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 Incoming links: Josiah Willard Gibbs

Boltzmann distribution 2026-09-28

At thermal equilibrium, the Boltzmann distribution assigns a state of energy

E_{i}

P_{i} = Z^{- 1} e^{- β E_{i}}

, where

β = (k_{B} T)^{- 1}

and the partition function

Z

normalizes the probabilities.

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Gibbs phenomenon 2026-10-14

The Gibbs phenomenon is the persistent overshoot near a jump discontinuity in ordinary truncated Fourier series. For this square wave, retain

M

nonzero odd modes:

Q_{M} (x) = \frac{4}{π} \sum_{k = 0}^{M - 1} \frac{s i n (( 2 k + 1 ) x )}{2 k + 1} .

M

grows, the first peak approaches the jump while its height tends to approximately

1.17897974

. This is an excess of about

0.17898

above the upper level

1

, or

8.949%

of the jump size

2

. The narrower region does not imply a vanishing maximum error. At the jump itself every

Q_{M} (0) = 0

. These observations are compatible with pointwise convergence and

L^{2}

convergence but exclude uniform convergence to the discontinuous square wave. ^[ref].

Figure 1.
More Fourier harmonics narrow a transition while leaving a persistent overshoot
. Left: the constructed square wave and sums of $3$ , $10$ and $50$ nonzero odd Fourier harmonics. Right: the same sums near the jump at zero, with each first positive peak marked. The highest frequency retained is $2 M - 1$ , so a count of $M$ nonzero terms differs from a cutoff at frequency $M$ . The limiting peak is approximately $1.17898$ , while the level above the jump is $1$ . Details.

For this particular sum, its derivative is

Q_{M}^{'} (x) = (2/ π) sin (2 M x) / sin x

away from multiples of

π

, so the first positive peak is exactly at

x = π / (2 M)

. There,

Q_{M} (\frac{π}{2 M}) = \frac{2}{M} \sum_{k = 0}^{M - 1} \frac{s i n ( π u _{k} )}{π u _{k}}, u_{k} = \frac{k + 1/2}{M} .

This is a midpoint Riemann sum tending to

2 Si (π) / π

, where

Si

is the sine integral. It gives a short explanation of the nonzero limiting excess.

The phenomenon matters whenever a sharp boundary is represented by a limited set of Fourier harmonics: a visible ripple can be a property of the approximation rather than an oscillation in the original profile. Josiah Willard Gibbs reported the phenomenon in his 1899 letter “Fourier's Series” in Nature, volume 59, page 606. This is a documented historical report, without a claim of first discovery. ^[ref].

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