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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...

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...

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.

Pore Size Distribution

Pore Size Distribution

In concrete, the pore size distribution significantly influences the material's properties. Capillary pores, markedly larger than gel pores, form a vast network within partially hydrated cement paste, reducing the concrete's strength and increasing its permeability. This heightened permeability leads to a greater risk of damage from environmental factors like freeze-thaw cycles and chemical attacks, with the extent of vulnerability also being tied to the water-to-cement ratio.
Adequate...

Entropy Change in Reversible Processes

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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Related Experiment Video

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

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Coarse Graining, Nonmaximal Entropy, and Power Laws.

Fernando C Pérez-Cárdenas¹, Lorenzo Resca², Ian L Pegg²

¹Vitreous State Laboratory, The Catholic University of America, Washington, DC 20064, USA.

Entropy (Basel, Switzerland)

|December 3, 2020

Summary

Coarse graining significantly impacts equilibrium state entropy, reducing it below theoretical maximums. Finer scales amplify this entropy drop and associated fluctuations, revealing predictable power-law relationships.

Keywords:

coarse graining entropy nonmaximal entropy power laws

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Area of Science:

Statistical Mechanics
Thermodynamics
Computational Physics

Background:

Entropy is a fundamental concept in thermodynamics, quantifying disorder and information.
Coarse-graining is a common technique to simplify complex systems by reducing their degrees of freedom.
Understanding the impact of coarse-graining on equilibrium entropy is crucial for theoretical and computational studies.

Purpose of the Study:

To investigate the effects of coarse-graining on the entropy of equilibrium states.
To demonstrate how coarse-graining leads to a predictable reduction in effective entropy.
To derive and validate power-law relationships governing coarse-graining, entropy gap, and fluctuations.

Main Methods:

Theoretical derivation of power-law relationships.
Numerical simulations using a two-dimensional lattice gas model.
Analysis of entropy evolution and fluctuations at different coarse-graining scales.

Main Results:

Coarse-graining introduces significant, predictable effects on equilibrium entropy.
Effective entropy typically decreases with increasing coarse-graining, deviating from the maximum entropy principle.
Two distinct power laws were derived and numerically validated, relating coarse-graining to entropy gap and fluctuation noise range.

Conclusions:

Coarse-graining introduces an 'effective entropy gap' that scales predictably with the graining level.
The observed power laws highlight the fundamental interplay between system scale and entropy in equilibrium states.
These effects vanish in the thermodynamic limit, reasserting the maximum entropy principle at macroscopic scales.