Equation 2 · Einstein's Random Walk and the Mathematics of Genetic Drift
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol f
f occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol x
x is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol t
t is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol D
D is an input to the expression that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →derivative
This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →Denominator: partial t
The complete quantity below the fraction bar; it must be nonzero for this division.
Denominator: partial x^2
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
Working through the statistical mechanics of a wall permeable to solvent but not to suspended particles, Einstein arrived at a differential equation for how the concentration f(x,t) of particles along one axis changes over time: . the diffusion equation, in a form still written the same way today [ 1 ] . Solving it for particles released from a point gives a spreading distribution whose variance grows linearly, not with the square root of time as a naive guess might suggest but as
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
Return to Einstein's Random Walk and the Mathematics of Genetic Drift