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Related Concept Videos

Entropy02:39

Entropy

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...
Entropy01:18

Entropy

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...
The Entropy as a State Function01:14

The 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...
Entropy and the Second Law of Thermodynamics01:20

Entropy 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...
Entropy and the Second Law of Thermodynamics01:26

Entropy and the Second Law of Thermodynamics

Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...
Entropy Change in Reversible Processes01:10

Entropy 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.

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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Shannon's monotonicity problem for free and classical entropy.

Dimitri Shlyakhtenko1, Hanne Schultz

  • 1Department of Mathematics, University of California, Los Angeles, CA 90095, USA. shlyakht@math.ucla.edu

Proceedings of the National Academy of Sciences of the United States of America
|September 21, 2007
PubMed
Summary

This study proves that the entropy of central limit sums for independent random variables is a monotone function of N. This finding applies to both classical and free probability theories, simplifying previous proofs.

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Area of Science:

  • Probability Theory
  • Mathematical Physics

Background:

  • Classical probability theory and Voiculescu's free probability theory are distinct frameworks for analyzing random variables.
  • Central limit theorems describe the behavior of sums of random variables.
  • Entropy is a measure of uncertainty in probability distributions.

Purpose of the Study:

  • To provide a unified and simplified proof for a theorem concerning the monotonicity of entropy in central limit sums.
  • To extend the applicability of the theorem to both classical and free probability settings.
  • To adapt and simplify existing arguments from classical probability theory.

Main Methods:

  • The study employs a unified proof technique applicable to both classical and free probability.
  • It adapts and simplifies the argument presented by Artstein, Ball, Barthe, and Naor for the classical case (n=1).
  • The core of the method involves analyzing central limit sums of random variables.

Main Results:

  • A theorem is proven stating that the entropy (or free entropy) of the n-tuple of central limit sums is a monotone function of N.
  • The proof is valid for both independent random variables (classical probability) and freely independent random variables (free probability).
  • The result generalizes and simplifies a known theorem for the classical case with n=1.

Conclusions:

  • The monotonicity of entropy in central limit sums is a robust property across different probability frameworks.
  • The simplified unified proof offers a more accessible understanding of this important result.
  • This work bridges classical and free probability theory through a common analytical tool.