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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...

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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...

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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.

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Updated: Jun 8, 2026

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

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Published on: May 27, 2020

Entropy of the hard-core model.

A Winkler¹, G Alsmeyer, O Rubner

¹Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 28, 2010

Summary

We analyzed hard-core model entropy across dimensions, proposing a scaling relation for 2D and 3D systems. This relation was validated using numerical simulations, offering insights into finite size effects relevant for glass-forming systems.

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Area of Science:

Statistical Mechanics
Condensed Matter Physics

Background:

The hard-core model is a fundamental system in statistical mechanics.
Understanding its entropy is crucial for various physical phenomena, including phase transitions and material properties.
Previous studies have focused on specific dimensions or approximations.

Purpose of the Study:

To analyze the entropy of the hard-core model considering dimension, system size, and particle density.
To propose a general scaling relation for the entropy in two and three dimensions.
To investigate the impact of finite size effects on hard-core model entropy.

Main Methods:

Analytical treatment inspired by the one-dimensional case and virial expansion.
Numerical simulations using a modified Widom's particle insertion method.
Comparison of theoretical scaling relations with numerical results.

Main Results:

A scaling relation for the entropy of the hard-core model in two and three dimensions was proposed.
Numerical results confirmed the proposed scaling relation.
Finite size effects were analyzed and discussed in relation to the findings.

Conclusions:

The proposed scaling relation provides a valuable tool for understanding hard-core model entropy in higher dimensions.
The study offers insights into the behavior of hard-core systems, with implications for atomic glass-forming systems.
Numerical validation strengthens the theoretical predictions and highlights the importance of finite size effects.