By Meghnad Saha

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**Additional info for A treatise on heat: Including kinetic theory of gasses [sic], thermodynamics and recent advances in statistical thermodynamics **

**Example text**

Proof: For each nonnegative integer r define the sets Er (k, n) = {x : − 1 ln m(xnk ) ∈ [r, r + 1)} n and hence if x ∈ Er (k, n) then r≤− 1 ln m(xnk ) < r + 1 n or e−nr ≥ m(xnk ) > e−n(r+1) . Thus for any r (− Er (k,n) 1 ln m(Xkn )) dm < (r + 1)m(Er (k, n)) n e−nr m(xnk ) ≤ (r + 1) = (r + 1) xn k xn ∈Er (k,n) k = (r + 1)e−nr ||A||n ≤ (r + 1)e−nr , where the final step follows since there are at most ||A||n possible n-tuples corresponding to thin cylinders in Er (k, n) and by construction each has probability less than e−nr .

By “the associated probability space” of a random process [A, m, X] we shall mean the sequence probability space (AT , BA T , m). It will often be convenient to consider the random process as a directly given random process, that is, to view Xn as the coordinate functions Πn on the sequence space AT rather than as being defined on some other abstract space. This will not always be the case, however, as often processes will be formed by coding or communicating other random processes. Context should render such bookkeeping details clear.

Consider next the dynamical system (AT , BA T , P, T ) and the random process formed by combining the dynamical system with the zero time sampling function Π0 (we assume that 0 is a member of T ). If we define Yn (x) = Π0 (T n x) for x = xT ∈ AT , or, in abbreviated form, Yn = Π0 T n , then the random process {Yn }n∈T is equivalent to the processes developed above. Thus we have developed three different, but equivalent, means of producing the same random process. Each will be seen to have its uses.

### A treatise on heat: Including kinetic theory of gasses [sic], thermodynamics and recent advances in statistical thermodynamics by Meghnad Saha

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