ln(Probability) Carrying the Information of a Conserved Quantity and Eigenvalue Equations

 In information theory, ln(Probability) is called information. We argue that in various physical problems this information is linked with a conserved quantity and a variable which changes. For example, in the Maxwell-Boltzmann case the conserved quantity is 1/Temperature and the variable e+V(x) = energy.  For a quantum free particle, the conserved quantity is momentum and the variable x. “e” may change subject to the conserved quantity 1/T and x may change subject to the conserved quantity p. Given that probability may be unnormalized, changes may be expressed in relative terms i.e. in terms of a derivative of ln(P) i.e. information. We argue that the derivative of ln(P) with respect to the variable which changes yields the value of the conserved quantity in special cases for which the conserved quantity is the only information governing changes in the probability distribution. Thus one has an eigenvalue equation:  d/dvariable ln(P) = conserved quantity. This implies that ln(P) = variable * conserved quantity. This may lead to two different kinds of conservation equations. In the MB case, e is the variable and 1/T the conserved quantity, but ln(P(e1)) + ln(P(e2)) = 1/T (e1+e2) = ln(P(e1+e2)). An eigenvalue equation allows for this conservation. In the quantum free particle case:   ln(P(k1,x)) + ln(P(k2,x)) where P=wavefunction = exp(ipx) (i.e. the square-root type probability is used), one has k1+k2=k3 with x the variable canceling out. (In the MB case, it was 1/T the conserved quantity canceling). Again, the eigenvalue equation allows for this conservation, this time of the conserved quantity. 

   We note that the eigenvalue equation is a special case and not only probability distributions are associated with it. For example, a power law distribution’s changes in its variable do not depend only on the conserved quantity 1/T in P(e/T)=power law.