3,994 research outputs found
Entropic Geometry from Logic
We produce a probabilistic space from logic, both classical and quantum,
which is in addition partially ordered in such a way that entropy is monotone.
In particular do we establish the following equation:
Quantitative Probability = Logic + Partiality of Knowledge + Entropy.
That is: 1. A finitary probability space \Delta^n (=all probability measures
on {1,...,n}) can be fully and faithfully represented by the pair consisting of
the abstraction D^n (=the object up to isomorphism) of a partially ordered set
(\Delta^n,\sqsubseteq), and, Shannon entropy; 2. D^n itself can be obtained via
a systematic purely order-theoretic procedure (which embodies introduction of
partiality of knowledge) on an (algebraic) logic. This procedure applies to any
poset A; D_A\cong(\Delta^n,\sqsubseteq) when A is the n-element powerset and
D_A\cong(\Omega^n,\sqsubseteq), the domain of mixed quantum states, when A is
the lattice of subspaces of a Hilbert space.
(We refer to http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-02-07.html
for a domain-theoretic context providing the notions of approximation and
content.)Comment: 19 pages, 8 figure
On the Origin of Gravity and the Laws of Newton
Starting from first principles and general assumptions Newton's law of
gravitation is shown to arise naturally and unavoidably in a theory in which
space is emergent through a holographic scenario. Gravity is explained as an
entropic force caused by changes in the information associated with the
positions of material bodies. A relativistic generalization of the presented
arguments directly leads to the Einstein equations. When space is emergent even
Newton's law of inertia needs to be explained. The equivalence principle leads
us to conclude that it is actually this law of inertia whose origin is
entropic.Comment: 29 pages, 6 figure
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
Entropic Dynamics
Entropic Dynamics is a framework in which dynamical laws are derived as an
application of entropic methods of inference. No underlying action principle is
postulated. Instead, the dynamics is driven by entropy subject to the
constraints appropriate to the problem at hand. In this paper we review three
examples of entropic dynamics. First we tackle the simpler case of a standard
diffusion process which allows us to address the central issue of the nature of
time. Then we show that imposing the additional constraint that the dynamics be
non-dissipative leads to Hamiltonian dynamics. Finally, considerations from
information geometry naturally lead to the type of Hamiltonian that describes
quantum theory.Comment: Invited contribution to the Entropy special volume on Dynamical
Equations and Causal Structures from Observation
Entropic Dynamics
I explore the possibility that the laws of physics might be laws of inference
rather than laws of nature. What sort of dynamics can one derive from
well-established rules of inference? Specifically, I ask: Given relevant
information codified in the initial and the final states, what trajectory is
the system expected to follow? The answer follows from a principle of
inference, the principle of maximum entropy, and not from a principle of
physics. The entropic dynamics derived this way exhibits some remarkable formal
similarities with other generally covariant theories such as general
relativity.Comment: Presented at MaxEnt 2001, the 21th International Workshop on Bayesian
Inference and Maximum Entropy Methods (August 4-9, 2001, Baltimore, MD, USA
Entropic force, noncommutative gravity and ungravity
After recalling the basic concepts of gravity as an emergent phenomenon, we
analyze the recent derivation of Newton's law in terms of entropic force
proposed by Verlinde. By reviewing some points of the procedure, we extend it
to the case of a generic quantum gravity entropic correction to get compelling
deviations to the Newton's law. More specifically, we study: (1) noncommutative
geometry deviations and (2) ungraviton corrections. As a special result in the
noncommutative case, we find that the noncommutative character of the manifold
would be equivalent to the temperature of a thermodynamic system. Therefore, in
analogy to the zero temperature configuration, the description of spacetime in
terms of a differential manifold could be obtained only asymptotically.
Finally, we extend the Verlinde's derivation to a general case, which includes
all possible effects, noncommutativity, ungravity, asymptotically safe gravity,
electrostatic energy, and extra dimensions, showing that the procedure is solid
versus such modifications.Comment: 8 pages, final version published on Physical Review
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