Notes on Probabilty and Statistical Mechanics

Barry Loewer

Francesco R. Ruggeri

In physics, there are deterministic theories such as classical mechanics and statistical theories such as classical statistical mechanics and quantum mechanics. Many interpretations are applied to statistical theories and there is sometimes debate as to the application of such theories in various physical scenarios. In a previous note, we considered cycling due to exp(iEt) in quantum mechanics and tried to argue that such cycling may also appear in statistical mechanics. We made use of the example of the ground state of the quantum oscillator which has a quantum mechanical density of the same form as the classical statistical mechanical result. In this note, we wish to consider the idea of cycling in statistical mechanics using some simple examples. In addition, we wish to also examine the notion of losing information to pass from a deterministic theory to a statistical one. Finally, we wish to consider some possible aspects of the classical statistical density in a potential. It seems that even in classical statistical mechanics, the picture presented by the ‘averaging’ may actually be quite different from the physical scenario which may contain deterministic behavior which is lost in the calculation of statistical results.

Gérard G. Emch

Francesco R. Ruggeri

A 2x2 matrix equation, linear in E and similar to the Dirac equation, was proposed in a series of earlier notes for the energy momentum equation. This allowed one to obtain an eigenvector of 1/sqrt(2) (sqrt(1+v), sqrt(1-v)) with c=1. It is believed that (1+v)/2 and (1-v)/2 are probabilities of this quantum mechanical type solution. It was also found that f1=(1+v)/2 and f2= (1-v)/2 could be obtained from maximizing entropy subject to (f1-f2)=v. Thus, there seems to be an equivalency between quantum mechanical statistics and regular statistical mechanics for this model. In addition, the model reproduces Schrodinger zitterbewegung. In this note, a 2x2 matrix model is proposed for a system of particles in a box with allowable energies of e1 and e2 . This system can be described by statistical mechanics, but the purpose of the note is to see if it can also be described in quantum mechanical terms with eigenvector (sqrt(f1), sqrt(f2)) with fi being the distribution of ei.. Doing so seems to give zitterbewegung results for a matrix |e1 0| |0 e2| suggesting that there may be periodic time dependent fluctuations in a statistical mechanical gas. It is possible that this zitterbewegung may be associated with a kind of oscillation in the hot gas. The expectation of the above matrix is E/N where E is the total energy of the gas and N the number of particles.

Marco Baldovin, Giacomo Gradenigo, Angelo Vulpiani +1 more

Francesco R. Ruggeri

A 2x2 matrix model of Einstein´s energy momentum equation appears to have links to statistical mechanics. First, it seems to suggest using dp (change in momentum) as a constant when counting energy levels in statistical mechanical calculations of the partition function. It also has further links to the de Broglie wavelength which is used to justify the use of equally spaced dp for energy state counting. It also appears that a 2x2 matrix model is consistent with Knuth´s statistical model. The two component positive energy eigenvector represents square roots of forward and backward motion probabilities. These probabilities maximize entropy. Thus, it appears that the 2x2 model, which uses quantum mechanical formalism, is also equivalent to a statistical mechanical model. In other words, at least in this model, there appears to be a link between statistical mechanics and quantum mechanics . It should be noted, however, that in this note very little has been shown about this link other than the equivalency of the energy eigenvector with maximum entropy. It is also suggested that zitterbewegung could lead to statistical fluctuations and that they may account for issues in statistical mechanics when interparticle separations reach the size of the de Broglie wavelength.