Speculation on the Max well-Boltzmann exp(-ei/T) From the View of Conservation of Momentum Part 3

Note: Oct. 4, 2025  One may see that the binomial factor form:  p(k)=n! / (k! (n-k)!) already leads to dp(i)/d k  (with k-->k+1)  = (n-2k-1)/ k p(k)  where  dk = 2 and so (n-2k-1) is proportional to    .5 dv v   where  v = (n-2k) dv.  The 1 in (n-2k-1) may be dropped because n and k are very large numbers. Thus, the idea of kinetic energy and probability as exp(-kinetic) energy are hinted at already in the statisical factorial function.

In Part 2, we noted that one may consider, in the nonrelativistic case, the creation of v through k dv units and (n-k) (-dv) units, where p =mov. We then noted that if the probability for a dv and (-dv) is the same, i.e. .5, then p(k) = n!/ (k! (n-k)!) .5 power n. One may  convert a binomial to a Gaussian in the n→ infinite limit, yielding p(v) = C exp(- C1 vv) or C exp(- C1 v dot v) in three dimensions. We noted that in an equilibrium system, the probability of v1 colliding with v2 to yield v3 and v4 should be time reversal invariant, suggesting that in a simple case, exp(-C1 v1v1) exp(-C1 v2v2) = exp(-C1 v3v3) exp(-C1 v4v4). This suggests that in equilibrium one has a conservation law v1v2 + v2v2 = v3v3 + v4v4. If this, however, is a real conservation condition, it holds whether or not there is equilibrium (as long as there are no physical processes which cause this conserved quantity to change, i.e. sound production etc.)  One may also note that if one considers special relativity, one may see the term .5mvv emerge from mocc/sqrt(1-vv/cc) in the v<<c limit. 

   We note that in the above scheme, the idea was to consider dv and (-dv) changes, which in the nonrelativistic case is analogous to dp changes. Thus, regardless of the velocity of a particle, one has a physical system which yields changes of dv or -dv. Now, changes in velocity are due to a push or pull, called a force by Newton. If one imagines that force is microscopically equivalent to a certain number of mo dv/N (N large) deliveries per sec, then F dt = dv m. This scheme then yields, upon multiplying by dx and dividing the LHS by dx,  Integral Fdx = .5 mvv  if the initial v is 0. 

  In other words, one sees how .5mvv appears in a nonrelativistic scheme by trying to interpret a force as the delivery of dp/N per sec. This delivery of momentum is consistent with the scheme of Part 2 in which a v is created by k dv units and (n-k) (-dv) units. In particular, one has the idea of action-reaction because each dv delivered by F may be linked to a -dv and -F. Thus, one may create v and -v and both should have the same probability. 

  Thus, we argue that Newtonian mechanics is implicitly contained in the statistical argument of Part 2. Speed is not a quantity, like the number of particles in a gas, but rather is changed dynamically through mechanics. . The Gaussian resulting from the binomial factor is associated with a conserved quantity C1 vv, but this in turn is linked to a physical scheme which delivers mo dv hits. The probability then becomes linked to a conserved quantity and this quantity is associated with work done creating v, i.e. .5movv (nonrelativistic case). In other words, stating that one may create v from k (dv)s and (n-k) (-dv)s really means that each dv or -dv process is linked with work (and a change in kinetic energy and this is directly linked to probability in an equilibrium situation. Thus, one should be able to obtain the C1 v v term (C1 v dot v in 3D) from “work” considerations which are linked to the notion of adding dv or mo dv units, which seems to be the case.