Abstract
According to Landauer’s principle, at least Joules are needed to erase a bit that stores information in a thermodynamic system at temperature T. However, the arguments for the principle rely on a regime where the equipartition principle holds. This paper, by exploring a simple model of a thermodynamic system using algorithmic information theory, shows the energy cost of transferring a bit, or restoring the original state, is Joules for a reversible system. The principle is a direct consequence of the statistics required to allocate energy between stored energy states and thermal states, and applies outside the validity of the equipartition principle. As the thermodynamic entropy of a system coincides with the algorithmic entropy of a typical state specifying the momentum degrees of freedom, it can quantify the thermodynamic requirements in terms of bit flows to maintain a system distant from the equilibrium set of states. The approach offers a simple conceptual understanding of entropy, while avoiding problems with the statistical mechanic’s approach to the second law of thermodynamics. Furthermore, the classical articulation of the principle can be used to derive the low temperature heat capacities, and is consistent with the quantum version of the principle.
Highlights
Background to the DebyeApproach to the Heat CapacityWhile the Einstein and the Debye approaches have the same allocation statistics, theDebye approach focuses on determining the energy of each mode of vibration, rather than the vibration of independent atoms
If the system is isolated and reversible, there cannot be an increase in entropy of the total system. This all makes sense as the number of bits needed to describe the initial state is the same as the number of bits needed to describe a later equilibrium or typical configuration provided reversibility is kept. When such a system trends to the equilibrium set of states, the thermodynamic entropy increases as stored energy bits and program bits become momentum bits, increasing the temperature
Because the Algorithmic Information Theory (AIT) approach recognizes that the heat capacity is a property of the instantaneous microstate of a system in the most probable set of states the relevant algorithmic parameter is the actual number of phonons n in each vibrational mode of a typical or equilibrium microstate
Summary
The laws of the universe that shift one state to another can be interpreted as computations on a real-world Universal Turing Machine (UTM) where the interaction between atoms and molecules can be seen as program instruction implemented on molecular computational gates. There are two constraints, reversibility must be maintained and the simulation, like a natural computation, must use instructions that have no end-markers, i.e., are self-delimiting With these real-world requirements in mind, the algorithmic entropy is the number of computational bits HU in the reversible self-delimiting program p undertaken by computer. The algorithmic entropy, the total number of bits in state s is the sum of bits in p, and the existing bits in the system that specify the initial state y. From the perspective of an on-going real-world computation, the net number of bits that define the state of interest is given by the initial bits, and the bit flows given by the shortest program that fully defines the state of interest and halts. Whenever the net number of bits is not consistent with the number obtained by a halting algorithm, these hidden bits need to be taken into account
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