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Related Concept Videos

Entropy02:39

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...
Entropy01:18

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...
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.
The Entropy as a State Function01:14

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...
Random Error01:04

Random Error

Random or indeterminate errors originate from various uncontrollable variables, such as variations in environmental conditions, instrument imperfections, or the inherent variability of the phenomena being measured. Usually, these errors cannot be predicted, estimated, or characterized because their direction and magnitude often vary in magnitude and direction even during consecutive measurements. As a result, they are difficult to eliminate. However, the aggregate effect of these errors can be...
Absolute Entropies and the Third Law of Thermodynamics01:23

Absolute Entropies and the Third Law of Thermodynamics

Ludwig Edward Boltzmann developed a definition for entropy, which stated that absolute entropy is proportional to the natural logarithm of the number of possible combinations of particles. Entropy stands alone among state functions as the only one whose absolute values can be determined.Consider a gas sample confined to a container. As the container expands, the energy levels of gas molecules become more closely spaced. This increases the number of available energy states, thereby increasing...

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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Eigenstate randomization hypothesis: why does the long-time average equal the microcanonical average?

Tatsuhiko N Ikeda1, Yu Watanabe, Masahito Ueda

  • 1Department of Physics, University of  Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 21, 2011
PubMed
Summary
This summary is machine-generated.

We introduce the eigenstate randomization hypothesis (ERH) to explain thermalization in isolated quantum systems. ERH provides a vanishingly small bound on thermalization errors and applies to systems where other theories fail.

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Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package
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Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

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Area of Science:

  • Quantum mechanics
  • Statistical mechanics
  • Condensed matter physics

Background:

  • Understanding thermalization in isolated quantum systems is crucial for statistical mechanics.
  • The eigenstate thermalization hypothesis (ETH) explains thermalization but has limitations.
  • A gap exists between long-time averages and microcanonical ensemble averages in some quantum systems.

Purpose of the Study:

  • To derive an upper bound for the difference between long-time average and microcanonical ensemble average of observables.
  • To propose and validate the eigenstate randomization hypothesis (ERH).
  • To establish the conditions under which isolated quantum systems thermalize.

Main Methods:

  • Derivation of an upper bound using analytical methods.
  • Numerical verification of the proposed hypothesis.
  • Analytical support for the eigenstate randomization hypothesis.

Main Results:

  • The eigenstate randomization hypothesis (ERH) is proposed and supported.
  • ERH implies that diagonal elements of observables in the energy eigenbasis fluctuate randomly.
  • ERH encompasses ETH and significantly reduces the thermalization error bound.
  • ERH is applicable to integrable systems where ETH fails.

Conclusions:

  • ERH provides a more general framework for understanding thermalization in quantum systems.
  • The validity of ERH dictates the applicability of the microcanonical description.
  • ERH offers new insights into the statistical properties of quantum observables.