Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Noncompartmental Analysis: Statistical Moment Theory00:56

Noncompartmental Analysis: Statistical Moment Theory

498
Noncompartmental analyses leverage statistical moment theory to examine time-related changes in macroscopic events, encapsulating the collective outcomes stemming from the constituent elements in play. Statistical moment theory is a mathematical approach used to describe the time course of drug concentration in the body without assuming a specific compartmental model. SMT provides insights into drug absorption, distribution, metabolism, and elimination by treating drug concentration versus time...
498
Entropy and the Second Law of Thermodynamics01:20

Entropy and the Second Law of Thermodynamics

5.3K
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...
5.3K
Entropy and the Second Law of Thermodynamics01:26

Entropy and the Second Law of Thermodynamics

262
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...
262
Principle of Moments01:20

Principle of Moments

2.6K
The principle of moments, also known as Varignon's theorem, is a fundamental concept in physics and engineering that describes the equilibrium of a rigid body under the influence of external forces. The principle states that the moment of a force about a point is equal to the sum of the moments of the components of the force about the same point.
The moment is calculated by multiplying the magnitude of the force by the perpendicular distance from the point of application to the point about...
2.6K
Entropy02:39

Entropy

37.8K
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...
37.8K
Entropy01:18

Entropy

3.8K
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...
3.8K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Research on a Regional Availability Evaluation Model for Road-Area High-Entropy Energy Based on Synergy Factors.

Entropy (Basel, Switzerland)·2026
Same journal

Atmospheric Turbulence Channel Modeling and Performance Analysis of a CO-ZP-OFDM Coherent Optical Communication System for UAV Air-to-Ground Scenarios.

Entropy (Basel, Switzerland)·2026
Same journal

Information Geometry and Asymptotic Theory for SMML Estimators.

Entropy (Basel, Switzerland)·2026
Same journal

Correlation Entropy and Power-Law Kinetics.

Entropy (Basel, Switzerland)·2026
Same journal

Research on the Contagion of Systemic Financial Risk Under the Impact of Climate Risks-From the Perspective of Complex Networks and Machine Learning.

Entropy (Basel, Switzerland)·2026
Same journal

The Statistical-Mechanical Meaning of the Wave Function of Quantum Mechanics.

Entropy (Basel, Switzerland)·2026

Related Experiment Video

Updated: Mar 29, 2026

15N CPMG Relaxation Dispersion for the Investigation of Protein Conformational Dynamics on the µs-ms Timescale
08:09

15N CPMG Relaxation Dispersion for the Investigation of Protein Conformational Dynamics on the µs-ms Timescale

Published on: April 19, 2021

6.3K

On Maximum Entropy Density Estimation with Relaxed Moment Constraints.

Thi Lich Nghiem1, Pierre Maréchal2,3

  • 1Informatics Department, Thuongmai University, 79 Ho Tung Mau Street, Cau Giay District, Hanoi 100000, Vietnam.

Entropy (Basel, Switzerland)
|March 28, 2026
PubMed
Summary
This summary is machine-generated.

This study presents a novel convex-analytic framework for Maximum Entropy density estimation with relaxed moment constraints. It simplifies complex infinite-dimensional problems into finite-dimensional dual formulations for practical application.

Keywords:
fenchel dualitymaximum entropymethod of momentsrelaxationspecies distribution

More Related Videos

Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy
11:15

Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy

Published on: June 27, 2013

34.5K
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

9.1K

Related Experiment Videos

Last Updated: Mar 29, 2026

15N CPMG Relaxation Dispersion for the Investigation of Protein Conformational Dynamics on the µs-ms Timescale
08:09

15N CPMG Relaxation Dispersion for the Investigation of Protein Conformational Dynamics on the µs-ms Timescale

Published on: April 19, 2021

6.3K
Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy
11:15

Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy

Published on: June 27, 2013

34.5K
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

9.1K

Area of Science:

  • Optimization Theory
  • Statistical Inference
  • Applied Mathematics

Background:

  • Maximum Entropy (MaxEnt) methods are crucial for density estimation under moment constraints.
  • Existing methods often struggle with uncertainty in empirical moments and continuous domains.
  • Relaxing constraints via convex penalties leads to complex infinite-dimensional optimization problems.

Purpose of the Study:

  • To develop a rigorous convex-analytic framework for relaxed Maximum Entropy density estimation on continuous domains.
  • To transform infinite-dimensional optimization problems into tractable, finite-dimensional dual formulations.
  • To provide a unified mathematical foundation for various relaxed MaxEnt methods.

Main Methods:

  • Formulation of MaxEnt as Kullback-Leibler divergence minimization.
  • Relaxation of moment constraints using convex penalty functions.
  • Application of convex integral functionals and Fenchel duality for problem transformation.
  • Derivation of primal-dual optimality conditions.

Main Results:

  • A rigorous convex-analytic treatment of relaxed MaxEnt problems in a functional setting.
  • Demonstration that infinite-dimensional primal problems admit finite-dimensional dual formulations under mild conditions.
  • Characterization of MaxEnt solutions in exponential form via the dual problem.
  • Unification of exact and relaxed moment constraints (e.g., box, quadratic) within a single framework.

Conclusions:

  • The proposed framework offers a mathematically sound foundation for relaxed Maximum Entropy methods.
  • The dual formulation simplifies complex problems, making them practically tractable.
  • This approach extends MaxEnt density estimation to continuous domains with robust handling of uncertainty in moment constraints.