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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 Entropy as a State Function

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...

Probability Histograms

Probability Histograms

A probability histogram is a visual representation of a probability distribution. Similar a typical histogram, the probability histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents. The vertical axis is labeled with probability. Each rectangular bar in the histogram is 1 unit wide, which suggests that the area under each bar equals the probability, P(x), where x is 1, 2, 3, and so on.

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...

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...

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Related Experiment Video

Updated: May 28, 2026

Optimization of Processing of Tiebangchui with Highland Barley Wine Based on the Box-Behnken Design Combined with the Entropy Method

Optimization of Processing of Tiebangchui with Highland Barley Wine Based on the Box-Behnken Design Combined with the Entropy Method

Published on: May 19, 2023

Communication: multicanonical entropy-like solution of statistical temperature weighted histogram analysis method.

Leandro G Rizzi¹, Nelson A Alves

¹Departamento de Física, FFCLRP, Universidade de São Paulo, Avenida Bandeirantes, 3900, 14040-901, Ribeirão Preto, SP, Brazil. lerizzi@usp.br

The Journal of Chemical Physics

|October 21, 2011

Summary

A new multicanonical update relation aids in calculating microcanonical entropy. This method uses inverse temperature estimates from the statistical temperature weighted histogram analysis method (ST-WHAM) for improved accuracy.

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Last Updated: May 28, 2026

Optimization of Processing of Tiebangchui with Highland Barley Wine Based on the Box-Behnken Design Combined with the Entropy Method

Optimization of Processing of Tiebangchui with Highland Barley Wine Based on the Box-Behnken Design Combined with the Entropy Method

Published on: May 19, 2023

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

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

Published on: December 4, 2017

Area of Science:

Statistical Mechanics
Computational Physics

Background:

Calculating microcanonical entropy (S(micro)(E)) is crucial for understanding thermodynamic systems.
Estimating inverse statistical temperature (β(S)) is a key step in entropy calculations.
Existing methods may face challenges with systems exhibiting strong free-energy barriers.

Purpose of the Study:

To propose a novel multicanonical update relation for computing microcanonical entropy.
To leverage the statistical temperature weighted histogram analysis method (ST-WHAM) for inverse temperature estimation.
To evaluate the performance of ST-WHAM in calculating S(micro)(E) from canonical measures, particularly in complex systems.

Main Methods:

Development of a multicanonical update relation.
Utilizing the statistical temperature weighted histogram analysis method (ST-WHAM) to obtain inverse statistical temperature estimates.
Comparison with traditional multicanonical simulation estimates.

Main Results:

The proposed multicanonical update relation enables the calculation of microcanonical entropy.
ST-WHAM provides reliable estimates of inverse statistical temperature.
The study demonstrates ST-WHAM's effectiveness in computing S(micro)(E) even in models with significant free-energy barriers.

Conclusions:

The novel multicanonical update relation offers an effective approach for microcanonical entropy calculation.
ST-WHAM is a valuable tool for estimating inverse temperature and computing entropy from canonical data.
This method shows promise for analyzing systems with complex energy landscapes.