The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the {\em Markov order}). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally "most random" distributions.
Deep Dive into Entropy: The Markov Ordering Approach.
The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant “additivity” properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the {\em Markov order}). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally “most random” distributions.
arXiv:1003.1377v5 [physics.data-an] 9 Nov 2013
Entropy 2010, 12, 1145-1193; doi:10.3390/e12051145
OPEN ACCESS
entropy
ISSN 1099-4300
www.mdpi.com/journal/entropy
Article
Entropy: The Markov Ordering Approach
Alexander N. Gorban 1,⋆, Pavel A. Gorban 2 and George Judge 3
1Department of Mathematics, University of Leicester, Leicester, UK
2Institute of Space and Information Technologies, Siberian Federal University, Krasnoyarsk, Russia
3Department of Resource Economics, University of California, Berkeley, CA, USA
⋆Author to whom correspondence should be addressed; E-mail: ag153@le.ac.uk.
Received: 1 March 2010; in revised form: 30 April 2010 / Accepted: 4 May 2010 /
Published: 7 May 2010 / Corrected Postprint 9 November 2013
Abstract: The focus of this article is on entropy and Markov processes. We study the
properties of functionals which are invariant with respect to monotonic transformations and
analyze two invariant “additivity” properties: (i) existence of a monotonic transformation
which makes the functional additive with respect to the joining of independent systems
and (ii) existence of a monotonic transformation which makes the functional additive with
respect to the partitioning of the space of states. All Lyapunov functionals for Markov
chains which have properties (i) and (ii) are derived. We describe the most general ordering
of the distribution space, with respect to which all continuous-time Markov processes are
monotonic (the Markov order). The solution differs significantly from the ordering given by
the inequality of entropy growth. For inference, this approach results in a convex compact
set of conditionally “most random” distributions.
Keywords: Markov process; Lyapunov function; entropy functionals; attainable region;
MaxEnt; inference
1.
Introduction
1.1.
A Bit of History: Classical Entropy
Two functions, energy and entropy, rule the Universe.
In 1865 R. Clausius formulated two main laws [1]:
Entropy 2010, 12
1146
1. The energy of the Universe is constant.
2. The entropy of the Universe tends to a maximum.
The universe is isolated. For non-isolated systems energy and entropy can enter and leave, the change
in energy is equal to its income minus its outcome, and the change in entropy is equal to entropy
production inside the system plus its income minus outcome. The entropy production is always positive.
Entropy was born as a daughter of energy. If a body gets heat ∆Q at the temperature T then for this
body dS = ∆Q/T. The total entropy is the sum of entropies of all bodies. Heat goes from hot to cold
bodies, and the total change of entropy is always positive.
Ten years later J.W. Gibbs [2] developed a general theory of equilibrium of complex media based on
the entropy maximum: the equilibrium is the point of the conditional entropy maximum under given
values of conserved quantities. The entropy maximum principle was applied to many physical and
chemical problems. At the same time J.W. Gibbs mentioned that entropy maximizers under a given
energy are energy minimizers under a given entropy.
The classical expression
R
p ln p became famous in 1872 when L. Boltzmann proved his
H-theorem [3]: the function
H =
Z
f(x, v) ln f(x, v)dxdv
decreases in time for isolated gas which satisfies the Boltzmann equation (here f(x, v) is the distribution
density of particles in phase space, x is the position of a particle, v is velocity). The statistical entropy
was born: S = −kH. This was the one-particle entropy of a many-particle system (gas).
In 1902, J.W. Gibbs published a book “Elementary principles in statistical dynamics” [4].
He
considered ensembles in the many-particle phase space with probability density ρ(p1, q1, . . . pn, qn),
where pi, qi are the momentum and coordinate of the ith particle. For this distribution,
S = −k
Z
ρ(p1, q1, . . . pn, qn) ln(ρ(p1, q1, . . . pn, qn))dq1 . . . dqndp1 . . . dpn
(1)
Gibbs introduced the canonical distribution that provides the entropy maximum for a given expectation
of energy and gave rise to the entropy maximum principle (MaxEnt).
The Boltzmann period of history was carefully studied [5]. The difference between the Boltzmann
entropy which is defined for coarse-grained distribution and increases in time due to gas dynamics, and
the Gibbs entropy, which is constant due to dynamics, was analyzed by many authors [6,7]. Recently,
the idea of two functions, energy and entropy which rule the Universe was implemented as a basis of
two-generator formalism of nonequilibrium thermodynamics [8,9].
In information theory, R.V.L. Hartley (1928) [10] introduced a logarithmic measure of information
in electronic communication in order “to eliminate the psychological factors involved and to establish
a measure of information in terms of purely physical quantities”. He defined information in a text of
length n in alphabet of s symbols as H = n log s.
In 1948, C.E. Shannon [11] generalized the Hartley approach and developed “a mathematical theory
of communication”, that is informat
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