Annus MirabilisEinstein in 1905
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brownian motion

The equation explained

δE=−∫0lKνδxdx\delta E = -\int_0^l K \nu \delta x d x
In wordsThe change in the energy is minus the sum, over the column from zero to its far end, of the force on one particle times the number of particles in unit volume times how far they are shifted.OverviewShifting the particles along the force takes energy out of the system, and the amount is the force times the number of particles moved times how far each moves, added up along the column.Full explanationThe liquid is taken to have cross-section 1 perpendicular to the XX-axis and to be bounded by the planes x=0x = 0 and x=lx = l. A slice at xx of thickness dxd x then has volume dxd x and holds νdx\nu d x particles. Each feels the force KK, and shifting the slice by δx\delta x lets that force do work KνδxdxK \nu \delta x d x on it. Work done by the force lowers the energy of the system, which is the minus sign, and the whole change is the sum of the slices, the integral from 0 to ll. The force may vary from place to place, and so may the number density and the shift, so all three stay inside the integral.Every stepKνdxK \nu d xThe force on the particles of one slice: νdx\nu d x particles in the slice, since the cross-section is 1, each feeling the force KK.KνδxdxK \nu \delta x d xThe work that force does when the slice is shifted by δx\delta x.δE=−∫0lKνδxdx\delta E = - \int_0^l K \nu \delta x d xAdded over the column and made negative, because work done by the force on the substance comes out of the system's energy. The printed form.Also

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