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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 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.

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...

Second Law of Thermodynamics

Second Law of Thermodynamics

In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, a significant conclusion regarding the relation between this property and spontaneity may be reached. In thermodynamic models, the...

Free Energy Changes for Nonstandard States

Free Energy Changes for Nonstandard States

The free energy change for a process taking place with reactants and products present under nonstandard conditions (pressures other than 1 bar; concentrations other than 1 M) is related to the standard free energy change according to this equation:

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Related Experiment Video

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Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Published on: June 15, 2018

Multifractality Meets Entanglement: Relation for Nonergodic Extended States.

Giuseppe De Tomasi^1,2, Ivan M Khaymovich³

¹T.C.M. Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom.

Physical Review Letters

|June 6, 2020

Summary

We found a connection between entanglement entropy and fractal dimension in many-body wave functions. Even nonergodic states can exhibit ergodic entanglement entropy, challenging previous assumptions.

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Area of Science:

Quantum Information Theory
Condensed Matter Physics
Statistical Mechanics

Background:

Entanglement entropy quantifies correlations in quantum systems.
Fractal dimensions describe the complexity and structure of wave functions.
Page's work established a benchmark for entanglement entropy in ergodic states.

Purpose of the Study:

To establish a relationship between entanglement entropy and the fractal dimension of many-body wave functions.
To generalize Page's result to sparse random pure states (SRPS).
To investigate the behavior of entanglement entropy in nonergodic and fractal states.

Main Methods:

Generalization of Page's result to sparse random pure states (SRPS).
Analytical and numerical calculations of entanglement entropy for SRPS.
Definition of SRPS with N^D nonzero elements in a Hilbert space of size N.

Main Results:

Entanglement entropy scales with fractal dimension D for small D (S ~ D ln N).
Entanglement entropy saturates at the thermal (Page) value for larger D (S ~ ln N_A).
Demonstration that nonergodic wave functions can exhibit ergodic entanglement entropy.

Conclusions:

A direct link between entanglement entropy and fractal dimension is established.
The study provides a scenario where entanglement entropy behaves ergodically despite a nonergodic wave function.
Results are generalized to Renyi entropies and multifractal states.