The 2x2 Dirac-like matrix linearization of the energy-momentum equation can be interpreted as a probability scenario. It was argued in earlier notes that this was in fact a quantum mechanical probabilistic approach as it leads to the Schrodinger zitterbewegung . We have also seen that the probabilities .5(1+v) and .5(1-v) associated with the energy eigenvector .5( sqrt(1+v), sqrt(1-v)) can also be obtained by finding a stationary solution of the entropy given by -k f1 ln(f1) -k f2 ln(f2) - w (f1-f2) where the constraint f1-f2=v has been used. The unnormalized results are f1=sqrt(1+v)/sqrt(1-v) and f2=1/f1, i.e. the two Doppler shift scenarios. Kaniadakis has developed a relativistic statistical model in which the Maxwell-Boltzmann distribution only holds in the nonrelativistic case. He instead uses for entropy: S= 1/2k Integral d3p [f(1-k)/(1-k) - f(1+k)/(1+k)] One can see that f1 and f2=1/f1 together with the constraint f1-f2 =v are also stationary solutions of this new relativistic entropy. In fact, we suggest that one could use the f1 and f2=1/f1 result to postulate Kaniadakis entropy expression such that it becomes the natural logarithm in the k->0 (nonrelativistic) limit. Kaniadakis’ new expression for the natural logarithm is the sum of the mathematical limits -ln(a)=lim x->0 [a-x -1]/x and ln(a) = lim x->0 [ax -1]/x. F1 and f2 from the 2x2 matrix model should already be relativistic expressions and so are not “nonrelativistic limits”. It turns out they provide stationary solutions to both Kaniadakis’ relativistic entropy expression and the normal S= -k f ln(f) expression which leads to the Maxwell-Boltzmann result. The experimental motivation for a new relativistic distribution is that relativistic data apparently does not match the tail the Maxwell-Boltzmann distribution.
The 2x2 Dirac-like matrix linearization of the energy-momentum equation can be interpreted as a probability scenario. It was argued in earlier notes that this was in fact a quantum mechanical probabilistic approach as it leads to the Schrodinger zitterbewegung . We have also seen that the probabilities .5(1+v) and .5(1-v) associated with the energy eigenvector .5( sqrt(1+v), sqrt(1-v)) can also be obtained by finding a stationary solution of the entropy given by -k f1 ln(f1) -k f2 ln(f2) - w (f1-f2) where the constraint f1-f2=v has been used. The unnormalized results are f1=sqrt(1+v)/sqrt(1-v) and f2=1/f1, i.e. the two Doppler shift scenarios. Kaniadakis has developed a relativistic statistical model in which the Maxwell-Boltzmann distribution only holds in the nonrelativistic case. He instead uses for entropy: S= 1/2k Integral d3p [f(1-k)/(1-k) - f(1+k)/(1+k)] One can see that f1 and f2=1/f1 together with the constraint f1-f2 =v are also stationary solutions of this new relativistic entropy. In fact, we suggest that one could use the f1 and f2=1/f1 result to postulate Kaniadakis entropy expression such that it becomes the natural logarithm in the k->0 (nonrelativistic) limit. Kaniadakis’ new expression for the natural logarithm is the sum of the mathematical limits -ln(a)=lim x->0 [a-x -1]/x and ln(a) = lim x->0 [ax -1]/x. F1 and f2 from the 2x2 matrix model should already be relativistic expressions and so are not “nonrelativistic limits”. It turns out they provide stationary solutions to both Kaniadakis’ relativistic entropy expression and the normal S= -k f ln(f) expression which leads to the Maxwell-Boltzmann result. The experimental motivation for a new relativistic distribution is that relativistic data apparently does not match the tail the Maxwell-Boltzmann distribution.
In a series of notes it was shown that a 2x2 matrix energy eigenvalue equation is naturally contained in Einstein´s 1905 energy momentum equation E2 = p2 + m2 . This equation, which appears to be related to the Dirac equation, but contains no spatial features, is linear in energy and describes bouncing of a particle back and forth with an overall average motion of v to the right. It appears to be quantum mechanical in nature as the energy matrix acts as an evolution operator, expectation values of operators are taken and one has an energy eigenvector representing the square roots of probability. Knuth [18 ] has used a model which yields directly probabilities for a particle bouncing back and forth with an average v to the right. It was shown that these probabilities, the squares of the terms of the energy eigenvector, can be obtained by maximizing Shannon´s entropy subject to the constraint the f1-f2= v, where f1 is the probability to move forward and f2, that to move backward. In this note, we try to find a link between the 2x2 matrix equation and the Schrodinger (or Dirac) equation in a region with a potential V(r). The Schrodinger equation, unlike the matrix equation, is spatial. Next, we try to see if there are any features of zitterbewegung that might apply in this case. In the 2x2 matrix model, Schrodinger´s zitterbewegung can be derived, so it seems reasonable to ask whether there might be zitterbewegung in the more complicated case of a bound state in a potential. Finally, we raise the issue of a possible link between the Schrodinger equation and statistical mechanics. In the 2x2 model, the forward and backward postulated motion of a particle was connected to maximum entropy. It would be interesting if a similar idea could be applied to bound states. We do not present any calculations related to such entropy maximization, but mention work in the literature in which Tsallis entropy is maximized subject to constraints to obtain the ground state wavefunction without solving the single particle temperature=0 Schrodinger equation.
Previously a 2x2 matrix equation, linear in energy, was obtained from Einstein’s energy momentum equation E2=p2+m2 which reproduced Schrodinger zitterbewegung [1]. A similar equation was applied to the case of a gas with two allowable energy levels and to a gas in a potential field, the goal beng to obtain zitterbewegung results in statistical mechanics. The starting point for applying the 2x2 matrix equation to statistical mechanics was the fact that the 2x2 matrix approach yields an eigenvector which is the square root of two probabilities which sum to one. In this note, we wish to see if we can extend ideas further. First, we wish to to see if there are stronger reasons for applying the 2x2 matrix approach to statistical mechanics, and second, we wish to find further analogies between the approach and statistical mechanics. We consider relationships between physical constraint equations and the 2x2 matrix approach., We also consider functions analogous to the rest mass and energy in the statistical case, where the analogous energy also represents the frequency in zitterbewegung.
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.
In a number of notes , a linear 2x2 matrix Dirac type equation was examined as a solution of Einstein’s 1905 energy momentum equation E2=p2+m2 (1). This model was able to reproduce Schrodinger zitterbewegung.To do this, however, one had to identify the 2x2 energy matrix H as being part of the evolution operator exp(-iHt). It was seen that the identity matrix together with H could transform any two vector into the eigenvector of H, which is taken as the stable solution. This idea of an evolution operator thus seems to be important. In addition, the model led to a 2-eigenvector which was interpreted as the square root of two components of probability, one in which a particle moves forward the other in which it moves backward Overall, the particle moves forward with a velocity of v along the x axis. Knuth has also proposed such a model. It was found that the probabilities which came from this quantum mechanical type matrix solution matched those obtained by maximizing statistical entropy. In a note , the 2x2 matrix model was applied to a system of classical particles which could have energy e1 or e2. It was argued that the 2x2 model predicted zitterbewegung for this statistical system. The object of this note is to explore whether the 2x2 model can be used to predict zitterbewegung in a statistical system with a potential.
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.
Classical statistical equilibrium appears to be based on a no-flow situation with a balance of pressure (including -dV/dx in the case of a potential present). It may be noted, however, that like quantum mechanics, a single particle has different probabilities of being at one x versus another. Furthermore, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions may be obtained using reaction balance which is linked to conservation of energy i.e. g(f(ei))g(f(ej)) = g(f(el))g(f(ek)) where ei+ej=el+ei=e. Taking ln of the former expression and linking to the latter yields f(ei), the probability distribution. g=1 for the MB case, f(e)/(1-f(ei)) for the FD and f(e)/(1+f(ei)) for the BE. In classical mechanics, one has the conservation equation: p(x)*p(x)/2m + V(x) = E. In quantum bound systems one has: KE(ave)(x) + V(x) = E which appears to be very similar, although it is based on ensemble averages and there is a momentum distribution P(p/x) at each x. A question which arises is: What is the “conservation of energy” equation for a particular p of the P(p/x) distribution? In this note, we argue the “potential” term associated with a component p*p/2m is related to the part of the potential which contributes to the “creation” of p. In particular, we consider V(x)= Sum over k V(k) exp(ikx) where each k term renders a kick of momentum k. One may create a momentum p using k and p-k for each value of k. We further argue that: p*p/2m P(p/x) + contributing portion of V(x) = E P(p/x) ((1)) for all p. Thus, in quantum equilibrium, which involves a flow, there is a “constant E” type of equation i.e. ((1)), different in form but similar in idea, to that of classical statistical mechanical equilibrium. In previous notes, we referred to this as cycling, but in this note we wish to consider it in terms of dynamics. We further note that in both classical statistical mechanics and quantum mechanics, the energy-probability p equation makes use of the overall average energy which is bT for the classical case and E for the quantum. (b is constant)