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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

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

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...

Third Law of Thermodynamics

Third Law of Thermodynamics

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.

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.

Standard Entropy Change for a Reaction

Standard Entropy Change for a Reaction

Entropy is a state function, so the standard entropy change for a chemical reaction (ΔS°rxn) can be calculated from the difference in standard entropy between the products and the reactants.

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Related Experiment Video

Updated: Dec 25, 2025

A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference

A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference

Published on: September 5, 2019

Chain Rule for the Quantum Relative Entropy.

Kun Fang¹, Omar Fawzi², Renato Renner³

¹Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom.

Physical Review Letters

|March 29, 2020

Summary

A new quantum relative entropy chain rule was proven, decomposing multipartite systems. This finding resolves a key problem in quantum channel discrimination, showing adaptive strategies offer no error rate advantage.

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Area of Science:

Quantum information theory
Mathematical physics
Information theory

Background:

Classical relative entropy has a chain rule for probability distributions.
Quantum relative entropy is crucial for quantifying information in quantum systems.
Asymptotic quantum channel discrimination is vital for reliable quantum communication.

Purpose of the Study:

To establish a chain rule inequality for quantum relative entropy.
To apply this new rule to solve an open problem in quantum channel discrimination.
To investigate the role of adaptive protocols in asymmetric channel discrimination.

Main Methods:

Development of a novel mathematical framework for quantum relative entropy.
Application of the derived chain rule to analyze error rates in quantum channel discrimination.
Comparison of adaptive and nonadaptive strategies within the established theoretical framework.

Main Results:

A rigorous chain rule inequality for quantum relative entropy has been proven.
The new chain rule enables the decomposition of relative entropy in multipartite quantum systems.
Adaptive protocols do not offer an improved error rate for asymmetric quantum channel discrimination compared to nonadaptive strategies.

Conclusions:

The proven quantum relative entropy chain rule provides a powerful tool for analyzing complex quantum systems.
The study resolves a significant open problem in quantum information theory, specifically in channel discrimination.
The findings indicate that for asymmetric quantum channel discrimination, adaptive strategies do not outperform nonadaptive ones.