Foundation · Explanatory preview

Exponentials and continuous scaling

The exponential function describes systems where the rate of change is proportional to current value, requiring a strictly dimensionless exponent.

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Why do exponential functions govern thermal distributions and random diffusion?

The exponential function e^u, often written exp(u), is the unique mathematical function whose derivative with respect to its argument equals the function value itself. Whenever a physical growth or decay rate is proportional to the amount already present, the resulting trajectory is an exponential.

A fundamental rule of dimensional physics is that the argument of any transcendental function, including the exponential, must be a dimensionless number. In Wien's radiation law exp(-beta nu / T), the product beta nu has dimensions of temperature, so dividing by temperature T yields a pure dimensionless ratio. In the Brownian spreading Gaussian exp(-x^2 / (4Dt)), x^2 has units of squared metres and 4Dt has units of (m^2/s)*s = m^2, ensuring the exponent is dimensionless.

Exponentials also arise naturally from the multiplication of independent probabilities across repeated random steps or subdivided spatial cells.

One worked example

f(x,t) = \frac{1}{\sqrt{4\pi Dt}} \exp\left(-\frac{x^2}{4Dt}\right)

Density f of x and t equals one over square root of four pi D t times exponential of minus x squared over four D t.

At the center x = 0, the exponential factor exp(0) = 1. At one standard deviation x = sqrt(2Dt), the exponent is -1/2 and exp(-0.5) approx 0.6065. At x = sqrt(4Dt), the exponent is -1 and exp(-1) approx 0.367879, showing symmetric bell-shaped decay.

A stopping point: The argument inside an exponential function must always be a dimensionless pure number.

Interactive construction: repeated proportional changes and dimensionless exponents

In linear change, an equal amount is added in every equal interval: y = y₀ + mt. In exponential change, the quantity is multiplied by an equal factor in every interval: y = y₀ · rⁿ. Because each change is proportional to the current amount, exponential functions naturally describe continuous growth and decay.

Select a step index n to inspect compounding:

Inspection at step n = 1

Formula: f(1) = (e^−0.5)¹ = e^−0.5

Fraction remaining: 0.606531 (60.65%)

Each 10-second interval scales the previous value by exactly e^−0.5 ≈ 0.606531.

Compounding decay steps with constant multiplier e⁻⁰·⁵
Step n	Elapsed time t (s)	Step multiplier	Fraction remaining	Percent
0	0	1.000000	1.000000	100.00%
1	10	e⁻⁰·⁵ ≈ 0.606531	0.606531	60.65%
2	20	e⁻⁰·⁵ ≈ 0.606531	0.367879	36.79%
3	30	e⁻⁰·⁵ ≈ 0.606531	0.223130	22.31%
4	40	e⁻⁰·⁵ ≈ 0.606531	0.135335	13.53%
5	50	e⁻⁰·⁵ ≈ 0.606531	0.082085	8.21%

Why exponents must always be dimensionless

You cannot evaluate e raised to three meters or five seconds, because the series definition e^u = 1 + u + u²/2! + … would require adding meters to square meters. In every physical law, dimensional quantities inside exponents are strictly cancelled by matching units:

Wien’s law (Light §4): e^−βν/T. The frequency ν (s⁻¹) and temperature T (K) are balanced by β = h/k (s·K), making βν/T dimensionless.
Brownian diffusion Gaussian (Brownian §4): e^−x²/(4Dt). The numerator x² has units m²; the denominator 4Dt has units (m²/s · s) = m², so x²/(4Dt) is purely dimensionless.

Textual summary of the construction

Equal steps in the independent variable produce equal multiplicative ratios in the dependent variable. The characteristic scale τ sets the interval over which the quantity changes by a factor of 1/e ≈ 0.367879. All physical exponents are dimensionless ratios of the independent variable to this characteristic scale.

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