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Entropy

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...

Entropy

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 Thermodynamics of Mixing

The Thermodynamics of Mixing

Mixing is a fascinating phenomenon in thermodynamics, particularly when considering the Gibbs energy of a mixture at constant temperature and pressure. This energy, denoted as G, tends to decrease during spontaneous mixing processes, offering insights into the composition changes that occur.Imagine two ideal gases, initially separated in different containers, with amounts nA and nB, respectively, both at a temperature T and pressure p. The chemical potentials of these gases have their 'pure'...

Entropy and the Second Law of Thermodynamics

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 Thermodynamics

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...

Gauss's Law: Problem-Solving

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...

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Numerical Evaluation of Gaussian Mixture Entropy.

Basheer Joudeh¹, Boris Škorić¹

¹Department of Computer Science and Mathematics, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands.

Entropy (Basel, Switzerland)

|May 4, 2026

Summary

We developed a new approximation method for calculating the differential entropy of Gaussian mixtures. Our h¯CPolyfit(X) method provides an accurate and efficient approximation, outperforming existing bounds, especially in high dimensions.

Keywords:

Gaussian mixture differential entropy entropy mixture distribution

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Area of Science:

Information Theory
Statistical Inference
Machine Learning

Background:

Differential entropy quantifies uncertainty in probability distributions.
Gaussian mixture models are widely used for density estimation.
Accurate entropy approximation is crucial for high-dimensional data analysis.

Purpose of the Study:

To develop an approximation method for the differential entropy of q-component Gaussian mixtures in Rn.
To introduce two specific approximation methods: h¯C,mTaylor(X) and h¯CPolyfit(X).
To evaluate the accuracy and efficiency of the proposed methods compared to existing bounds.

Main Methods:

Developing a novel approximation technique for differential entropy.
Implementing Taylor series expansion for h¯C,mTaylor(X).
Utilizing polynomial fitting for h¯CPolyfit(X).

Main Results:

h¯C,mTaylor(X) offers a computationally simple lower bound for differential entropy.
h¯CPolyfit(X) provides a highly accurate and efficient approximation.
h¯CPolyfit(X) demonstrates superior accuracy and conjectured resilience in high dimensions compared to existing methods.

Conclusions:

The proposed h¯CPolyfit(X) method offers a significant advancement in approximating differential entropy for Gaussian mixtures.
This method is particularly valuable for applications involving high-dimensional data.
The study provides a robust tool for information-theoretic analysis in complex statistical models.