Diffusion equation

The following equation describes the diffusion process of a substance present in non-uniform concentration.

Here, c is the substance concentration, t is time, and D is the diffusion coefficient. Or, if we focus on a single particle microscopically, and this particle is at the origin r = 0 of space at time t = 0, the equation for the probability density P ( r , t |0,0) of being at point r in space after time t is:

is called the diffusion equation. This equation can be easily integrated when the diffusion coefficient is isotropic and constant, and in spherical coordinates it is given by

The normal distribution is as follows. The standard deviation L , which indicates the spread of the distribution, is

L = 2
and increases in proportion to the square root of time in the distance dimension, which is called the diffusion distance.

Source: Morikita Publishing "Chemical Dictionary (2nd Edition)" Information about the Chemical Dictionary 2nd Edition

不均一な濃度で存在する物質の拡散過程を記述する次の方程式．

ここに，cは物質濃度，tは時間，Dは拡散係数である．または微視的に1個の粒子に注目し，この粒子が時間t ＝ 0で空間の原点r ＝ 0にあったとき，時間tの後，空間のr点にある確率密度P(r,t|0,0)に関する方程式，

を拡散方程式という．この式において拡散係数が等方的でかつ一定であるときには容易に積分でき，球座標を用いると，

という正規分布になる．分布の広がりを示す標準偏差Lは

L ＝ 2
になり，距離の次元で時間の平方根に比例して増加し，これを拡散距離という．

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