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Related Concept Videos

Entropy Change in Reversible Processes01:10

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.
Graphs of Equations in Two Variables01:30

Graphs of Equations in Two Variables

An equation with two variables, typically written in the form y = f(x) or Ax + By = C, describes a relationship between quantities represented by x and y. Each solution to such an equation is an ordered pair (x, y) that satisfies the equation when substituted. These pairs can be represented graphically to understand the variables' relationship visually.A common technique for constructing the graph of a two-variable equation is to create a value table. Begin by choosing several values for the...
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...
Graphs of Polar Equations01:17

Graphs of Polar Equations

The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
Graphs of Functions01:30

Graphs of Functions

Graphs of functions provide a visual representation of how output values change in response to varying inputs. Each point on the graph corresponds to an ordered pair, where the x-coordinate (independent variable) determines the horizontal position and the y-coordinate (dependent variable) determines the vertical position. Linear functions like y = x give a straight line, indicating a constant rate of change.Nonlinear functions display more complex behaviors. Even power functions generate...
Graphs of Two-Variable Functions01:27

Graphs of Two-Variable Functions

A weather map provides a practical example of a function of two variables. Across a wide region such as the United States, temperatures vary from one location to another. Each location can be identified by two geographic coordinates: longitude and latitude. Since a single temperature value is assigned to each coordinate pair, the situation can be represented mathematically as a function with two inputs and one output.In mathematical notation, longitude and latitude can be labeled as x and y,...

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Updated: Jun 23, 2026

Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit
05:30

Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit

Published on: September 8, 2023

Quantum ergodicity on graphs.

S Gnutzmann1, J P Keating, F Piotet

  • 1School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom.

Physical Review Letters
|May 14, 2009
PubMed
Summary
This summary is machine-generated.

We estimate deviations from quantum graph eigenfunction equidistribution in the high-energy limit. Our findings refine understanding of quantum chaotic systems and predict convergence rates for large graphs.

Related Experiment Videos

Last Updated: Jun 23, 2026

Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit
05:30

Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit

Published on: September 8, 2023

Area of Science:

  • Quantum mechanics
  • Mathematical physics
  • Spectral graph theory

Background:

  • Equidistribution of eigenfunctions is a key concept in quantum chaos.
  • Understanding this phenomenon on quantum graphs is crucial for theoretical physics.
  • Previous estimates for quantum chaotic systems have limitations.

Purpose of the Study:

  • To investigate the equidistribution of eigenfunctions on quantum graphs in the high-energy limit.
  • To provide an estimate for deviations from equidistribution in large, well-connected graphs.
  • To develop a criterion for the asymptotic emergence of equidistribution.

Main Methods:

  • Utilizing an exact field-theoretic expression derived from a supersymmetric nonlinear sigma model.
  • Performing a saddle-point analysis of the field-theoretic expression.
  • Comparing theoretical predictions with numerical tests on specific graph examples.

Main Results:

  • An estimate of deviations from equidistribution for large, well-connected quantum graphs.
  • A criterion determining when equidistribution emerges asymptotically.
  • A refined prediction for the rate of convergence, differing from previous assumptions in some cases.

Conclusions:

  • The study provides a significant refinement of previous estimates for quantum chaotic systems.
  • The developed theory offers new insights into the behavior of eigenfunctions on quantum graphs.
  • Numerical examples support the theoretical predictions, validating the approach.