Related Experiment Video
Updated: Jun 26, 2026
11:00
Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
Published on: July 19, 2016
An equivalent condition for a sequence to be the sequence of partial entropies
1Department of Mathematics, Cracow University of Economics, Rakowicka 27, 31-510 Kraków, Poland. Fryderyk.Falniowski@im.uj.edu.pl
Chaos (Woodbury, N.Y.)
|January 7, 2009
Summary
A sequence represents partial entropies from a dynamical system if it is non-negative, increasing, and has decreasing increments. This finding characterizes entropy sequences in dynamical systems.
Area of Science:
- Dynamical Systems Theory
- Information Theory
- Ergodic Theory
Background:
- Dynamical systems generate complex behaviors over time.
- Entropy quantifies the uncertainty or information content in a system.
- Partial entropies, derived from partitions, offer insights into system dynamics.
Purpose of the Study:
- To establish a precise mathematical characterization for sequences representing partial entropies.
- To identify the necessary and sufficient conditions for a sequence to be an entropy sequence of a dynamical system.
Main Methods:
- Analysis of properties of entropy sequences.
- Development of a criterion based on sequence monotonicity and increment behavior.
- Proof of necessity and sufficiency for the identified conditions.
Main Results:
- A sequence is an entropy sequence if and only if it is non-negative.
- A sequence is an entropy sequence if and only if it is increasing.
- A sequence is an entropy sequence if and only if it has decreasing increments.
Conclusions:
- The characterization provides a direct test for identifying entropy sequences.
- This work connects properties of sequences to the underlying structure of dynamical systems.
- The findings contribute to a deeper understanding of information dynamics in physical systems.
Related Concept Videos
01:14The Entropy as a State Function
Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
01:10Entropy Change in Reversible Processes
In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
01:18Entropy
The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
02:39Entropy
Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
01:29Sequences
Sequences are fundamental mathematical objects consisting of ordered lists of numbers that follow a specific rule or pattern. Sequences are critical in various mathematical concepts, including calculus, series, and number theory. They can model real-world phenomena such as population growth, financial investments, and physical processes like the diminishing height of a bouncing ball.Each number in a sequence is referred to as a term. Typically, the terms are denoted as a1, a2, a3,…, where the...
01:20Entropy and the Second Law of Thermodynamics
The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
The relation between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...