Related Experiment Video
Updated: Jul 18, 2026
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
Retrieving the Green's function of the diffusion equation from the response to a random forcing.
1Center for Wave Phenomena and Department of Geophysics, Colorado School of Mines, Golden, Colorado 80401, USA. rsnieder@mines.edu
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 13, 2006
Summary
Researchers can retrieve the Green's function for diffusion equations by correlating random solutions, similar to wave propagation methods. This technique allows for extracting system properties from ambient noise, applicable to various diffusive phenomena.
Area of Science:
- Physics
- Applied Mathematics
- Geophysics
Background:
- Green's function retrieval is established for wave propagation using noise correlation.
- Wave equations for acoustics and elasticity are time-reversal invariant, enabling Green's function extraction.
- The diffusion equation lacks time-reversal invariance.
Purpose of the Study:
- To demonstrate Green's function retrieval for the diffusion equation using ambient fluctuations.
- To show that correlation of random solutions can yield the diffusion Green's function.
- To explore applications of this retrieval method in diffusive systems.
Main Methods:
- Correlating solutions of the diffusion equation excited by random sources.
- Recording these solutions at different spatial locations.
- Analyzing the correlation functions to extract the Green's function.
Main Results:
- The Green's function for the diffusion equation can be successfully retrieved by correlating random solutions.
- This method is effective even though the diffusion equation is not time-reversal invariant.
- The technique leverages ambient fluctuations for Green's function extraction.
Conclusions:
- Ambient noise correlation is a viable method for Green's function retrieval in diffusive systems.
- This approach extends Green's function recovery beyond time-reversal invariant systems.
- Potential applications span fluid dynamics, electromagnetism, contaminant transport, and wave scattering.
Related Concept Videos
01:26Vector Forms of Green’s Theorem
The study of fluid motion often involves understanding how local rotational behavior relates to global circulation. In the context of a pond with pollutants, direct measurement of water movement along an irregular shoreline can be impractical. Green’s Theorem in vector form provides an alternative by relating the circulation around a closed boundary to properties of the flow within the enclosed region.Measurements of water velocity at different points define a continuous vector field that...
01:27Green’s Theorem
Green’s Theorem establishes a relationship between a line integral around a closed plane curve and a double integral over the region enclosed by that curve. It applies to a vector field F(x, y) = 〈P(x, y), Q(x, y)〉, where P and Q have continuous first partial derivatives on an open set containing the region.Let C be a positively oriented, simple, closed, piecewise smooth curve, and let R be the plane region bounded by C. Green’s Theorem states that\begin{equation*}\oint_C P\,dx+Q\,dy =\iint_R...
01:27Extended Versions of Green’s Theorem
Green’s Theorem connects the circulation of a vector field around a closed curve with the behavior of the field across the region enclosed by that curve. It provides a way to replace a line integral around a boundary with a double integral over the interior region, making it especially useful in plane geometry, fluid flow, and vector calculus.Although Green’s Theorem is often introduced using simple regions without gaps, it can also be applied to regions made from several simple parts. This...
01:25Poisson's And Laplace's Equation
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.
01:11Types of Responses of Series RLC Circuits
A second-order differential equation characterizes a source-free series RLC circuit, marking its distinct mathematical representation. The complete solution of this equation is a blend of two unique solutions, each linked to the circuit's roots expressed in terms of the damping factor and resonant frequency.
01:28Debye–Huckel–Onsager Conductance Equation
The Debye-Hückel-Onsager equation is a cornerstone of physical chemistry, providing a method to determine the molar conductance (Λm) and molar conductance at infinite dilution (Λ°m) for uni-univalent electrolytes.Uni-univalent electrolytes are electrolytes that dissociate in solution to produce one cation with a +1 charge and one anion with a –1 charge per formula unit.This equation addresses two crucial phenomena: the asymmetry effect and the electrophoretic effect. According to this equation,...