By Bopp F.
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Extra resources for Thermostatistik
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An opposite collision increases both fdω and F1 dω1 by one. In order to find the total increase of fdω resulting from all collisions of m-molecules with m1 -molecules, during time dt, one must first express in terms of ξ, η, ζ, ξ1 , η1 , ζ1 , ϑ, and ∊. Since We leave the letters f′, , we have: and dλ in the equation, remembering however that one must consider their arguments to be functions of ξ, η, ζ, ξ1 , η1 , ζ1 , ϑ, ∊ and also dλ through the differentials of the latter angles. As is well known, one can show that dλ = sin ϑdϑd∊ (cf.
A surface element of area σ 2 dλ is thereby cut out from the surface of each of these spheres. Since all the spheres are rigidly attached to the corresponding molecules, all these surface elements move a distance gdt relative to the specified m1 -molecule. A collision of the specified kind occurs whenever one of these surface elements touches the center of a specified m1 -molecule, which is of course possible only if the angle ϑ between the directions of the lines C1 C and OK is acute. Each of these surface elements traverses by its relative motion toward the m1 -molecule an oblique cylinder of base σ 2 dλ and height g cos ϑdt.
In fact, when Maxwell’s state has arisen from some other state, the exact recurrence of that other state will take place after a sufficiently long time (cf. the second half of §6). Thus one can have a state arbitrarily close to the Max-wellian state which finally is transformed into a completely different one. It is not a defect that the minimum theorem is tied to the assumption of disorder, rather it is a merit that this theorem has clarified our ideas so that one recognizes the necessity of this assumption.