8 V O L . 3 , D O C . 1 0 a B O L T Z M A N N S P R I N C I P L E If there are many particles present in a single liquid, then the right-hand side of the equation indicates the distribution density of the particles as a function of depth. Perrin tested this relation and found it confirmed.[8] From this relation the law of Brownian motion can very easily be deduced. For, from it, it immediately follows that the mean height of a particle above the vessel bottom is equal to . Now, however, because of its greater density the particle drops downward, according to Stokes’s law, by in the time if signifies the liquid’s viscosity coefficient and P, the radius of the (spherically shaped) particle. But in the same time , as a consequence of the irre- gularity of the molecular thermal process, [it] is also shifted a distance upwards or downwards, where positive and negative values for appear equally frequently so . A particle that, before time has elapsed, is located at height z, is, after has elapsed, at the height . As the distribution law of all the particles should not depend on time, the mean value of must be equal to , therefore, , or for sufficiently small is negligible and . This is the familiar law of Brownian motion, which has likewise been confirmed by experience.[9] The just-described example of a particle suspended in a liquid offers a fitting depiction of Boltzmann’s conception of irreversible processes. For, if we imagine a particle suspended in such a tall vessel, and that it is so much heavier than the dis- placed liquid that the expression for probability W is very small, even at a height z just barely above the bottom of the vessel when compared to the value for , then very rarely will the particle rise much from the bottom, once it has z ze N RT -(μ μ0)gz –------ zd e N RT -(μ μ0)gz –------ zd ----------------------------------------- RT N ------ - 1 g( μ μ0) ----------------------- = D g( μ μ ) 6πηP -----------------------τ0 = τ η [p. 9] τ Δ Δ Δ is 0 = τ τ z D Δ + z′ = z2 z′2 z D Δ)2 + ( z2 = τ, D2 0 = = Δ2 2zD RT N ------ - 1 3πηP -------------- τ⋅ = = W0 z 0=
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