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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Gas Transport Problem

I have a question that I thought would be a very simple effusion calculation that sent me down a massive rabbit hole after I tried to look it up in Landau's Physical Kinetics and so now I'm coming to ...
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Derivation of heat equation

What are the main physical laws to derive the following heat equation: $$u_t -\Delta u=f(t,x)?$$ I'm wondering about the interpretation of the Laplacian $\Delta$ and its role in heat process.
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What is the Carrè du Champ operator and what is it used for in the heat equation

Looking in a number of mathematical papers dealing with Markov semigroups and heat kernels, very often the Carrè du Champ operator appears that is defined as a bilinear form based on the infinitesimal ...
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Physical interpretation of a PDE

I'm not a physicist and I would like to understand the physical meaning of the following equation: $$u_t (t,x)=-\Delta^2 u(t,x)+f(t,x).$$ This is a $4^{th}$ order parabolic equation similar to the ...
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Ambipolar diffusion and sheaths in a bounded plasma

I am currently working through an introductory text book on plasma physics, and I have encountered two topics that I separately understand but seem to be at odds with one another. In a quasi neutral ...
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Factor of 1/3 in diffusion constant

In Feynman's Lectures on Physics, it says that the diffusion constant for a diffusive gas may be written as $$D=\frac{1}{3}lv$$ where $D$ is the diffusion constant, $l$ is the mean free path between ...
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Considering the diffusion of a single particle

Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero As I quoted through the Wikipedia, The diffusion ...
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Normal diffusion and dynamical chaos

Are there any central results/theorems which concern the implication that a dynamical system which is chaotic (in the sense of a largest positive Lyapunov exponent) will exhibit normal diffusion? By '...
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231 views

Rayleigh scattering light intensity

Assume a single ray of unpolarized light propagating through an opaque medium (water). Due to absorption and Rayleigh scattering the intensity decreases and light gets scattered/diffused before hiting ...
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Factor of 3 in Photon Diffusion coefficient

From definition of Diffusion coefficient: $$D = c/3(\mu_a+\mu_s),$$ where $c$ is the speed of light front, $\mu_a$ is absorption coefficient and $\mu_s$ is scattering coefficient. I wonder where does ...
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Boundary condition for a bulk-surface and bulk-bulk diffusion reaction system

Consider this simple example below and the corresponding geometries. I simplified these equations from the real system. Geometry 1 The first geometry is a sphere. Inside this sphere a species $b(t,...
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The convective analog to Fourier's equation of conduction?

The differential form of the thermal conduction law is given by $J=-\kappa\nabla T$ where $\kappa$ is the thermal conduction coefficient, or in the one-dimensional case, $J=-\kappa \frac{\partial ...
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Diffusion equation with walls (if possible with gravity), analytical solution

The solution of diffusion equation $$ \partial_t\rho=D\nabla^2\rho$$ with a point source $$ \rho(0,z)=\delta(z)$$ is in 1 dimension $$ \rho(t,z)=\frac{1}{\sqrt{4\pi Dt}}e^{-\frac{z^2}{4Dt}}$$ My ...
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Why is dry soil hydrophobic? Bad gardener paradox

When I forget to water my plants, and their soil becomes very dry, during the next watering I can see that the soil becomes hydrophobic. I can even see pockets of air between the repelled blob of ...
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Influence of air pressure on degassing of sparkling water

We all experience that sparkling water in a closed bottle will degas for a certain time and as the amount of degassed CO2 increases, this process slows down. My question is: can I slow down the ...
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How are drawing common tangents to get diffusion different for concave and convex phase fields?

Whenever there is a concept of diffusion involved and we need to decipher what is going to happen we tend to draw common tangent s in the phase field diagrams and decide which direction the diffusion ...
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Where does differntial equation of covariance matrix come from?

Does anyone know where the differential equation of covariance matrix comes from? $$\frac{dC(t)}{dt}=AC(t)+C(t)A^T+D$$ where $C$ is the covariance matrix, $A$ is the drift matrix, and $D$ is the ...
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Is $A\cos(kx-wt)$ a valid solution to the diffusion equation? [closed]

I know that for the wave equation $$\frac{\partial^2\psi}{\partial t^2}=c^2\frac{\partial^2\psi}{\partial x^2} \;, $$ we could always plug in the ansatz $A\cos(kx-\omega t)$, since equations of this ...
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How much air could an activated charcoal bag or baking soda box actually purify / deodorize?

Someone I know recently bought a set of small stylish and expensive cloth bags (roughly paperback book sized) filled with activated bamboo charcoal, that you place in a room and it purportedly ...
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50 views

Why does the Einstein relation hold in the derivation of the Goldman equation?

The Einstein relation $D = \mu k_B T$ is derived by assuming an equilibrium between the drift current and the diffusion current. Knowing this I would assume, that the relation is only valid under this ...
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Why does wavelength coherence affect diffusion?

Not to be confused with the relationship between wavelength and photon localization. But, laser light is is able to stay concentrated over a vast distance, much more so than every-day lamp light. ...
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Why do we interpret the first term of the Fokker-Planck equation as drift?

With the derivation of the Fokker-Planck equation we get: $$\frac{\partial}{\partial t}P(x,t)=-\frac{\partial}{\partial x}(A(x,t)P(x,t))+\frac{1}{2}\frac{\partial^2}{\partial x^2}(B(x,t)P(x,t))$$ We ...
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Where does the Master equation for the derivation of the Fokker-Planck equation come from?

I'm participating in an introductory course for biophysics. We briefly discussed the derivation of the Fokker-Planck equation and used the so-called Master equation as a starting point. $$ \frac{\...
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How to model mass diffusion with half life

I am in the process of writing a model for the diffusion of a chemical in an aqueous medium. Typically, to do this, one would use the diffusion equation $$\frac{\partial}{\partial t}U=\gamma \nabla ^...
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Does diffusion MRI measure diffusion or osmosis?

I am trying to understand the physical property which is measured in diffusion weighted imaging (DWI) and diffusion tensor imaging (DTI). I read that these methods estimate the apparent diffusion ...
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diffusion of magnetic field and Joule heating effects

Since the magnetic induction equation is $${\frac {\partial {\boldsymbol {B}}}{\partial t}}=\nabla \times ({\boldsymbol {v}}\times {\boldsymbol {B}})+\eta \nabla ^{2}{\boldsymbol {B}}$$ I want to ...
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Calculating the reflection and solute permeability for a diffusion cell

Here is the question, at which I have some problem. 'The three characteristic parameters $\sigma$, $L_p$ and $\omega$ can be determined from two experiments; in the first experiment 4.6 ml of water ...
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Solution to diffusion equation of a random walk

In my class of statistical physics, we studied the classic problem of random walk for the discrete case. In the end, we made the changes necessary for the master equation to be in the continuous ...
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Does a mixture of two different gases increase in entropy in a thermally isolated system?

I know gas will increase in entropy as it goes toward equilibrium, but what if there are multiple gases of different densities?
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Intrpretting questions on Fickian diffusion

I am considering the Zimm model for polymer dynamics, and have come across a question Find an expression for the time it takes for the polymer to diffuse a distance equal to its contour length $L=...
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Emptying tank through diffusion

Water ($40 ^{\circ}$, $1.0\, {\rm atm}$) slowly and steadily evaporates into nitrogen at the same temperature and pressure. from the bottom of a cylindrical tank as shown in the figure below. A stream ...
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Diffusion in a Semi-Infinite Medium: Breakaup of a Duhamel Convolution Integral

For a diffusion problem in a semi-infinite domain with a transient boundary condition, the temperature profile can be obtained from Duhamel's Principle as $$ \theta(r,t)= \int_0^t \frac{\partial \phi}...
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Water equilibrium while dying of thirst in a confined space

I'm trying to understand what happens with the water while a person is dying from thirst in a deep, narrow mine shaft. Assuming he has dry food for a long time, and some liters of water. Dry air is ...
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Diffusion equation with external forces

I am given the diffusion equation including the external forces as follows: $$\frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2} - \frac{F}{\gamma} \frac{\partial c}{\partial x}$$ ...
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Expression for “rotational diffusivity”; orientation random-walk of thin rod-like particles?

From this answer and from the Stokes-Einstein equation the diffusivity of a particle of radius $R$ in a fluid of viscosity $\eta$ is $$D=\frac{k_B T}{6 \pi \eta R}$$ where $\xi=6 \pi \eta R$ is a ...
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Difference between diffusion and interface mobility?

I have had this question when reading about the growth of ferrite (in steel) that is controlled by carbon diffusion and by interface mobility of Carbon. I would like to know the difference between ...
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Pouring Soda At A Constant Angle And Minimizing Filltime While Accounting For $\mathrm{CO}_2$

I was pouring soda, when the act made me think of a question. If you have an unopended $2L$ bottle of soda, and you wish to fill up a generic cylindrical glass cup - if you hold the $2L$ bottle at a ...
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Why doesn't the diffusivity of a particle in a fluid depend on the particle's density?

From this answer and from the Stokes-Einstein equation the diffusivity of a particle of radius $R$ in a fluid of viscosity $\eta$ is $$D=\frac{k_B T}{6 \pi \eta R}$$ where $\xi=6 \pi \eta R$ is a ...
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what's the difference between viscous flow and diffusion

For a one-componet gas flow in a channel. There are a pressure i.e density difference between two ends of the channel. There will be a mass flux. My question is how to describe this flux? it's a ...
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The influence of a temperature gradient on the growth of dendrites in a solid liquid interface

Not sure if this is the correct place to ask... My book states that, for a Solid Liquid interface in which the liquid has an increasing positive temperature gradient (increasing temperatures in ...
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Why and how do holes in the valence band of the p-type material of a p-n junction diffuse through to the valence band of the n-type material

tl;dr -- See the question title I quote from this Wikipedia article on the depletion region: By definition, the N-type semiconductor has an excess of free electrons (in the conduction band) ...
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Why don't we use Leibniz integral rule when solving Diffusion equation using the Fourier transform?

My question concerns the solution to the diffusion equation: $$\frac{\partial{p(x,t)}}{\partial{t}}=D\frac{\partial^2{p(x,t)}}{\partial{x}^2}~.\tag{1}\label{1}$$ I have a question about the solution ...
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What formula can serve as an approximate estimate of the time taken for the smell of a perfume to reach somebody?

I am in an attempt to calculate the time required for the smell of a bottle of perfume to reach a person's nose $10$m away. Real life experience tells me that it takes several seconds. I tried to work ...
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Classification of 2D time dependent diffusion equation

I was trying to classify the following PDE: $$\frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{x^2}}+\frac{\partial^2{u}}{\partial{y^2}}$$ where $u = u(x,y,t)$. I was originally using ...
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Rand-walk/Brownian-motion on 2D lattice [closed]

I started to learn stochastic processes this year. Only had two classes, but I already have some problem. We learned about Einstein's and Langevin's description of Brownian-motion and now I need to ...
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Diffusion equation Lagrangian: what is the conjugate field?

Morse and Feshbach state without elaboration that the diffusion equation for temperature or concentration $\psi$ and its "conjugate" $\psi^*$ (quotation marks theirs) has Lagrangian density: $$L=-\...
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Diffusion stopping current?

In the forward bias when electrons enter the p-type region of the semiconductor, the charge concentration is increased, and since the movement of electrons is faster than the movement of holes, it ...
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on diffusion process in semiconductor devices

Consider a PN junction. Say that the P and the N sides are just brought into contact and the diffusion process has started. We say that as the diffusion occurs, the charge carriers leave behind ...
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How to fit mean squared displacement data from nanoparticle tracking analysis?

I am trying to independently analyze raw nanoparticle tracking analysis (NTA) data, but my diffusion coefficients calculated directly from x and y pixel values only moderately correlate with reported ...
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Commutation of differential operators with boundary conditions

First post ever. Let's see how this goes... My question concerns the commutation of differential operators in the presence of boundary conditions. If it is of any help, this is relevant to me in the ...