If one tosses a die or coin, different outcomes occur with a probability p(i) where i is the outcome. This probability may be measured by performing trial runs. For N trial runs, where N is very large, one might expect Np(outcome i) results of an outcome i. In other words, p(outcome i) is both the probability and describes the trial run. Even in the Maxwell-Boltzmann case, one has a probability p(ei) where ei is kinetic energy (no V(x)) and reaction balance p(e1)p(e2) = p(e3)p(e4) represents trial runs. Thus the trial run is again characterized by p(ei) as ei’s contribution. It is, however, possible to have coupling of probabilities in certain physical situations even though p(outcome i) still exists. In such a case, the number of outcome i results for N large trials is not N p(outcome i). In other words, p(i) does not fully describe the trial run. There may be other physical rules in place. If elastic scattering of particles is the way trial runs occur, e1+e2 = e3 + e4 represents the salient information in the problem and even if there is nonindependence of p(outcome i) in the scattering, one should at least obtain a function Q(p(e1)) such that Q(p(e1)) = -e1/T. Q(p(ei)), however, need not be p(ei). We consider the physical rule given by Pauli stating that only one particle may fit in an energy level (no spin). Thus there is either 0 or 1 particle in the level. This should inhibit scattering, so the trial run is no longer simply described by p(ei). Rather the probability to scatter into level ej is (1-n(ej)) and so one has: n(e1)[1-n(e3)] n(e2)[1-n(e4)]= n(e3)[1-n(e1)]n(e4)[1-n(e2)] where n(ej) is the average occupancy, a number between 0 and 1. The opposite of only 0 or 1 particle in a state ei, is to have any possible number of particles in this state and an enhancement, not a hindrance given by 1+p(ej). This is only a guess, but experiment shows that bosons follow this rule. Given a definition of trial runs in terms of the inhibition and enhancement factors, one may do two things. First, since the trial rule seems to involve different numbers of particles in a state associated with a p(ei), one may consider sums of p(ei) (power number of particles) when describing the system statistically (i.e. grand canonical scheme). This approach gives rise to linear terms (a+bp(ei)) which may in turn be formulated in terms of a factorial/arrangement equation which may be maximized subject to the constraints Sum over i ei n(ei) and Sum over i n(ei). We note that this maximization is not simply in terms of all degrees of freedom in the system, but is specifically linked with the rules associated with the trial runs and the constraints. We note, however, that issues seem to arise with the factorial scheme in the boson case. Thus, overall we argue that a single particle probability and its trial run scheme are key to establishing the statistical behaviour of a system. (We also note that single particle independent probability, which is an approximation, also affects the final results.)
If one tosses a die or coin, different outcomes occur with a probability p(i) where i is the outcome. This probability may be measured by performing trial runs. For N trial runs, where N is very large, one might expect Np(outcome i) results of an outcome i. In other words, p(outcome i) is both the probability and describes the trial run. Even in the Maxwell-Boltzmann case, one has a probability p(ei) where ei is kinetic energy (no V(x)) and reaction balance p(e1)p(e2) = p(e3)p(e4) represents trial runs. Thus the trial run is again characterized by p(ei) as ei’s contribution. It is, however, possible to have coupling of probabilities in certain physical situations even though p(outcome i) still exists. In such a case, the number of outcome i results for N large trials is not N p(outcome i). In other words, p(i) does not fully describe the trial run. There may be other physical rules in place. If elastic scattering of particles is the way trial runs occur, e1+e2 = e3 + e4 represents the salient information in the problem and even if there is nonindependence of p(outcome i) in the scattering, one should at least obtain a function Q(p(e1)) such that Q(p(e1)) = -e1/T. Q(p(ei)), however, need not be p(ei). We consider the physical rule given by Pauli stating that only one particle may fit in an energy level (no spin). Thus there is either 0 or 1 particle in the level. This should inhibit scattering, so the trial run is no longer simply described by p(ei). Rather the probability to scatter into level ej is (1-n(ej)) and so one has: n(e1)[1-n(e3)] n(e2)[1-n(e4)]= n(e3)[1-n(e1)]n(e4)[1-n(e2)] where n(ej) is the average occupancy, a number between 0 and 1. The opposite of only 0 or 1 particle in a state ei, is to have any possible number of particles in this state and an enhancement, not a hindrance given by 1+p(ej). This is only a guess, but experiment shows that bosons follow this rule. Given a definition of trial runs in terms of the inhibition and enhancement factors, one may do two things. First, since the trial rule seems to involve different numbers of particles in a state associated with a p(ei), one may consider sums of p(ei) (power number of particles) when describing the system statistically (i.e. grand canonical scheme). This approach gives rise to linear terms (a+bp(ei)) which may in turn be formulated in terms of a factorial/arrangement equation which may be maximized subject to the constraints Sum over i ei n(ei) and Sum over i n(ei). We note that this maximization is not simply in terms of all degrees of freedom in the system, but is specifically linked with the rules associated with the trial runs and the constraints. We note, however, that issues seem to arise with the factorial scheme in the boson case. Thus, overall we argue that a single particle probability and its trial run scheme are key to establishing the statistical behaviour of a system. (We also note that single particle independent probability, which is an approximation, also affects the final results.)
Addendum July 19, 2021 The Fermi-Dirac and Bose-Einstein statistical cases, like the Maxwell-Boltzmann case, may be formulated in terms of a reaction balance between a regular two body elastic collision and its time reversed reaction: e1+e2 = e3 + e4 and f(e1)[1-/+f(e3)]f(e2)[1-/+f(e4)] = f(e3)[1-/+ f(e1)] f(e4)[1-/+ f(e2)]. Both the Fermi-Dirac and Bose-Einstein cases reduce to the Maxwell-Boltzmann case under certain conditions. An unusual feature of bound state quantum mechanics is that there is a spatial density (or probability to be at x = P(x)), just as in classical statistical mechanics, but also a wavefunction W(x) such that P(x)=W(x)W(x). In other words, W(x) for the bound state is like a square root of probability. Furthermore, there is a P(p)=a(p)a(p) where W(x)=Sum over p a(p)exp(ipx). In this note, we try to argue that P(p), which is based on the measurement of momenta of particles knocked out of an ensemble of identical single particle bound states, does not have physical meaning within the bound state, unlike P(p,x) which has physical meaning for a Maxwell-Boltzmann gas both inside the gas and for particles released from a small hole in the gas wall. This forces one to create another function to “link” the two physically observed probabilities P(x) and P(p).
The Maxwell-Boltzmann (MB), Fermi-Dirac FD and Bose-Einstein BE distributions may be obtained by maximizing the ln of energy level degeneracies subject to the constraint of Sum over i ei ni = Eave in the MB case (canonical distribution) and this constraint together with a second one Sum over i ni = Nave (grand canonical case) as shown in (1) page 182 . In particular, in the MB case, a single energy level ei with ni particles is said to have the degeneracy (gi) (power ni) / ni ! (which overcounts). This maximization process leads to the weight gi exp(-ei/T). Setting gi=1, however, leads to the same weight exp(-ei/T) so the MB factor is not a result of a single particle level degeneracy i.e. the direction in which the momentum vector points. To shed physical light on this situation, one may consider two particle scattering in the MB, FD and BE cases. In all three cases, e1+e2 = e3+e4 (with e=pp/2m), but only in the MB case does n(e1)n(e2)=n(e3)n(e4). This suggests that the single energy level ei degeneracy levels ei actually play a physical role in two body scattering for the FD and BE cases, but not in the MB case. Thus, calculating probabilities based on counting still involves the physics related to the interactions in the problem. Degeneracies may exist and may or may not need to be part of the counting of arrangements. As a specific example, we note that in (1) (gi) (power ni) / ni ! is used to count degenerate arrangements for the MB case (on page 182). This same expression is used in (1) on page 194 to describe a method by Fowler and Darwin to obtain the MB single particle probability exp(-ei/T). In that case, gi is taken to be a parameter which is ultimately set to 1 and not the degeneracy of the single particle level. This begs the question: When should single energy ei degeneracy be included in the counting of states?
In a previous note (1), we argued that even though photons and particles with rest mass both have the same quantum free particle probability (wavefunction) exp(-iEt+ipx), they have different spins, spin 1 (photon) and spin ½ (fermion like nucleons, electron). We argued that one may find equations for spin matrices by identifying an inherent force associated with the object. For a particle with rest mass, we argued that one chooses rest mass, mo, which is proportional to the force, inertia. For a photon, one chooses the electric field. The key is then to find an equation which is linear in the inherent force and contains p, E (for special relativistic effects and geometry). For a particle with rest mass, the Lorentz invariant -EE + cc p dot p = -momo cccc is linearized as Dirac showed. For a photon, one seeks an equation linear in the electric field. One may linearize d/dt partial (.5eo El dot El + .5/uo B dot B) + 1/uo grad dot (El x B) = 0, where El=electric field and B=magnetic field. For a particle with rest mass, one must retain the identity of the rest mass when forces interact. If one has two identical mo’s with the same center-of-mass, one must somehow distinguish between the two to deliver an impulse hit to only one. This may be done by using a magnetic field and having one particle have spin up, and the other, spin down. This leads to the Pauli exclusion principle. Using the Pauli exclusion principle in a two body elastic scattering problem, one may easily establish time reversal balance by: p(e1)p(e2) {1-p(e3)}{1-p(e4)} = p(e3)p(e4) {1-p(e1)}{1-p(e2)} ((1)). From this one may obtain the Fermi-Dirac distribution. The question becomes: What does one do for photons? We argue that photons do not interact in the same manner as two particles with rest mass which elastically scatter. Instead we consider photons interacting with electrons in atoms. We note that in the case of particles with rest mass, there is a grouping of two energies, i.e. e1,e2 on the LHS side of ((1)), and e3,e4, on the RHS. We keep this notion of grouping, but note that e1, and e2 are not being converted into e3, e4 photons. Rather, all four types of photons are present and may interact with the electrons in atoms. Thus, we consider the sum of the probabilities for the e1,e2 grouping, namely (e1,e2) + (e1,e2) e3 + (e1,e2) e4 + (e1,e2) e3, e4. We argue that the sum of probabilities for this set of groupings should equal the sum of probabilities for the grouping of (e3,e4), i.e. (e3,e4)+ (e3,e4) e1 + (e3,e4) e2 + (e3,e4) e1,e2. This is equivalent to: p(e1)p(e2) {1+p(e3)}{1+p(e4)} = p(e3)p(e4) {1+p(e1)}{1+p(e2)}. From this, the Bose-Einstein distribution follows.
In classical statistical mechanics, although one has averages of a system, a single particle within the system has properties which are sharply defined at any x,t, even though these properties change stochastically, i.e. through interactions. A quantum free particle, on the other hand, has information based on the Lorentz invariant -Et+px, multiplied by i to create periodicity which indicates an inherent information, not one based on the outcome of trial runs or interactions. The probability is exp(-iEt+ipx). If E and p are considered as constant, then there are independent probabilistic fluctuations in x and t, which means there may be corresponding probabilistic interactions, both impulse ones (one dimensional reflection/refraction, n-slits) and temporal ones (Rabi frequency). In such a case, one has interference and the form exp(ipx) or exp(-iEt) governing the statistics, not a maximization of entropy linked to interaction/outcome results. Nevertheless, the quantum particle interacts with fields, it seems, which give rise to p or E changes. It seems the key idea is that one has stochastic uncertainty before an interaction occurs and hence interference. The alternative is that such stochastic uncertainty does not exist before a specific interaction, i.e. the inherent probability exp(-iEt) or exp(ipx) has no effect on the interaction. This is not to say that there cannot be probabilities before the interaction. In a Maxwell-Boltzmann gas, there are probabilities P(e1) and P(e2) for particles with e1 and e2, but a specific e1 and e2 have no uncertainty. The uncertainty is in the outcome of the reaction. Another example is a two state electron in an atom which absorbs and emits photons. The rules for time evolution are that the ground state is represented by exp(-iEot), but at some stochastic t value, it changes to exp(-iE1t) and vice versa. The ground state absorbs a photon and the excited emits. The two states are clearly defined by the differing interactions which govern their time dependence which is taken to be simply the changing of Eo to E1 and vice versa. This problem is then based on probabilities based on outcomes of an interaction and maximization of entropy subject to a constraint is employed. There is no initial interference.
Classical mechanics in the ideal gas scenario and quantum mechanics make use of a set of energy levels which a single particle may occupy. Thus the problem becomes to determine the probability associated with ei i.e. P(ei). In this note we take the approach that relative probabilities depend on the energy gap in question. For example, consider a quantum two level system where ei values are provided by the time-independent Schrodinger equation. From the point of view of statistics, these are simply a set of ei values. Their associated physics is contained in the Schrodinger equation. For the quantum two level system one must find the relative probabilities PR(e1) and PR(e2). Next, imagine a quantum system with many levels and assume for the sake of argument there are levels e1,e2 and ei,e(i+1) such that e2-e1=e(i+1)-ei. If the mechanism is the same for jumping from e1 to e2 and from ei to e(i+1) (i.e. absorbing a photon of the same energy), one might argue that the relative probabilities PR(e2)/PR(e1) = PR(e(i+1))/PR(ei). From this one may obtain the Maxwell-Boltzman result exp(-ei/T)). Note that levels do not need to be adjacent. For example in a large level system one might consider e1 and e3 and any ei and ej such that e3-e1 = ej - ei. In some situations this may need to be modified (e.g. fermions and bosons have enhancements and there may be level related issues), but as a first step we argue that insisting on relative probabilities being dependent on ei-ej has important consequences. Furthermore, in such a case the normalization constant for these relative probabilities i.e. C(T)= Sum over i exp(-ei/T) is important. Taking ln(C(T) and then taking the derivative d/dT yields an expression proportional to the average energy. An expression for dE/dT dT, however, would lead to a second order differential equation in ln(C(T)). If one introduces a new function S such that E-TS = -Tln(C), then one may show that dE/dT = T dS/dT i.e. entropy S emerges.
In previous notes, we argued that one may introduce a probability into Newtonian elastic scattering if one assumes that for an incident (e1,e2) energies and (p1,p2) momenta vectors, any (ei,ej), (pi,pj) which satisfy conservation of momentum and energy have equal probability. We then found that for a single particle, the relevant probability is exp(-iEt+ip dot r). This is a unit modulus complex number which is Lorentz invariant, and exp(i p dot r) is time reversal invariant, i.e. p→-p and r→-r yield the same value. The question then becomes: How does one use exp(-iEt+ipx) or for time-independent problems, exp(ipx)? This is not a usual probability linked to quantity, e.g. N P(reflect) = the number of particles which reflect. exp(ipx) is a probability to have a certain momentum, but there is no reason it cannot carry a weight, i.e. a math factor. The question is: What does such a factor mean physically? Secondly, exp(ipx) depends on x (space), but exp(ipx) is not the probability to be at x, but it is somewhat related in that for any x and a given p, the modulus has the same value, suggesting equal probability to be at any x. This mapping, however, really only holds for a single exp(ipx). It seems that one can say two things about exp(ipx). First, if it is a probability for a momentum p, then one should be able to write p exp(ipx) or function(p)exp(ipx). Secondly, one should be able to write AND (multiply) and OR (add) statements. The idea of an OR or addition of exp(ipx)s is important, because one may be able to write an equation for constant kinetic energy: -1/2m grad dot grad W(x) = numberW. The general solution should be a W(x) which contains a sum of all exp(ipx) type solutions which have this kinetic energy. (In the relativistic case, one may consider d/dt d/dt W - cc grad dot grad W = -momocccc W.) The question we consider in this note is: Given that one may write W and grad W, what can only really do with these? They do not directly represent physical observables. Furthermore, if one wishes to impose continuity on W and gradW, which is reasonable mathematically, one should have a physical argument to justify it, and we suggest that this is a little complicated. Ultimately, we try to argue that one may consider continuity of W as well grad W in certain cases based on the association with a classical quantity probability equation. We also suggest one may relate W* grad W which may be generalized to W* grad W + (grad W*) W, as discussed in Part 3 and provides a link to classical probabilities.