Questions tagged [diffusion]
Diffusion is the net movement (spreading out) of molecules or atoms down a concentration gradient: from a region of high concentration to a region of low concentration.
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Question about diffusion current in semiconductor?
I'm reading semiconductor physics by Neamen and when the disscusion about the diffusion current density he wrote this (I'll cite the image from the book because it has an important graph that will ...
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Fluid viscosity, mass diffusion and Navier-Stokes equation
With the increase of fluid viscosity, mass diffusion of a fluid decreases. Then how the diffusion term in Navier-Stokes equation has a dominant effect at high viscosity? Also how the mass convection ...
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Can oxygen 'leach' along a wire? [closed]
[If this is not the right SE for this question, apologies.]
Automobile gasoline engine controls use oxygen sensors in the exhaust stream. The Wikipedia article on Oxygen Sensors notes that (some) ...
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Solving the generalized Smoluchwski equation
The generalization of the Smoluchowski equation for $N$ interacting particles reads
$$\frac{\partial w}{\partial t} = \sum_{i, j=1}^N \nabla_i \cdot D_{ij} (\nabla_j - \bar{F}_j/kT) w,$$
where $D_{ij}$...
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Charge diffusion
Considering the following current in flat Minkowski space
$$J^\mu=nu^\mu -\sigma \Delta^{\mu\nu}\partial_\nu \mu,$$ where $n$ and $\sigma$ depend on chemical potential $\mu$:
$$\mu=\mu_0+\varepsilon \...
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How does a diffusive wave behave differently than a traditional wave at an interface?
Traditional wave phenomena (I.E. Electromagnetic, Acoustic etc.) can be described by the Hyperbolic wave equation $\bigtriangledown^2\psi(r,t) + \frac{d^2\psi(r,t)}{dt^2} = q(r,t)*e^{i\omega t}$. ...
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Carbon dioxide Diffusion in solids vs Liquid, such as water or solid-state plasticized polymer
Carbon dioxide say when released from a gas pump into water contained into an uncovered beaker will diffuse and also solublize in water and form carbonic acid. But what about the dehydration of ...
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Connection between diffusion and non-integrable 1D spin chains
My question concerns non-integrable (à la Bethe) 1D spin chains.
Consider, for example, the 1D non-integrable Ising model
\begin{equation}
H = \sum_{i \in \mathbb{Z}}\sigma_{i}^{z} \sigma_{i+1}^{z} + ...
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Chapman–Enskog theory and a Noether's invariant of energy?
So in this answer:
Folks figured out thermodynamics before statistical mechanics. In particular, we had thermometers. People measured the "hotness" of stuff by looking at the height of a ...
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What's the difference between the mass equilibrium equation and the diffusion equation?
Was wondering whether the mass equilibrium ODE and the diffusion equation PDE stem from the same physical concepts (at least when the mass represents Chemical Pollution):
In my own words, the "...
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Diffusion from an instantaneous spherical source with a continuous spherical sink
I assume this is a solved problem, but I cannot find the solution in some common sources, so I am asking here. Suppose you have diffusion of a molecule (diffusion coefficient $D$) from an ...
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Under what conditions is mean square displacement $\text{MSD}(t)=4Dt+v^2t^2$ a valid asymmetric random walk model?
I am reading the paper Actin dynamics drive microvillar motility and clustering during brush border assembly by Meenderink et al. (2019). In this paper, the authors fit the mean square displacement (...
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Survival function of particles with different starting points
I am studying diffusion equations to model the diffusion of fluorescence particles into a cell monolayer. The easiest way is to model them starting from a point source (x = 0 at t = 0 -> $x_\mathrm{...
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On the derivation of Fick's first law from first principles
I risk repeating a question that has been asked too many times, but ... is there a rigorous (but not too complicated) way to derive Fick's first law?
In this post the answer starts from what is ...
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Does diffusion of a spring depend on the spring’s equilibrium length?
Say we perform a random walk on $\mathbb{Z}$ with two equal particles that are connected by a spring with spring constant $K$ and equilibrium length $a$, where at each time step each particle has ...
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Partition function for a 1 dimensional polymer
A one-dimensional polymer (a chain), made of (N + 1) monomers, is diffusing on top of a one-dimensional lattice having a lattice constant a = 1. The i-th monomer (i = 0, 1, 2,..., N) is located at a ...
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Kirkendall effect: equilibrium vacancy concentration and pair interaction energy
I am reading the following article on Kirkendall effect leading to the Formation of a Hollow Binary Alloy Nanosphere: A Kinetic Monte Carlo Study. I am unable to understand or find in references the ...
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Diffusion problem: number of droplets in the air
I'm having a big brain block here and want to check if my reasoning is correct. I'm considering a room of volume $V$, with $p$ persons, during the time $dt$. We want to know the number of droplets ...
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What real life applications does diffusion limited aggregation have?
While studying DLA - diffusion limited aggregation - in the field of fluid dynamics I started wondering what real life application does this phenomena have? We observe fractals in experiments such as ...
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Sense for a viscoelastic term of Maxwell type
I go reference the book of Jan Prüss (Evolutionary Integral Equations and Applications, 2012) as needed. In this book (pg. 128) it is explored some concepts of stress and strain considering the ...
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Relation between time it take to diffuse and temperature
I was reading Reif's Statistical physics. In problem 7 in chapter 2, It says
"Consider a thin copper wire stretched out along the $x$-axis. A few of the copper atoms, located near $x=0$, are ...
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3D information from shadows
I was holding a 1 inch diameter pipe recently on a sunny morning. I noticed that the turning the pole at various orientations changed the shadow on the ground. The closer portion of the pipe to the ...
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Relation between the diffusion coefficients
My course is giving me a strange relation I have difficulties to understand. In the context of thermodynamics and evolution, we define:
The mass diffusion flux: (Newton's law)
$$J_{M_d} = -D\frac{dc}{...
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Simulating diffusion for particles undergoing Brownian motion
I am trying to do a computer simulation as part of a research project of the diffusion of a few particles undergoing brownian motion along a two-dimensional canvas. I am having some holes in my ...
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Do pressure gradients affect diffusion speed for gases dissolved in liquids (scuba offgassing question)
This question comes from a scuba forum where we are discussing the technical scuba technique of breathing 100% O2 during stops at shallow depths to speed up the offgassing of nitrogen (N2) from body ...
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Where can I find/derive these equations?
I am currently investigating the behavior of the smoke and I will need to account for the velocity of the smoke as soon as the flame is smothered. There is also the behavior where the smoke expands in ...
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Generalized random walk process: Fokker-Planck equation
The diffusion equation
$$
\frac{\partial p(x,t)}{\partial t} = D \frac{\partial^2 p(x,t)}{\partial x^2}
$$
can be derived from a simple random walk on a line.
For example, if the probability of ...
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Understanding the Markovian and Gaussian approximations in Brownian motion
I am trying to understand in the original derivation of Brownain diffusion, where does the assumption of Markovian and Gaussian nature factor in.
In Albert Einstein's original work on Brownian motion ...
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Diffusion of gas mixtures through a membrane
Suppose we have a mixture of gases $k = 1,2 \ldots$ in a large volume with constant pressure and temperature (via external pressure and temperature control).
We immerse a small volume separated via a ...
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Where does the negative sign in the Cahn-Hillard model of diffusion come from?
In the Cahn-Hillard model of diffusion, motion of a particle field phi is defined by the diffusion of a potential mu
For electrodynamics, phi can be thought of as charge, M as the conductivity of the ...
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Diffusion Equation vs. Gradient Flow
I am trying to understand the connection between the 'common' diffusion equation and the evolution equation of the concentration as a gradient flow of the free energy.
So let's simply assume an ideal ...
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What is the entropy associated with a diffusion process?
Is it possible to calculate entropy from the solution of a diffusion equation(with natural boundary conditions) by using the formula of Shanon entropy? Could anyone help me to understand the entropic ...
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Is diffusion a nonequlibrium process?
We know that the diffusion equation can be viewed as the continuum limit of a discreet unbiased random walk.
My question is the following:
Is diffusion (or an unbiased random walk) a non-equilibrium ...
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Is it possible to define concepts like free energy in the diffusion process?
How does the idea of free energy (which we derive from the canonical partition function) fit in the domain of non-equilibrium processes like diffusion?
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Henry's Law - When scuba diving, why is breathing pure $\rm O_2$ not the same as going to outer space?
In Short
I had a discussion with my dive instructor the other day. He said that for "decompression dives" you can cut down decompression time to 1/3 if you switch to pure $\rm O_2$ at 6m.
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Doubts about how to apply Fick's law to gas
According to Fick's law the flux of gas through a membrane depends on the concentration gradient and then on the pressure gradient:
$$\Delta P = P_a - P_b$$
where $P_a$ is the pressure of container $A$...
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Interdifussion (binary diffusion) of Gases Boundary Condition Question
I was trying to solve this 1D unsteady state diffusion problem of two gases being mixed in a finite length of a tube over time (each end is blocked and impermeable).
$$
\frac{\partial{C}}{\partial{t}}...
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Estimate diffusion coefficient from 2D projections of 3D trajectories?
Given the projections x(t) y(t) of the trajectories of several particles moving in 3D in a water solution, is there a way to estimate the diffusion coefficient of the 3D diffusion, if we assume it ...
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How should I set the boundary condition of a diffuse equation in a potential well?
I want to solve the initial value problem
of a particle diffusing in a well on a disk.
Paticle density$\rho(\vec{r},t)$satisfy the following equation:
$$
\partial\rho/\partial t=\nabla\cdot(\nabla V(r)...
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Can mass transfer occur without a concentration difference? (analogous to the question, can energy transfer occur without a temperature difference?)
The driving force for diffusion of mass is a concentration gradient. However, mass can be transported both in a diffusive and a convective (due to the flow of a fluid) way. By definition, the combined ...
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Random walk in finite VERSUS infinite space: Probability density functions and their interpretation
I am studying the probability density function of a random walk in a confined geometry (2D-BOX). I am also comparing this probability density function to its equivalent in infinite two-dimensional ...
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Properties of random-walk in infinite and finite two-dimensional space: probability of two particles being in the same location at time t
I have been told that one of the property of the continuous-time random walk in two dimensions is that:
$$\int_{Z} \, G(z, t | p_1) \, G(z, t | p_2) \,dz = \,G(p_1,p_2,2t)$$
where ...
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How can NO charge carriers exist in the depletion region?
I learned that no charge carriers exist in the depletion region of a PN junction due to the balance between the diffusion current and the drift current due to the electric field created by charged ...
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Singularities/infinities of continuity equation in polar coordinate
I encountered a bit of a difficulty in solving the continuity equation for polar coordinates. For a "fluid" or density of particles moving radially outwards with constant velocity, its flux ...
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Probability distribution of particle diffusion system with a source and absorbing boundaries
Consider a simple 1D particle diffusion process described by the SDE $dx=\sigma dW$, where $dW$ is a Wiener process. The forward Fokker-Planck equation can then be written as
$$
\frac{\partial P(x,t)}{...
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If the inclination of a cloud of oxygen in outer space is to diffuse, then how do nebulae form?
If a cruise ship-size object in outer space were surrounded by a spherical cloud of oxygen, and there were no other bodies exerting significant gravitational force in the vicinity, would the cloud of ...
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What is the mathematical derivation for no diffusion term in the mass continuity equation of the Navier-Stokes/Euler equations?
In this post the fact that the mass continuity equation in a mixture of gases has no diffusion term, i.e.,
$$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{v})=0$$
has been discussed. ...
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What resources are you aware of and can recommend me to study and model spectral diffusion?
I am looking for textbooks, papers, etc. that cover this topic in some detail and would allow me to develop a simple model to use in my BSc thesis. I'm having a hard time finding the resources I need. ...
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Brownian motion and multi-scale stochastic processes
The Stokes-Einstein equation for the diffusion coefficient of small colloidal particles in suspension is canonically derived under the assumption that the primary motion of the particle is ...
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Differences in interactions of light with different mediums
I am trying to figure out this problem, I apologize in advance if the question is silly.
If i am correct sunsets are red because red light doesnt get diffounded as much as blue light, which eventually ...
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