Solving for a Probability Distribution P(ei) from a Simpler Constraint for a Subset of P(ei) Derived from a Constraint Over All P(ei)

Solving for a Probability Distribution P(ei) from a Simpler Constraint for a Subset of P(ei) Derived from a Constraint Over All P(ei) Francesco R. Ruggeri Hanwell, N.B. Dec. 26, 2025 In (1), it is suggested that there are two formal mathematical approaches for solving for a probability distribution. The first is the uniform probability approach, called the principle of indifference in (1) which apparently goes back to Laplace. The second is maximization of Shannon’s entropy subject to constraints (Jaynes), if constraints exist. In this case, it is assumed that in general the constraint applies to the entire probability distribution p(ei). We argue here that for a certain kind of constraint, i.e. one which implies the conservation of the variable ei upon which p(ei) depends, one may consider a subset of p(ei) which are then linked to a uniform probability distribution through a product p(ei)p(ej)=p(ei+ej) for an unnormalized p(ei). This then represents another way in which to solve for a distribution. In other words, one is not forced to use maximization of entropy subject to constraint in such a case. Two Methods for Solving for a Probability Distribution In (1), two formal math approaches are given for solving for a probability distribution p(ei). The first is called the principle of indifference and is associated with a uniform distribution. This applies to a coin or die toss as there is no constraint which provides any bias. If a constraint exists, assumed to involve all p(ei)’s, then there is bias and one must use a different approach. In (1), the maximization of entropy subject to constraints (Jaynes) is then used, i.e. one maximizes: - Sum over i p(ei) ln(p(ei)) subject to a constraint ((1)) In certain cases, the constraint is linked to a conserved quantity, in particular the argument of p(ei), ei. We argue that this conserved quantity is now associated with a physical interaction problem. For example, one may have (as in the Maxwell-Boltzmann distribution case), the constraint: Sum over i p(ei) ei = Etotal/N where N is the number of particles ((2)) In such a case, ((2)) implies conservation of energy, but this requires the notion of a physical interaction and more than one p(ei). The simplest scenario is 2-body elastic scattering. The point is that the constraint ((2)) which applies to all p(ei)’s leads to a simpler situation (uniform distribution) when applied to products of p(ei)s, i.e. p(ei)p(ej) with p(ei) being a subset of the entire set of p(ei)’s. In particular, if considers a specific energy E1, then 0 <= ei <= E1 and so only p(0) to p(E1)’s are considered ((3)) ((3)) is a subset of all p(ei)’s. Secondly, one may apply the notion of a uniform distribution to: p(ei)p(ej) for ei+ej = E1. ((4)) As a result, even though a constraint exists in the problem, namely Sum over i ei p(ei) = Etotal/N, due to conservation of energy ei implied by the constraint, one may reduce the problem to a subset of p(ei)s for which p(ei)p(ej) for ei+ej=E1 represents a uniform distribution. This means that for p(ei) unnormalized: p(ei) p(ej) = p(ei+ej) for ei+ej=E1 or p(ei) = C exp(-ei/T) ((5)) for a p(ei) which falls with rising ei. Thus, one does not always have to consider a math approach to finding p(ei) for all p(ei)s together with all p(ei)’s considered in a constraint and then maximizing entropy subject to the constraint. A constraint which implies conservation of ei leads to simplification which applies to a subset of p(ei)’s, but still solves for the functional form of p(ei). Conclusion In conclusion, in (1) it is noted that there are two general mathematical approaches to solving for a probability distribution. The first is the principle of indifference which means that there is no bias towards any p(ei), i.e no constraint. This is the case for a coin or die toss and p(ei) = constant. The second case involves maximizing Shannon’s entropy -Sum over i p(ei) ln(p(ei)) subject to a constraint(s) which involves all p(ei)s in general. We suggest that if the constraint implies a conservation of ei, the variable of p(ei), then one need not consider maximization subject to a global constraint, but may consider a subset of p(ei)s i.e. 0<= ei <= E1. In such a case, p(ei)p(ej) = p(ei+ej) for ei+ej=E1. Thus, one has a uniform distribution within this subset of p(ei)’s, but that is enough for one to solve for p(ei) = C exp(-ei/T) to have a distribution which falls with rising ei. (Note, one may then change E1 to different values, so exp(-ei/T) holds in general.) References 1. Conrad, K. Probability Distributions and Maximum Entropy https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.pdf