A note on the attainability of states by equalizing processes

Zylka, Christian

doi:10.1007/bf00529057

Cited by 25 publications

(20 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is now clear that K must be star-shaped with respect to e = (1/3, 1/3, 1/3) because the three lines joining pairs of extreme points satisfy n 1 = n 2 = 1 − n 1 − n 2 there. 8 Likewise, ch(K) for a three-level system is defined by at most seven extreme points (see Appendix A). Each generation approaches e more closely, and no extreme points are generated after the third generation.…”

Section: The Set Of Points Kmentioning

confidence: 99%

“…Note that the set K is non-convex, 8 since not all of the points in ch(K) can be reached by sequences of B ij applied to n 0 . Per the fundamental theorem of linear programming, the minimum W d over the convex feasible region ch(K) is found either at a single extreme point of ch(K) or along an edge of ch(K) joining two or more extreme points of ch(K).…”

Section: The Set Of Points Kmentioning

confidence: 99%

See 1 more Smart Citation

Maximal energy extraction under discrete diffusive exchange

2015

View full text Add to dashboard Cite

Waves propagating through a bounded plasma can rearrange the densities of states in the six-dimensional velocity-configuration phase space. Depending on the rearrangement, the wave energy can either increase or decrease, with the difference taken up by the total plasma energy. In the case where the rearrangement is diffusive, only certain plasma states can be reached. It turns out that the set of reachable states through such diffusive rearrangements has been described in very different contexts. Building upon those descriptions, and making use of the fact that the plasma energy is a linear functional of the state densities, the maximal extractable energy under diffusive rearrangement can then be addressed through linear programming.

show abstract

Section: The Set Of Points Kmentioning

confidence: 99%

Section: The Set Of Points Kmentioning

confidence: 99%

Maximal energy extraction under discrete diffusive exchange

2015

View full text Add to dashboard Cite

show abstract

“…The minimal forward-invariant non-convex (but directionally convex) sets were introduced in [54] and studied for chemical kinetics in [53] and for Markov chains (master equation) in [67].…”

Section: A3 Greedy Peelingmentioning

confidence: 99%

General H-theorem and Entropies that Violate the Second Law

Gorban

2014

Entropy

View full text Add to dashboard Cite

H-theorem states that the entropy production is nonnegative and, therefore, the entropy of a closed system should monotonically change in time. In information processing, the entropy production is positive for random transformation of signals (the information processing lemma). Originally, the H-theorem and the information processing lemma were proved for the classical Boltzmann-Gibbs-Shannon entropy and for the correspondent divergence (the relative entropy). Many new entropies and divergences have been proposed during last decades and for all of them the H-theorem is needed. This note proposes a simple and general criterion to check whether the H-theorem is valid for a convex divergence H and demonstrates that some of the popular divergences obey no H-theorem. We consider systems with n states A i that obey first order kinetics (master equation). A convex function H is a Lyapunov function for all master equations with given equilibrium if and only if its conditional minima properly describe the equilibria of pair transitions A i ⇌ A j . This theorem does not depend on the principle of detailed balance and is valid for general Markov kinetics. Elementary analysis of pair equilibria demonstrate that the popular Bregman divergences like Euclidian distance or Itakura-Saito distance in the space of distribution cannot be the universal Lyapunov functions for the first-order kinetics and can increase in Markov processes. Therefore, they violate the second law and the information processing lemma. In particular, for these measures of information (divergences) random manipulation with data may add information to data. The main results are extended to nonlinear generalized mass action law kinetic equations.

show abstract

“…Already simple examples demonstrate that the sets of distributions which are accessible from a given initial distribution by Markov processes with equilibrium are, in general, non-convex polytopes [62,85] (see, for example, the outlined region in Figure 4, or, for particular graphs of transitions, any of the shaded regions there). This non-convexity makes the analysis of attainability for continuous time Markov processes more difficult (and also more intriguing).…”

Section: Continuous Time Kineticsmentioning

confidence: 99%

Entropy: The Markov Ordering Approach

Gorban

Judge

2010

Entropy

View full text Add to dashboard Cite

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

show abstract

A note on the attainability of states by equalizing processes

Cited by 25 publications

References 2 publications

Maximal energy extraction under discrete diffusive exchange

Maximal energy extraction under discrete diffusive exchange

General H-theorem and Entropies that Violate the Second Law

Entropy: The Markov Ordering Approach

Contact Info

Product

Resources

About