Speculation on Newton's Second Law In Terms of Information

Historically Newtonian mechanics appeared before the statistical mechanical treatment of a Maxwell-Boltzmann (MB) gas. In this note, however,  we initially investigate a MB gas in terms of information which we define as -ln(density(x)). In equilibrium there are no macroscopic changes in time so there is only one variable namely x in one dimension. The density may be changed so it is variable while a potential V(x) is not. Thus we argue that there should be a fixed piece of information based on x in the system i.e. V(x). We then argue (as in (1)) that equilibrium involves minimizing the information so we suggest:   -ln(density(x)) = C V(x). This ultimately leads to the form d(x) = C1 exp(-V(x)/T).

  Next we focus on a Newtonian system, but unlike the MB gas two variables x and t are needed to describe the macroscopic changes for the particle even though x suffices to describe V(x). {In particular one may send in a particle with an initial speed at xo and it will move through x,t in a certain manner. Sending another particle with the same initial speed at xo at a later time leads to the same motion. Thus we argue there is a given V(x) function in the system. } We also suggest there should exist a density d(x) and an information given by -ln(d(x)). One may see from (2) that a Newtonian system may be described by a density(x) even though it is usually only described by x(t), velocity, acceleration etc. The density is linked to a variable quantity of “information” in the system. Given the presence of V(x) we divide the x axis into equally spaced dx units. Thus d/dx the translation generator applies to V(x). For the moving object, however, one has a corresponding dt(x) for each equal dx and the dt’s are usually not the same. In other words the ratio dx/dt = function(x), but it is dt which changes.  We place dt in the denominator so that it mimics the time translational generator d/dt.

     One cannot simply equate function(x) and C V(x) because function(x) does not change as d/dx, rather it changes because dt values change. Thus we suggest using d/dt to express its changes. As a result we propose:   d/dt (dx/dt) = C dV/dx ((1)) as both sides of the equation must be vectors.  Now d/dt= dx/dt d/dx so both sides  of ((1)) may be expressed as functions of x. One then obtains function(x) d/dx function(x) = C dV/dx where function(x) = dx/dt. We call this v(x). v(x) is adjustable so we equate it with information i.e.  -ln(d(x)) where d(x) is density as there is no reason to have two different functions of x. Thus d(x)=C/v(x). In other words density(x) is proportional to the “information” dt at each x. In (2) a density is ascribed to a Newtonian system using dt as the definition of density. Here we wish to link density to information= -ln(d(x)).

   As a result one finds that ((1)) involves information squared unlike the MB gas which involves  information -ln(d(x)) to the first power. As a result:    d/dx ( v(x)v(x))  = C dV/dx which may be integrated to yield Newton’s conservation of energy:   .5v(x)v(x)+ C V(x) = constant. In other words we suggest it may be possible to obtain Newton’s second law using ideas of information.

There is further loss of information as the constant applies to all x points so the the information from .5v(x)v(x) combines with CV(x) to eliminate x information. (x information, however, is present in each term i.e. .5v(x)v(x) and CV(x)).