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?
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?
An interesting result arises in information theory, namely that if one considers a microstate with large N trials, each representing a state i with probability p(i), then even though this is only one of many possible microstates (each being a permutation of each other) it carries information about all of the permutations based on the average occupancy number. In particular, the probability of a microstate may be written as: Product over i p(ei) [power Np(ei) ] for large N. This holds for any microstate and so all have the same probability. Given that all of the microstates are permutations in the large N case, this probability is equivalent to 1/ {number of permutations} where Number of permutations = N! / { Product over i n(ei) }. We point out the above example, because it seems there are at least two main strategies in statistical mechanics. One is to begin with a set of average occupancy numbers n(ei/T) whose specific form is unknown. The idea then seems to be to seek degeneracy in the system, possibly among the arrangement of the n(ei)’s or degeneracy within single levels ei and to maximize the number of degenerate arrangements in order to account for as little information as possible. This approach may be used to calculate the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions. In the case of the latter two, degeneracy in the single energy levels ei is key. One then maximizes the ln of the degenerate arrangements subject to constraints such as Sum over i ei n(ei) and Sum over i n(ei) = N. The degeneracies themselves are associated with momentum degeneracies yielding the same energy. A completely different approach, it seems, is to consider exp(-(E-uN)/T) as representing the probability of a state with energy E and N particles and to use either the grand canonical ensemble i.e. sum the various exp(-(E-uN)/T)’s. These may then be broken into single particle sums using E(N) = sum over i ei n(ei) for n(ei) variable or sum over i ei for N fixed. Even though momentum degeneracy still exists, it is not part of the n sum of exp(-(E-uN)/T) factors which gives a single energy level probability distribution with no maximization necessary. Thus we point out that exp(-(E-uN)/T) for microstates seems to incorporate information about degeneracies without them needing to be explicitly added. On the other hand, if one starts with average n(ei) (without knowing their form), one must explicitly consider degeneracies in order to maximize so that there is no unnecessary information in the system. Both approaches should yield the same result. We consider the Bose-Einstein example of a photon gas as an example.
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.)
The Maxwell-Boltzmann MB distribution is often derived by maximizing the number of possible arrangements of particles with different energies subject to the constraint of a fixed number of particles and fixed energy. Using the canonical approximation that only average energy is conserved and using Stirling’s approximation for large number factorials leads to the Maxwell-Boltzmann distribution. The ln of total number of arrangements using Stirling’s approximation is linked to a math form called Shannon’s entropy i.e. - Sum over i f(ei) ln(f(ei)). Interestingly this same MB result may be obtained through reaction balance which makes no use of the idea of arrangements or entropy or maximization. It is concerned with including all possible physical reactions (for example two body reactions between all energy particles) subject to conservation of energy. In this sense it deals directly with the dynamics of the problem while the arrangement approach does not. The dynamical approach considers all possible reactions with notions of independence or lack of information and ensures the system remains unchanged. Why are they the same? We argue that it is because the “recipe” for the probability of two body reactions namely n(e1)n(e2) is of the same form as the leading factor of n(e1)! n(e2)!. In the case of quantum mechanics one may apply the dynamical approach to obtain the time-independent Schrodinger equation. In such a case one does not need to think in terms of “maximizing” any quantity although one considers all possible reactions with a potential V(x). a(p) factors in W(x) = wavefunction = Sum over p a(p)exp(ipx) are interrelated, but retain their identity after all possible interactions occur. This we argue is the idea for the eigenvalue equation. At the same time, Shannon’s entropy, which is directly linked to the notion of arrangements, is applied to a bound state and possibly linked to thermodynamics i.e. dE = TdS -PdV which may be applied to an adiabatic expansion. Shannon’s entropy is created from existing probabilities in this case which do not arise from explicitly maximizing entropy subject to constraints as the probabilities follow directly from the Schrodinger equation. One might ask why Shannon’s entropy has any relevance if it is the dynamics which determines the value of W(x) and a(p). It seems, however, that one may knock out a particle in a state p with probability P(p)= a*(p)a(p) or find a particle at x with probability P(x)=W*(x)W(x). These are classical type probabilities linked to the quantum dynamical probabilities a(p) and W(x). It seems Shannon’s entropy is only applied to P(x) and P(p) or a product P(x)P(p) which may give it physical meaning in a quantum bound state. We examine these ideas in this note.
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 Newtonian mechanics there exists a continuous x i.e. for a conservative force: .5mv(x)v(x) + V(x) = E with x taking any value. This is associated with the idea of dx→0. As a result one may choose an x and x+dx with dx→0 and v(x) and v(x+dx) become arbitrarily close in value. It is almost as if their identities merge. In (1) entropy and average energy are calculated using the Maxwell-Boltzmann MB factor exp(-e/T) and a continuous energy. The results E= NT (k=1) and S = -ln(T/To) with To being an integration constant show that one must have T>To because T→0 does not lead to S→0. Thus there is a problem if one uses continuous energy. In other words continuous energy is rejected.. (1) then considers an equally spaced energy i.e. e=bn, n=0,1,2,3 … and calculates energy and entropy again. In this case S→0 as T→0. Thus (1) concludes that it is entropy which leads to the discretization of energy In this note we try to argue that discretization of energy is already implied by the MB factor. The MB factor exp(-energy/T) where T is a math parameter called temperature represents a probability. In other words, different energies have different probabilities i.e. one must distinguish between them. Having two energy values merge is a problem now because one has a second parameter T which may also go to zero i.e. e2-e1→0 and T→0 leads to trouble because exp(-e2/T) does not tend to zero as it should if e1=0. We argue that the underlying reason for e1 and e2 being discretely different is that one cannot use the probability/T formalism i.e. MB factor unless interactions occur. We argue that such two body interactions cannot ultimately transfer or remove a de→0 and keep e1 and e2 distinct. In other words the interaction/collision is ultimately linked to a finite energy transfer even though T→0. As a result energy should be discretized in an equilibrium scenario. This argument does not make use of entropy explicitly although implicitly having exp(-e2/T)-->0 as T→0 (say for a 2 level system) means that the particle is in state e1 which may be associated with an entropy of 0 if one ignores internal features of this state. We also note that the continuous energy result E=NT, and not just the entropy, leads to trouble because as T becomes extremely tiny E which is an average of say e1 and e2 with e1=0 as an example approaches T in magnitude. The smaller T becomes the smaller e2 becomes which e2 and e1 no longer have separate identities, but in such a case one does not have an equilibrium scenario either. In other words exp(-e/T) implies interactions between distinguishable states with respect to energy.
Classical statistical mechanics is often presented in terms of the maximization of entropy (ln(# arrangements) or ln( total probability for a set of arrangements)) subject to a constraint. In a previous note (1), we argued that such a process seems to be associated with a distributive variable which is a scalar i.e.is additive such that a product of probabilities Product over i n(ei) of different values of this variable Sumover i ei = E is equivalent to a single probability of E i.e. n(E). We applied this idea to both a Maxwell-Boltzmann gas and a 3D random walk. This distributive variable changes in the stochastic process of the problem. For example for an MB gas, there are elastic collisions so energy is the distributive variable. For an MB gas in a potential V(x), there is potential and kinetic energy of a single particle, but this is not regarded as a stochastic process in the classical view. In (2) we argued that a distributed variable is not the only type of equilibrium which may exist, even in the classical world. We considered the example of a beam of light which reflects and refracts at x=0 (medium along the y axis). Energy is conserved even at the photon level, but there is an unknown probability namely that for reflection “a” because the refraction probability is then 1-a. Thus energy conservation is not enough. We argued that in order to solve for this problem in an equilibrium or steady state manner, one needed to deal with square roots of fluxes/intensities in order to obtain two equations to solve for B,C i.e. square root fluxes at x=0 for the reflected and refracted rays in terms of A, the square root flux for the incident beam. We argued these fluxes were associated with eigenfunctions of the translation generator i.e. exp(ipx). Thus a dynamic equilibrium is introduced (i.e. quantum mechanics) which does not use a distributed variable i.e. does not use maximization of entropy. Rather it is associated with eigenfunctions of the generator of the dynamical motion involved. It is this dynamical variable (e.g. p) which changes in this stochastic process even if in direction only. A question arises: When does one use one type of equilibrium approach versus the other? We attempt to answer this in this note.