Statistical States and Their Relationship to Constraints Part II

 In Part I, we argue that in statistics, probability is often taken to be proportional to the number of states consistent with some constraints. For example, for the case of a Maxwell-Boltzmann gas of n particles with total energy E, the constraint is on energy and so one considers various energy configurations. In other words, energy is physically being distributed because this is what changes in a collision. One may argue that e=pp/2m, so momentum is also changing, but the equilibrium problem is associated with distributing energy, not momentum. The equilibrium process gives rise to a probability function in terms of the constraint variable, in this case single particle energy, i.e. e=pp/2m. The fact that one may write energy in terms of the magnitude of momentum does not mean that one is dealing with the change in the momentum vector. Such a change is different because it includes orientation changes with no magnitude change. In other words, the constraint dictates how changes to a single particle variable occur (e.g. single particle energy) and this determines the states available.

     We stress this point, because it is possible to write the constraint for the Maxwell-Boltzmann gas as:  Sum over i=1,n  p(i)p(i) = 2mE  where one has n particles moving in 1-dimension, so p(i) is the momentum of the ith particle. Mathematically this is an equation of an n-sphere. If one sets p(n) to a particular value:  Sum over i=1,n-1 p(i)p(i) = 2mE-p(n)p(n). This is the equation of an n-1 sphere. In (1), it is argued that one may think of {p(i)} as constituting an n-1 vector. We argue against this because there is no physical constraint of a vector in this problem. This is a mathematically created constraint. In collisions, single particle energy changes and this creates the equilibrium. In other words, one considers 2-body elastic scattering and ignores momentum completely in obtaining exp(-e/T). Thus, on physical grounds, we argue that one does not have a physical vector and should not use a constraint linked with such a vector. Arguing that P(E-p(n)p(n)) is proportional to the surface area of an n-1 sphere makes use of a constraint which is not part of the physical distribution process of the problem. It is a stronger constraint, so to speak, and leads to fewer allowable states, i.e. (E-e) power ( (n-3)/2) versus (E-e) power (n-2).One does not physically have a vector constrained to a surface. 

  One may, however, use the procedure of (1) to describe the probability of a vector component for an n-vector. This is a different problem from the Maxwell-Boltzmann one. The result for n=2 for the general result of (1) yields the probability of a vector component (px,py) for a fixed magnitude as given in (3).. A physical vector may be associated with motion constrained to a surface.

   Thus, we argue that one must be careful about interpreting constraints in a problem and determining states. Given a constraint in one variable, it may be written in terms of another, e.g. an energy constraint may be written in terms of momentum squared of a number of particles, but to argue that the momentum components of different 1-dimensionally moving particles constitute a vector is introducing a constraint which does not exist in the problem.