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Journal of Physics A Mathematical and General

Isothermal Maxwell daemon: numerical results in a simplified model

Vladislav Čápek, Jiří Bok

Published Nov 6, 1998• 16 citations
Publisher

Abstract

A recently suggested simple model of an open microscopic system interacting with a bath and behaving like an isothermal Maxwell daemon is simplified and treated numerically. Results obtained confirm expectations that the system can, at the cost of thermal energy of the bath, spontaneously and in a cyclic process, pump particles from a particle reservoir to states with even higher energy. This contradicts the usual opinion based on the second law of macroscopic thermodynamics. For this effect, the interaction with the bath cannot be treated as weak. The role of dephasing owing to this interaction is illustrated.

DOI
10.1088/0305-4470/31/44/004

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A recently suggested simple model of an open microscopic system interacting with a bath and behaving like an isothermal Maxwell daemon is simplified and treated numerically. Results obtained confirm expectations that the system can, at the cost of thermal energy of the bath, spontaneously and in a cyclic process, pump particles from a particle reservoir to states with even higher energy. This contradicts the usual opinion based on the second law of macroscopic thermodynamics. For this effect, the interaction with the bath cannot be treated as weak. The role of dephasing owing to this interaction is illustrated.
The second law of thermodynamics places constraints on state transformations. It applies to systems composed of many particles, however, we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath. Is there a second law of thermodynamics in this regime? Here, we find that for processes which are approximately cyclic, the second law for microscopic systems takes on a different form compared to the macroscopic scale, imposing not just one constraint on state transformations, but an entire family of constraints. We find a family of free energies which generalize the traditional one, and show that they can never increase. The ordinary second law relates to one of these, with the remainder imposing additional constraints on thermodynamic transitions. We find three regimes which determine which family of second laws govern state transitions, depending on how cyclic the process is. In one regime one can cause an apparent violation of the usual second law, through a process of embezzling work from a large system which remains arbitrarily close to its original state. These second laws are relevant for small systems, and also apply to individual macroscopic systems interacting via long-range interactions. By making precise the definition of thermal operations, the laws of thermodynamics are unified in this framework, with the first law defining the class of operations, the zeroth law emerging as an equivalence relation between thermal states, and the remaining laws being monotonicity of our generalized free energies.
Maxwell's daemon is a popular personification of a principle connecting information gain and extractable work in thermodynamics. A Szilard Engine is a particular hypothetical realization of Maxwell's daemon, which is able to extract work from a single thermal reservoir by measuring the position of particle(s) within the system. Here we investigate the role of particle statistics in the whole process; namely, how the extractable work changes if instead of classical particles fermions or bosons are used as the working medium. We give a unifying argument for the optimal work in the different cases: the extractable work is determined solely by the information gain of the initial measurement, as measured by the mutual information, regardless of the number and type of particles which constitute the working substance.
A fully standard quantum model of a particle interacting with a single-mode phonon system under the influence of a thermodynamic bath is considered. Numerically exact solution shows that, for very specific values of parameters involved, the phonon mode cooperating with the particle becomes able to respond to particle hops and thus to suppress the back particle transfer. The particle becomes free at the end of the process, during which it can be transferred prevailingly in one direction only, even going uphill in energy, at the cost of just the thermal energy of the single bath. This behaviour is due to the fact that both the particle and the particle + oscillator density matrices differ, in the stationary situations and for at least intermediate oscillator coupling to the bath, from the respective canonical forms.
A macroscopic system acting as a heat reservoir affords a simplified description in terms of a collection of independent modes. Although in most cases this model is unable to account for the behavior of the macroscopic system itself, it provides a consistent desciption of the macroscopic degrees of freedom acting as a dissipation mechanism for a microscopic system. This is found to be a consequence of the statistical properties of bath quantities relevant to the dynamics of the microscopic system. Furthermore, these properties ensure that the unphysical features of the independent bath model do not affect its performance as a heat reservoir.