Abstract
The resolution of the Maxwell’s demon paradox linked thermodynamics with information theory through information erasure principle. By considering a demon endowed with a Turing-machine consisting of a memory tape and a processor, we attempt to explore the link towards the foundations of statistical mechanics and to derive results therein in an operational manner. Here, we present a derivation of the Boltzmann distribution in equilibrium as an example, without hypothesizing the principle of maximum entropy. Further, since the model can be applied to non-equilibrium processes, in principle, we demonstrate the dissipation-fluctuation relation to show the possibility in this direction.
Highlights
Thermodynamics, on the other hand, is constructed upon firmly established empirical and operational evidence on macroscopic objects[8]
We need the notion of probability in the consideration to bridge thermodynamics and statistical mechanics and it should be introduced through operations
Information processing can be seen as a physical operation, since once information is encoded in a physical state any computational manipulation is realized as an operation on the state[19,20]
Summary
Let us clarify first what we mean by “Maxwell’s demon”, as sometimes this can be a source of confusion. The demon is an entity that can measure and change the energy levels of particles, and manipulate/process information encoded in memory registers (cells). We consider the NOT (flipping) operation on a particle as an elementary process of the noise, Eq (3) is a condition against a small number of random NOT operations on P This definition of equilibrium is associated with the stationarity of the principal system and the memory tape, rather than the largest likeliness of the state as in Jaynes’ argument[22]. The cells of the tape can store d values from 0 to d − 1, and there are d possible states for particles, Φ 0, Φ 1, ..., Φ d−1, whose energy levels are E0, E1, ..., Ed−1, respectively. This relation holds for any pairs of i and j, pi ∝ exp(− Ei/kBT) for all i
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