Annus MirabilisEinstein in 1905
Reading preferences

Reading preferences

Keep every explanation and worked example on the page. Nothing moves or plays by itself, and the larger instruments load only when you ask.

Line length
Type size
Contrast
Paragraph spacing

light quanta

The equation explained

δ∫0∞φ(ϱ,ν)dν=0,\delta \int_0^\infty \varphi(\varrho, \nu) d \nu = 0,
In wordsUnder any small alteration of the spectrum, the change in the sum over every frequency of the entropy per unit volume and unit frequency, taken at the energy density there, is zero.OverviewFor black radiation the entropy is a maximum at the given energy: no small change in how the energy is spread over the frequencies raises it, so its first-order change is zero.Full explanationSection 3 defines black radiation by a maximum principle: ϱ\varrho is the function of ν\nu for which the entropy is a maximum at a given energy. The paper writes this as a variation, marked δ\delta: alter the density at each frequency by a small amount δϱ\delta \varrho, and at a maximum the entropy does not change to first order. The volume is fixed, so the factor vv is dropped and the condition is put on the integral itself. It is to hold for every alteration that leaves the energy unchanged, the condition of the next display.Every step∫0∞φ(ϱ,ν)dν\int_0^\infty \varphi(\varrho, \nu) d \nuThe entropy per unit volume: the integral of the preceding display, with the volume held fixed.φ(ϱ+δϱ,ν)−φ(ϱ,ν)≈∂φ∂ϱδϱ\varphi(\varrho + \delta \varrho, \nu) - \varphi(\varrho, \nu) \approx \frac{\partial \varphi}{\partial \varrho} \delta \varrhoChange the density at each frequency by a small amount δϱ\delta \varrho. To first order the entropy density changes by its rate of change with the density times that amount.δ∫0∞φ(ϱ,ν)dν=0\delta \int_0^\infty \varphi(\varrho, \nu) d \nu = 0At a maximum no such small change raises or lowers the entropy to first order: the printed condition, to hold whenever the energy is unchanged (the next display).Also

Read it in the paper