123 research outputs found
Comment on "Consistency, amplitudes, and probabilities in quantum theory"
In a recent article [Phys. Rev. A 57, 1572 (1998)] Caticha has concluded that
``nonlinear variants of quantum mechanics are inconsistent.'' In this note we
identify what it is that nonlinear quantum theories have been shown to be
inconsistent with.Comment: LaTeX, 5 pages, no figure
Entropic Dynamics, Time and Quantum Theory
Quantum mechanics is derived as an application of the method of maximum
entropy. No appeal is made to any underlying classical action principle whether
deterministic or stochastic. Instead, the basic assumption is that in addition
to the particles of interest x there exist extra variables y whose entropy S(x)
depends on x. The Schr\"odinger equation follows from their coupled dynamics:
the entropy S(x) drives the dynamics of the particles x while they in their
turn determine the evolution of S(x). In this "entropic dynamics" time is
introduced as a device to keep track of change. A welcome feature of such an
entropic time is that it naturally incorporates an arrow of time. Both the
magnitude and the phase of the wave function are given statistical
interpretations: the magnitude gives the distribution of x in agreement with
the usual Born rule and the phase carries information about the entropy S(x) of
the extra variables. Extending the model to include external electromagnetic
fields yields further insight into the nature of the quantum phase.Comment: 29 page
Opinion Dynamics of Learning Agents: Does Seeking Consensus Lead to Disagreement?
We study opinion dynamics in a population of interacting adaptive agents
voting on a set of complex multidimensional issues. We consider agents which
can classify issues into for or against. The agents arrive at the opinions
about each issue in question using an adaptive algorithm. Adaptation comes from
learning and the information for the learning process comes from interacting
with other neighboring agents and trying to change the internal state in order
to concur with their opinions. The change in the internal state is driven by
the information contained in the issue and in the opinion of the other agent.
We present results in a simple yet rich context where each agent uses a Boolean
Perceptron to state its opinion. If there is no internal clock, so the update
occurs with asynchronously exchanged information among pairs of agents, then
the typical case, if the number of issues is kept small, is the evolution into
a society thorn by the emergence of factions with extreme opposite beliefs.
This occurs even when seeking consensus with agents with opposite opinions. The
curious result is that it is learning from those that hold the same opinions
that drives the emergence of factions. This results follows from the fact that
factions are prevented by not learning at all from those agents that hold the
same opinion. If the number of issues is large, the dynamics becomes trapped
and the society does not evolve into factions and a distribution of moderate
opinions is observed. We also study the less realistic, but technically simpler
synchronous case showing that global consensus is a fixed point. However, the
approach to this consensus is glassy in the limit of large societies if agents
adapt even in the case of agreement.Comment: 16 pages, 10 figures, revised versio
Bayesian Probabilities and the Histories Algebra
We attempt a justification of a generalisation of the consistent histories
programme using a notion of probability that is valid for all complete sets of
history propositions. This consists of introducing Cox's axioms of probability
theory and showing that our candidate notion of probability obeys them. We also
give a generalisation of Bayes' theorem and comment upon how Bayesianism should
be useful for the quantum gravity/cosmology programmes.Comment: 10 pages, accepted by Int. J. Theo. Phys. Feb 200
Maximum Entropy and Bayesian Data Analysis: Entropic Priors
The problem of assigning probability distributions which objectively reflect
the prior information available about experiments is one of the major stumbling
blocks in the use of Bayesian methods of data analysis. In this paper the
method of Maximum (relative) Entropy (ME) is used to translate the information
contained in the known form of the likelihood into a prior distribution for
Bayesian inference. The argument is inspired and guided by intuition gained
from the successful use of ME methods in statistical mechanics. For experiments
that cannot be repeated the resulting "entropic prior" is formally identical
with the Einstein fluctuation formula. For repeatable experiments, however, the
expected value of the entropy of the likelihood turns out to be relevant
information that must be included in the analysis. The important case of a
Gaussian likelihood is treated in detail.Comment: 23 pages, 2 figure
On The Complexity Of Statistical Models Admitting Correlations
We compute the asymptotic temporal behavior of the dynamical complexity
associated with the maximum probability trajectories on Gaussian statistical
manifolds in presence of correlations between the variables labeling the
macrostates of the system. The algorithmic structure of our asymptotic
computations is presented and special focus is devoted to the diagonalization
procedure that allows to simplify the problem in a remarkable way. We observe a
power law decay of the information geometric complexity at a rate determined by
the correlation coefficient. We conclude that macro-correlations lead to the
emergence of an asymptotic information geometric compression of the statistical
macrostates explored on the configuration manifold of the model in its
evolution between the initial and final macrostates.Comment: 15 pages, no figures; improved versio
Is Tsallis thermodynamics nonextensive?
Within the Tsallis thermodynamics' framework, and using scaling properties of
the entropy, we derive a generalization of the Gibbs-Duhem equation. The
analysis suggests a transformation of variables that allows standard
thermodynamics to be recovered. Moreover, we also generalize Einstein's formula
for the probability of a fluctuation to occur by means of the maximum
statistical entropy method. The use of the proposed transformation of variables
also shows that fluctuations within Tsallis statistics can be mapped to those
of standard statistical mechanics.Comment: 4 pages, no figures, revised version, new title, accepted in PR
Information-Geometric Indicators of Chaos in Gaussian Models on Statistical Manifolds of Negative Ricci Curvature
A new information-geometric approach to chaotic dynamics on curved
statistical manifolds based on Entropic Dynamics (ED) is proposed. It is shown
that the hyperbolicity of a non-maximally symmetric 6N-dimensional statistical
manifold M_{s} underlying an ED Gaussian model describing an arbitrary system
of 3N degrees of freedom leads to linear information-geometric entropy growth
and to exponential divergence of the Jacobi vector field intensity, quantum and
classical features of chaos respectively.Comment: 8 pages, final version accepted for publicatio
Microtubules: Montroll's kink and Morse vibrations
Using a version of Witten's supersymmetric quantum mechanics proposed by
Caticha, we relate Montroll's kink to a traveling, asymmetric Morse double-well
potential suggesting in this way a connection between kink modes and
vibrational degrees of freedom along microtubulesComment: 2pp, twocolum
Entropy Distance: New Quantum Phenomena
We study a curve of Gibbsian families of complex 3x3-matrices and point out
new features, absent in commutative finite-dimensional algebras: a
discontinuous maximum-entropy inference, a discontinuous entropy distance and
non-exposed faces of the mean value set. We analyze these problems from various
aspects including convex geometry, topology and information geometry. This
research is motivated by a theory of info-max principles, where we contribute
by computing first order optimality conditions of the entropy distance.Comment: 34 pages, 5 figure
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