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Related Concept Videos

Atomic Nuclei: Nuclear Spin State Population Distribution01:14

Atomic Nuclei: Nuclear Spin State Population Distribution

Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.
Carrier Transport01:21

Carrier Transport

The generation of electrical current in semiconductors is fundamentally driven by two mechanisms: drift and diffusion. These processes are essential for the functionality and performance of semiconductor-based devices.
Drift Current:
The drift of charge carriers is started by an external electric field (E). Charged particles, such as electrons and holes, experience an acceleration between collisions with lattice atoms. For electrons, this results in a drift velocity (vd) given by:
Atomic Nuclei: Nuclear Relaxation Processes01:23

Atomic Nuclei: Nuclear Relaxation Processes

In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis. This...
The Thermodynamics of Mixing01:28

The Thermodynamics of Mixing

Mixing is a fascinating phenomenon in thermodynamics, particularly when considering the Gibbs energy of a mixture at constant temperature and pressure. This energy, denoted as G, tends to decrease during spontaneous mixing processes, offering insights into the composition changes that occur.Imagine two ideal gases, initially separated in different containers, with amounts nA and nB, respectively, both at a temperature T and pressure p. The chemical potentials of these gases have their 'pure'...
Atomic Nuclei: Nuclear Spin State Overview01:03

Atomic Nuclei: Nuclear Spin State Overview

NMR-active nuclei have energy levels called 'spin states' that are associated with the orientations of their nuclear magnetic moments. In the absence of a magnetic field, the nuclear magnetic moments are randomly oriented, and the spin states are degenerate. When an external magnetic field is applied, the spin states have only 2 + 1 orientations available to them. A proton with = ½ has two available orientations. Similarly, for a quadrupolar nucleus with a nuclear spin value of one, the...
Electrostatic Boundary Conditions01:16

Electrostatic Boundary Conditions

Consider an external electric field propagating through a homogeneous medium. When the electric field crosses the surface boundary of the medium, it undergoes a discontinuity. The electric field can be resolved into normal and tangential components. The amount by which the field changes at any boundary is given by the difference between the field components above and below the surface boundary.
The surface integral of an electric field is given by Gauss's law in integral form and is related to...

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Related Experiment Video

Updated: Jun 20, 2026

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets
06:26

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets

Published on: May 15, 2017

Thermalization in the mixed-field Ising model: An occupation-number perspective.

Isaías Vallejo-Fabila1, Fausto Borgonovi2,3, Felix M Izrailev4,5

  • 1University of Connecticut, Department of Physics, Storrs, Connecticut 06269, USA.

Physical Review. E
|June 19, 2026
PubMed
Summary

We studied thermalization in quantum and classical spin models using occupation numbers. Classical ergodicity deviations decay algebraically, providing bounds for quantum thermalization.

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Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses
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Published on: June 7, 2018

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Last Updated: Jun 20, 2026

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets
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Published on: May 15, 2017

Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses
08:55

Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses

Published on: June 7, 2018

Area of Science:

  • Statistical Mechanics
  • Quantum Physics
  • Condensed Matter Theory

Background:

  • Occupation number is crucial for understanding thermalization and statistical distributions (Fermi-Dirac, Bose-Einstein, Boltzmann).
  • In spin systems, it quantifies magnetization sublevel populations.
  • Probing thermalization in isolated quantum spin models is challenging due to Hilbert space size.

Purpose of the Study:

  • To investigate the onset of thermalization in a quantum spin-1 Ising model and its classical counterpart.
  • To establish a quantitative criterion for ergodicity in interacting spin systems.
  • To determine bounds for the approach to thermal equilibrium in quantum models.

Main Methods:

  • Analyzed the occupation number dynamics in a 1D quantum spin-1 Ising model with transverse and longitudinal fields.
  • Utilized the classical counterpart to overcome finite-size limitations in quantum simulations.
  • Tracked individual spin dynamics on Bloch spheres and applied random matrix theory.

Main Results:

  • Thermalization in the quantum model was assessed by the convergence of occupation number averages to microcanonical predictions.
  • Classical ergodicity deviations were found to decay algebraically with system size (power-law scaling).
  • This scaling provided a quantitative bound for the approach to thermal equilibrium in the quantum system.

Conclusions:

  • The study establishes a link between classical ergodicity and quantum thermalization.
  • Algebraic decay of classical ergodicity provides a practical method to bound quantum thermalization.
  • The findings offer insights into the dynamics and equilibrium properties of isolated spin systems.