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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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How to derive the backward Fokker-Planck equation from a forward Fokker-Planck equation (with state-dependent diffusion coefficient)?

I am interested in a system with state-dependent diffusion coefficients. This paper by Lau and Lubensky derives the correct Forward FPE in this case: $$\partial_tP(x,t) = \frac{\partial}{\partial x} ...
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Thermal diffusion equation with heat per unit length?

Let us say we have a metal rod. Along that rod there is a rate of heat generation of H per unit length. If we assume we are in the steady state then I would expect us (from the thermal diffusion ...
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Calculating the mutual diffusion coefficient of two spheres

I have several trajectories of two spheres and I'm trying to obtain the mutual diffusion coefficient of the two spheres as a function of their separation. I'm wondering if there is a nice way to do ...
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Nuclear Fission- One-group diffusion equation - Compute arbitrary constants

Nuclear Fission. In the one-group neutron diffusion equation it is possible to find an expression for the neutron flux for a specified geometry. For example, for an infinite slab the neutron flux can ...
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114 views

Do electrons really diffuse when a temperature gradient is applied?

In many websites and books, it is generally said that the charge carriers, be it electrons or holes, diffuse through the considered material when a temperature gradient is applied. However I have ...
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4k views

Characteristic time for heat diffusion

I often see that one can write the characteristic time (or time scale) of a diffusive process as: $\tau = \frac{L^2}{d}$ where $L$ is the characteristic length and $d$ is the diffusion coefficient. ...
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Diffusion of perfume according to fick's law

Fick's first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low ...
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18k views

How does smoke move in the air and how can I direct it so it will go to a place I want it to go?

Let us assume there is a closed room with 2 people, only one of them is smoking cigarette. What equation would describe the smoke from the cigarette spreading? Is it “diffusion”? If so, what are the ...
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131 views

Chemical Potential of Interacting Particles

I'm interested in studying the diffusion of (classical) particles that have some interaction with each other. More specifically, let the potential energy of any two of particles separated by a ...
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489 views

Difference between and diffusion and heat equations?

I read everywhere that diffusion and heat equations are similar. The same differential equations can be solved for both. Consider a finite one-dimensional diffusion or heat transfer where one end is ...
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What is the difference between the diffusion equation and the heat equation?

I know that the diffusion equation is a more general version of the heat equation. But what is the exact difference informally?
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Helium in an old mylar balloon

As a sealed mylar foil balloon ages, it deflates. I assume that this is mostly due to the monoatomic helium escaping through the envelope material. (In much the same manner as hydrogen can escape ...
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Is there any way to measure or calculate the diffuse reflection of a polymeric fibre?

I want to estimate the diffuse reflection of a polymer fibre. Is there any equation, for example, can be used to determine this value?
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21 views

Exponent of Lewis number in coupled heat-mass transfer problem

I am writing a small model of the coupling of heat and mass transfers in the nose and mouth. By solving the heat and mass balance equations, Lewis number expression appears that allows for coupling ...
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1answer
114 views

How can the analytical solution of the diffusion equation be used for a series of $N$ positions?

Given the exact solution to the diffusion equation: $$C(x,t) = \frac{1}{\sqrt{4 \pi D t}} \exp\left[-\frac{x^2}{4 D t}\right]$$ I am unsure as how it can be applied to a 1D series, as this equation ...
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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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Solving the diffusion equation

I am trying to clarify the relation between random walk and diffusion, and the source book proposes the following which I can't get. Starting from the diffusion equation $$ \frac{\partial C}{\partial ...
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187 views

Diffusion of electromagnetic fields

In magnetotelluric method, inside earth EM field has diffusion equation as opposed to wave equation in air. Physically diffusion of atoms is intuitive. But what does diffusion of EM field mean ...
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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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1answer
55 views

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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303 views

On the derivation of diffusion coefficient in thermodynamic terms

I'm trying to express the diffusion coefficient $D$, regarding particles suspended in a liquid, in terms of thermodynamic parameters. In this, some evident problems were encountered when trying to ...
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225 views

Speed of spontaneous mixing of different gases

Suppose we have a rectangular box divided into two equal cubic parts by a vertical impenetrable wall. Part 1 of the box contains a standard state mixture of $(1-x)$ mole of gas $A$ (e.g. Oxygen) and $...
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169 views

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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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 reason why Prandtl number is order unity for gases?

Is there a physical reason behind the fact that for gases the thermal diffusivity is on the same order of magnitude as kinematic viscosity (and as such a Prandtl number of order unity) and if so what ...
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Schrödinger equation derivation and Diffusion equation

I am aware of the debate on whether Schrödinger equation was derived or motivated. However, I have not seen this one that I describe below. Wonder if it could be relevant. If not historically but for ...
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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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Why do we need Diffusion currents to explain semiconductor current flow?

Why do we need the idea of carrier concentrations to explain current flow? Can we simply not associate the disparity in carrier concentrations between two samples to a disparity in relative charge ...
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236 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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65 views

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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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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815 views

1D drift-diffusion equation with single absorbing boundary

If we have just the simple diffusion equation (in 1D): $$ \frac{\partial P(x,t)}{\partial t} = D \frac{\partial^2 P(x,t)}{\partial x^2} $$ with an absorbing boundary at x=0 and initial condition $P(x,...
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985 views

Does diffusion current in semiconductor always exist?

According to what I understand, diffusion current is caused by the change in concentration of charge carriers in semiconductor (free electrons and holes) from higher concentration region to lower ...
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1answer
841 views

Diffusion and Drift currents in forward bias

Why do the current in diode in forward bias configuration called "Diffusion current" even if this current is a resultant of external voltage supply which create electric field in diode ,hence should ...
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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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1answer
535 views

How do I set up the tridiagonal matrix for a heat diffusion with layers of different thermal diffusivity?

I have Scala code that recreates the Crank-Nicolson solutions for the diffusion equations, and matches 'Excel for Scientists and Engineers' (Joe Billo, Wiley). However, I would like to be able to ...
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Spin drift velocity?

I am currently reading this Phys Rev paper by H C Torrey. In this paper, he derives the Bloch equations with an additional diffusion term. He says that the current density is given by $$\mathbf j_{\...
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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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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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1answer
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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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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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1answer
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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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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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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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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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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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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. ...