Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

3.5K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
3.5K
Divergence and Stokes' Theorems01:06

Divergence and Stokes' Theorems

2.3K
The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
2.3K
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

2.8K
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.
2.8K
Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion03:48

Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion

29.7K
Although gaseous molecules travel at tremendous speeds (hundreds of meters per second), they collide with other gaseous molecules and travel in many different directions before reaching the desired target. At room temperature, a gaseous molecule will experience billions of collisions per second. The mean free path is the average distance a molecule travels between collisions. The mean free path increases with decreasing pressure; in general, the mean free path for a gaseous molecule will be...
29.7K
Distribution of Molecular Speeds01:27

Distribution of Molecular Speeds

4.2K
The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This predictable distribution of molecular speeds is known as the Maxwell-Boltzmann distribution. The distribution of molecular speeds in liquids is comparable to that of gases but not identical and can help to understand the phenomenon of the boiling and vapor pressure of a liquid. Consider that a molecule requires a...
4.2K
Gauss's Law: Problem-Solving01:10

Gauss's Law: Problem-Solving

2.2K
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...
2.2K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Thyroid Function Parameters, Thyroid Status, and Vitamin B12 Deficiency in Hospitalized Adults: A Retrospective Cross-Sectional Study in Eastern India.

Cureus·2026
Same author

Coexistence of Anaemia and Common Morbidities Among Children in India Below the Age of Five Years: Evidence From the National Family Health Survey-5 (2019-21).

Cureus·2026
Same author

Pathogen-mediated priming induces intergenerational immunity against spot blotch in wheat.

World journal of microbiology & biotechnology·2026
Same author

Anti-CRISPR-mediated continuous directed evolution of CRISPR-Cas9 in human cells.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same author

Machine learning framework for cost effective deep mutational scanning through targeted substitution profiling.

BMC bioinformatics·2026
Same author

Gold nanoparticle-based electrophoresis-free colorimetric detection method for allele-specific PCR-SNP genotyping.

Analytical biochemistry·2026

Related Experiment Video

Updated: Sep 30, 2025

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
06:55

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

Published on: September 26, 2016

8.0K

Extreme value statistics and arcsine laws for heterogeneous diffusion processes.

Prashant Singh1

  • 1International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India.

Physical Review. E
|March 16, 2022
PubMed
Summary
This summary is machine-generated.

This study analyzes heterogeneous diffusion models with power-law diffusion coefficients. We derive exact probability distributions for extreme values and passage times, revealing distinct behaviors compared to standard Brownian motion.

More Related Videos

The Diffusion of Passive Tracers in Laminar Shear Flow
08:01

The Diffusion of Passive Tracers in Laminar Shear Flow

Published on: May 1, 2018

8.7K
Measurement of Particle Size Distribution in Turbid Solutions by Dynamic Light Scattering Microscopy
09:16

Measurement of Particle Size Distribution in Turbid Solutions by Dynamic Light Scattering Microscopy

Published on: January 9, 2017

14.5K

Related Experiment Videos

Last Updated: Sep 30, 2025

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
06:55

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

Published on: September 26, 2016

8.0K
The Diffusion of Passive Tracers in Laminar Shear Flow
08:01

The Diffusion of Passive Tracers in Laminar Shear Flow

Published on: May 1, 2018

8.7K
Measurement of Particle Size Distribution in Turbid Solutions by Dynamic Light Scattering Microscopy
09:16

Measurement of Particle Size Distribution in Turbid Solutions by Dynamic Light Scattering Microscopy

Published on: January 9, 2017

14.5K

Area of Science:

  • Statistical Physics
  • Non-equilibrium Systems
  • Complex Dynamics

Background:

  • Heterogeneous diffusion, characterized by a spatially varying diffusion coefficient D(x), is prevalent in biological and physical systems.
  • A power-law model D(x) ~ |x|^{-α} exhibits anomalous scaling of mean-squared displacement and weak ergodicity breaking.
  • Understanding extreme value statistics and passage times is crucial for characterizing complex diffusion processes.

Purpose of the Study:

  • To derive exact probability distributions for extreme spatial displacement and the time of maximum displacement in a power-law heterogeneous diffusion model.
  • To analyze the statistical properties and exact distributions of residence time and last-passage time for this model.
  • To contrast the statistical behaviors of heterogeneous diffusion with standard Brownian motion.

Main Methods:

  • Analytical derivation of probability distributions for maximum displacement, time of maximum displacement, residence time, and last-passage time.
  • Investigation of a power-law diffusion model D(x) ~ |x|^{-α} for all α > -1.
  • Extensive numerical simulations to validate analytical findings.

Main Results:

  • Exact probability distributions for maximum displacement M(t) and its time of occurrence t_m(t) derived for all α.
  • Exact distributions for residence time t_r(t) and last-passage time t_l(t) computed for all α.
  • Demonstrated significant differences in these distributions between heterogeneous diffusion (α ≠ 0) and Brownian motion (α = 0), unlike the universal arcsine laws for Brownian motion.
  • Identified a critical exponent α_c ≈ -0.3182, below which the residence time distribution shows a maximum at t/2, and at or above which it shows minima.

Conclusions:

  • The power-law heterogeneous diffusion model exhibits rich statistical properties distinct from Brownian motion, particularly in extreme value and passage time statistics.
  • The derived exact distributions provide a comprehensive understanding of these complex diffusion phenomena.
  • The critical exponent α_c highlights a fundamental change in the process dynamics based on the diffusion heterogeneity.