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Related Concept Videos

Gaussian Elimination: Problem Solving01:30

Gaussian Elimination: Problem Solving

Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
Gauss's Law01:07

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.
Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a uniform...
Gauss's Law: Planar Symmetry01:27

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
Gauss's Law: Problem-Solving01:10

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...

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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
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Gaussian equilibration.

Lorenzo Campos Venuti1, Paolo Zanardi

  • 1Department of Physics and Astronomy and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089-0484, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 16, 2013
PubMed
Summary
This summary is machine-generated.

Quantum systems equilibrate probabilistically. This study introduces Gaussian equilibration for quasifree Fermi systems, showing observables follow a Gaussian distribution with errors scaling as O(L(-1/2)).

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Area of Science:

  • Quantum mechanics
  • Statistical physics
  • Many-body systems

Background:

  • Finite quantum systems equilibrate probabilistically under unitary evolution.
  • Time fluctuations of observables in many-body systems are typically small, scaling exponentially with system size.
  • Quasifree Fermi systems, characterized by quadratic Hamiltonians and observables in Fermi operators, are a key focus for studying equilibration.

Purpose of the Study:

  • To investigate the temporal fluctuations of observables in quasifree Fermi systems.
  • To establish a connection between quantum equilibration dynamics and classical models.
  • To introduce and analyze the concept of Gaussian equilibration.

Main Methods:

  • Proving bounds on temporal fluctuations (ΔA(2)) for observables.
  • Mapping quantum equilibration dynamics to a generalized classical XY model at infinite temperature.
  • Analytical and numerical methods to verify the Gaussian equilibration conjecture, including studies on the quantum XY model and tight-binding models.

Main Results:

  • A bound on temporal fluctuations ΔA(2) is established for quasifree Fermi systems.
  • Equilibration dynamics are successfully mapped to a classical XY model.
  • The conjecture of Gaussian equilibration is supported, predicting a relative error of O(L(-1/2)) for observables, where L is the single-particle space dimension.
  • The variance is shown to be discontinuous at the transition between quasifree and nonintegrable models.

Conclusions:

  • Gaussian equilibration provides a framework for understanding the probabilistic equilibration of observables in quasifree Fermi systems.
  • The findings offer insights into the statistical properties of quantum systems approaching equilibrium.
  • The study highlights the role of system integrability in the behavior of quantum observables near transitions.