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A Statistical Mechanical Interpretation of Algorithmic Information Theory (Paperback, 1st ed. 2019)
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A Statistical Mechanical Interpretation of Algorithmic Information Theory (Paperback, 1st ed. 2019)
Series: SpringerBriefs in Mathematical Physics, 36
Expected to ship within 12 - 17 working days
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This book is the first one that provides a solid bridge between
algorithmic information theory and statistical mechanics.
Algorithmic information theory (AIT) is a theory of program size
and recently is also known as algorithmic randomness. AIT provides
a framework for characterizing the notion of randomness for an
individual object and for studying it closely and comprehensively.
In this book, a statistical mechanical interpretation of AIT is
introduced while explaining the basic notions and results of AIT to
the reader who has an acquaintance with an elementary theory of
computation. A simplification of the setting of AIT is the
noiseless source coding in information theory. First, in the book,
a statistical mechanical interpretation of the noiseless source
coding scheme is introduced. It can be seen that the notions in
statistical mechanics such as entropy, temperature, and thermal
equilibrium are translated into the context of noiseless source
coding in a natural manner. Then, the framework of AIT is
introduced. On this basis, the introduction of a statistical
mechanical interpretation of AIT is begun. Namely, the notion of
thermodynamic quantities, such as free energy, energy, and entropy,
is introduced into AIT. In the interpretation, the temperature is
shown to be equal to the partial randomness of the values of all
these thermodynamic quantities, where the notion of partial
randomness is a stronger representation of the compression rate
measured by means of program-size complexity. Additionally, it is
demonstrated that this situation holds for the temperature itself
as a thermodynamic quantity. That is, for each of all the
thermodynamic quantities above, the computability of its value at
temperature T gives a sufficient condition for T to be a fixed
point on partial randomness. In this groundbreaking book, the
current status of the interpretation from both mathematical and
physical points of view is reported. For example, a total
statistical mechanical interpretation of AIT that actualizes a
perfect correspondence to normal statistical mechanics can be
developed by identifying a microcanonical ensemble in the framework
of AIT. As a result, the statistical mechanical meaning of the
thermodynamic quantities of AIT is clarified. In the book, the
close relationship of the interpretation to Landauer's principle is
pointed out.
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