4 research outputs found
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.Comment: 50 pages, 4 figures, Postprint version. More detailed discussion of
the various entropy additivity properties and separation of variables for
independent subsystems in MaxEnt problem is added in Section 4.2.
Bibliography is extende
Thermodynamic Tree: The Space of Admissible Paths
Is a spontaneous transition from a state x to a state y allowed by
thermodynamics? Such a question arises often in chemical thermodynamics and
kinetics. We ask the more formal question: is there a continuous path between
these states, along which the conservation laws hold, the concentrations remain
non-negative and the relevant thermodynamic potential G (Gibbs energy, for
example) monotonically decreases? The obvious necessary condition, G(x)\geq
G(y), is not sufficient, and we construct the necessary and sufficient
conditions. For example, it is impossible to overstep the equilibrium in
1-dimensional (1D) systems (with n components and n-1 conservation laws). The
system cannot come from a state x to a state y if they are on the opposite
sides of the equilibrium even if G(x) > G(y). We find the general
multidimensional analogue of this 1D rule and constructively solve the problem
of the thermodynamically admissible transitions.
We study dynamical systems, which are given in a positively invariant convex
polyhedron D and have a convex Lyapunov function G. An admissible path is a
continuous curve along which does not increase. For x,y from D, x\geq y (x
precedes y) if there exists an admissible path from x to y and x \sim y if
x\geq y and y\geq x. The tree of G in D is a quotient space D/~. We provide an
algorithm for the construction of this tree. In this algorithm, the restriction
of G onto the 1-skeleton of (the union of edges) is used. The problem of
existence of admissible paths between states is solved constructively. The
regions attainable by the admissible paths are described.Comment: Extended version, 31 page, 9 figures, 69 cited references, many minor
correction