically) the homogeneous ensemble cannot be decomposed into mixtures either of pure states or other mixtures; (b) the value 〈A〉 of any observable is not given by any particular measurement result but by the average of an ensemble of measurement results, that is, 〈 A〉=∑ i a i/N , where A is the Hermitian operator representing the observable, a i the measurement result of A from the ith member of the ensemble, and N→∞, that is, the number of members of the ensemble; and (c) in a thermodynamic equilibrium state, molecules do not move—each molecule has a zero value of momentum. In Part II, the demon is shown to be incapable of accomplishing his task because this requires the sorting of air molecules into swift and slow and in thermodynamic equilibrium there are no such molecules—all the molecules are at a standstill. It is worth noting that this conclusion would not be generally valid if the ensemble for ρ were not homogeneous, that is, if the ensemble were a statistical mixture of different subensembles. But then, the ensemble would not be subject to the laws of physics.