68 research outputs found
Entropy? Honest!
Here we deconstruct, and then in a reasoned way reconstruct, the concept of
"entropy of a system," paying particular attention to where the randomness may
be coming from. We start with the core concept of entropy as a COUNT associated
with a DESCRIPTION; this count (traditionally expressed in logarithmic form for
a number of good reasons) is in essence the number of possibilities---specific
instances or "scenarios," that MATCH that description. Very natural (and
virtually inescapable) generalizations of the idea of description are the
probability distribution and of its quantum mechanical counterpart, the density
operator.
We track the process of dynamically updating entropy as a system evolves.
Three factors may cause entropy to change: (1) the system's INTERNAL DYNAMICS;
(2) unsolicited EXTERNAL INFLUENCES on it; and (3) the approximations one has
to make when one tries to predict the system's future state. The latter task is
usually hampered by hard-to-quantify aspects of the original description,
limited data storage and processing resource, and possibly algorithmic
inadequacy. Factors 2 and 3 introduce randomness into one's predictions and
accordingly degrade them. When forecasting, as long as the entropy bookkeping
is conducted in an HONEST fashion, this degradation will ALWAYS lead to an
entropy increase.
To clarify the above point we introduce the notion of HONEST ENTROPY, which
coalesces much of what is of course already done, often tacitly, in responsible
entropy-bookkeping practice. This notion, we believe, will help to fill an
expressivity gap in scientific discourse. With its help we shall prove that ANY
dynamical system---not just our physical universe---strictly obeys Clausius's
original formulation of the second law of thermodynamics IF AND ONLY IF it is
invertible. Thus this law is a TAUTOLOGICAL PROPERTY of invertible systems!Comment: 27 pages, 11 figures. Published in the journal "Entropy" in June
2016. Abstracts from referee's reports quoted right after the abstrac
Computation and construction universality of reversible cellular automata
An arbitrary d-dimensional cellular automaton can be constructively embedded in areversible one having d+1 dimensions. In particular, there exist computation- and construction-universal reversible cellular automata. Thus, we explicitly show a way of implementing nontrivial irreversible processes in a reversible medium. Finally, we derive new results for the bounding problem for configurations, both in general and for reversible cellular automata
When--and how--can a cellular automaton be rewritten as a lattice gas?
Both cellular automata (CA) and lattice-gas automata (LG) provide finite
algorithmic presentations for certain classes of infinite dynamical systems
studied by symbolic dynamics; it is customary to use the term `cellular
automaton' or `lattice gas' for the dynamic system itself as well as for its
presentation. The two kinds of presentation share many traits but also display
profound differences on issues ranging from decidability to modeling
convenience and physical implementability.
Following a conjecture by Toffoli and Margolus, it had been proved by Kari
(and by Durand--Lose for more than two dimensions) that any invertible CA can
be rewritten as an LG (with a possibly much more complex ``unit cell''). But
until now it was not known whether this is possible in general for
noninvertible CA--which comprise ``almost all'' CA and represent the bulk of
examples in theory and applications. Even circumstantial evidence--whether in
favor or against--was lacking.
Here, for noninvertible CA, (a) we prove that an LG presentation is out of
the question for the vanishingly small class of surjective ones. We then turn
our attention to all the rest--noninvertible and nonsurjective--which comprise
all the typical ones, including Conway's `Game of Life'. For these (b) we prove
by explicit construction that all the one-dimensional ones are representable as
LG, and (c) we present and motivate the conjecture that this result extends to
any number of dimensions.
The tradeoff between dissipation rate and structural complexity implied by
the above results have compelling implications for the thermodynamics of
computation at a microscopic scale.Comment: 16 page
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