[Paper Review] A statistical mechanical interpretation of algorithmic information theory

Kohtaro Tadaki|ArXiv.org|Jan 28, 2008

Neural Networks and Applications4 citations

TL;DR

This paper establishes a statistical mechanical interpretation of algorithmic information theory by introducing thermodynamic quantities—such as free energy, entropy, energy, and specific heat—into the framework of program-size complexity. It reveals that temperature acts as a compression rate for all thermodynamic quantities, including itself, leading to fixed-point theorems on compression rate that reflect the self-referential nature of algorithmic randomness.

ABSTRACT

We develop a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities, such as free energy, energy, statistical mechanical entropy, and specific heat, into algorithmic information theory. We investigate the properties of these quantities by means of program-size complexity from the point of view of algorithmic randomness. It is then discovered that, in the interpretation, the temperature plays a role as the compression rate of the values of all these thermodynamic quantities, which include the temperature itself. Reflecting this self-referential nature of the compression rate of the temperature, we obtain fixed point theorems on compression rate.

Motivation & Objective

• To develop a rigorous statistical mechanical interpretation of algorithmic information theory using thermodynamic concepts.
• To interpret program-size complexity and algorithmic randomness through the lens of equilibrium statistical mechanics.
• To clarify the physical meaning of thermodynamic quantities like free energy, entropy, and specific heat in the context of algorithmic information theory.
• To explore the self-referential role of temperature as a compression rate for all thermodynamic quantities, including itself.
• To establish a formal correspondence between algorithmic information theory and statistical mechanics, particularly via the partition function and canonical ensemble.

Proposed method

• Introduce thermodynamic quantities—free energy, energy, entropy, and specific heat—into algorithmic information theory using program-size complexity.
• Define the partition function $ Z(T) = \sum_{p \in \text{dom}\,U} 2^{-|p|/T} $, which matches the form of Chaitin's $ \Omega^D $ when $ D = 1/T $.
• Model the system as a canonical ensemble with probability distribution $ R(p) = \frac{1}{Z(T)} 2^{-|p|/T} $, corresponding to the Gibbs measure in statistical mechanics.
• Derive expressions for free energy $ F(T) = -T \log_2 Z(T) $, expected program length $ E(T) $, entropy $ S(T) = \frac{1}{T}E(T) + \log_2 Z(T) $, and specific heat $ C(T) = E'(T) $.
• Establish a mapping between algorithmic information theory and statistical mechanics by identifying microcanonical ensembles and using canonical ensemble formalism.
• Use the self-referential nature of temperature as a compression rate to derive fixed-point theorems on compression rate.

Experimental results

Research questions

• RQ1 How can thermodynamic quantities such as free energy, entropy, and specific heat be meaningfully defined within algorithmic information theory?
• RQ2 What is the physical interpretation of the partition function $ Z(T) $ in the context of program-size complexity and algorithmic randomness?
• RQ3 How does the temperature $ T $ function as a compression rate for thermodynamic quantities, including itself?
• RQ4 What fixed-point theorems emerge from the self-referential nature of temperature as a compression rate?
• RQ5 To what extent can a complete statistical mechanical interpretation of algorithmic information theory be constructed with perfect correspondence to equilibrium statistical mechanics?

Key findings

• The partition function $ Z(T) $ in algorithmic information theory matches the form of Chaitin’s $ \Omega^D $ when $ D = 1/T $, establishing a direct link to algorithmic randomness.
• The free energy is given by $ F(T) = -T \log_2 Z(T) $, and the expected program length is $ E(T) = \frac{1}{Z(T)} \sum_{p \in \text{dom}\,U} |p| \, 2^{-|p|/T} $, with $ E(L,N) = E(T(L,N)) $.
• The statistical mechanical entropy $ S(E(L,N),1) $ coincides with the Shannon entropy of the distribution $ R(p) $, confirming consistency with information-theoretic entropy.
• The specific heat is defined as $ C(T) = E'(T) $, and it captures the sensitivity of expected program length to temperature changes.
• Temperature $ T $ acts as a compression rate for all thermodynamic quantities, including itself, leading to fixed-point theorems on compression rate.
• The formalism realizes a perfect correspondence with equilibrium statistical mechanics, with all thermodynamic quantities derived from the canonical ensemble and partition function.