By Ludwig Boltzmann
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Extra resources for Lectures on Gas Theory (Dover Books on Physics)
Example text
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.