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Related Concept Videos

Propagation of Uncertainty from Random Error00:59

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
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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.
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Propagation of Uncertainty from Systematic Error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this particular...
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Consider a scenario where a wave propagates from a string of low linear mass density to a string of high linear mass density. In such a case, the reflected wave is out of phase with respect to the incident wave, however the...
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A common physical example of wave propagation with radial symmetry is the ripple formed when a stone is dropped into a still pond. The disturbance originates at a central point and travels outward as a circular wave. As the radius of the wavefront increases, the same initial energy is distributed along a progressively larger circumference. Consequently, the amplitude, or height, of the wave decreases with distance from the center. This decay behavior cannot be captured by simple sine or cosine...
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Related Experiment Video

Updated: Jul 15, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Exact propagator for generalized Ornstein-Uhlenbeck processes.

F Mota-Furtado1, P F O'Mahony

  • 1Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom. f.motefurtado@rhul.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 16, 2007
PubMed
Summary

This study derives a general formula for particle movement under momentum-dependent diffusion, crucial for understanding anomalous diffusion dynamics. The findings provide exact expressions for probability distributions and particle displacement correlations.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Non-equilibrium Systems

Background:

  • The Ornstein-Uhlenbeck process is a fundamental model for Brownian motion.
  • Generalizations are needed to describe systems with momentum-dependent diffusion.
  • Anomalous diffusion requires advanced analytical techniques.

Purpose of the Study:

  • Derive a closed-form propagator for a generalized Ornstein-Uhlenbeck process with momentum-dependent diffusion.
  • Analyze the behavior of the system for different diffusion function forms.
  • Obtain analytic expressions for probability distributions and correlations.

Main Methods:

  • Solving the Fokker-Planck equation with a momentum-dependent diffusion coefficient D(p) ~ |p|^(-alpha).
  • Utilizing modified Bessel functions for the closed-form propagator.
  • Applying the derived propagator to calculate specific physical quantities.

Main Results:

  • A general closed-form expression for the propagator was derived using modified Bessel functions.
  • The standard Gaussian propagator is recovered for alpha=0.
  • Analytic expressions for probability distributions and correlation coefficients were obtained for D(p) ~ |p|^(-1).
  • An exact expression for the proportionality constant in anomalous diffusion of mean-square displacement at short times was found.

Conclusions:

  • The derived propagator provides a powerful tool for studying systems with momentum-dependent diffusion.
  • The results offer insights into anomalous diffusion phenomena.
  • The study establishes a framework for analyzing complex stochastic processes in physics.