There is a focus in classical statistical mechanics on maximizing an entropy function subject to constraints to obtain probability values. For a coin or die probabilities are given, yet one may still calculate a Shannon’s entropy. In the case of a Maxwell-Boltzmann (MB) gas, one may obtain probabilities using energy conservation and products of probabilities and then show this is equivalent to maximizing an entropy function subject to an energy constraint. We argue that the energy conservation approach yields an energy conservation equation and a product of probabilities representing a certain total energy equated to a product of different probabilities representing the same total energy. The two equations are linked to solve for probability in terms of energy and this is formally the same as a maximization problem subject to conservation of energy. A point we make is that this approach is based on a kind of static conservation, even in the presence of a potential V(x). In quantum mechanics, we argue one one has fluxes even though there is also conservation of energy. For light moving in the positive x direction which reflects and refracts (with each photon retaining its energy) there is both energy conservation and flux. The energy conservation equation in terms of flux/intensity terms is given in (1) as: AA/c1 = BB/c1 + CC/c2 where A,B,C refer to the incident, reflected and refracted rays. This single equation cannot solve for probabilities B,C in terms of A and there is not a separate product of probabilities equation as in the MB case. Rather quantum mechanics seems to be solved using square root fluxes as dynamical probabilities. In particular, the energy conservation equation is equivalent to two square root flux equations A+B=C and p1A -p1B = p2C where E=pc. Thus probabilities are determined through dynamics not static conservation or entropy maximization. In an earlier note we concluded that exp(ipx), a quantum free particle wavefunction, is also like a square root flux, and also acts as a dynamic probability. These ideas may be extended to a quantum bound state which we discuss in the note. Thus the approach to finding probabilities in the classical statistical mechanics and quantum realms seem to be completely different. One uses static conservation laws in the former and square root flux continuity/dynamical probability in the latter.
There is a focus in classical statistical mechanics on maximizing an entropy function subject to constraints to obtain probability values. For a coin or die probabilities are given, yet one may still calculate a Shannon’s entropy. In the case of a Maxwell-Boltzmann (MB) gas, one may obtain probabilities using energy conservation and products of probabilities and then show this is equivalent to maximizing an entropy function subject to an energy constraint. We argue that the energy conservation approach yields an energy conservation equation and a product of probabilities representing a certain total energy equated to a product of different probabilities representing the same total energy. The two equations are linked to solve for probability in terms of energy and this is formally the same as a maximization problem subject to conservation of energy. A point we make is that this approach is based on a kind of static conservation, even in the presence of a potential V(x). In quantum mechanics, we argue one one has fluxes even though there is also conservation of energy. For light moving in the positive x direction which reflects and refracts (with each photon retaining its energy) there is both energy conservation and flux. The energy conservation equation in terms of flux/intensity terms is given in (1) as: AA/c1 = BB/c1 + CC/c2 where A,B,C refer to the incident, reflected and refracted rays. This single equation cannot solve for probabilities B,C in terms of A and there is not a separate product of probabilities equation as in the MB case. Rather quantum mechanics seems to be solved using square root fluxes as dynamical probabilities. In particular, the energy conservation equation is equivalent to two square root flux equations A+B=C and p1A -p1B = p2C where E=pc. Thus probabilities are determined through dynamics not static conservation or entropy maximization. In an earlier note we concluded that exp(ipx), a quantum free particle wavefunction, is also like a square root flux, and also acts as a dynamic probability. These ideas may be extended to a quantum bound state which we discuss in the note. Thus the approach to finding probabilities in the classical statistical mechanics and quantum realms seem to be completely different. One uses static conservation laws in the former and square root flux continuity/dynamical probability in the latter.
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.
In classical statistical mechanics, e.g. a Maxwell-Boltzmann (MB) gas, one tries to find the probability distribution of a particular quantity such as particle energy pp/2m. Thus there is a distribution in a quantity, making it the stochastic variable. This quantity, however, is subject to a conservation law i.e. elastic two particle scattering. In a previous note (1), we considered the case of reflection/refraction and called momentum the stochastic variable and noted that it too was linked with conservation. In fact, we argued that both the MB gas and reflection/refraction followed the same stochastic law: P(variable1) P(variable 2) = P(variable 1 + variable 2) whether variable 1 is a scalar or vector. In (2), however, we noted that exp(ipx), a free particle wavefunction, treats p as a parameter with x being the “probability” variable. This seems a little unusual because p, which is normally a variable in an MB gas and even in a quantum bound state i.e. a*(p)a(p) where W(x)=wavefunction = Sum over p a(p) exp(ipx) takes on the role of a parameter i.e. p=hbar/wavelength. Why this duality? Why is p a parameter for “x” of the probability distribution and yet at the same time a variable associated with conservation of momentum? In this note, we argue that unlike the distribution of a defining property of a particle such as momentum or energy, one may have a distribution associated with probability itself. This we argue is the case for a reflection/refraction problem which applies to both light and a particle. In the case of light, there is an interface of two media with different indices of refraction and in the case of a particle an interface with two constant potentials. In both cases, total energy of the photon/particle is constant, but there are two choices or probabilities, reflection or refraction, but they are linked with different times. A single photon or particle can only undergo one at a time. One knows exactly the two values of momentum, but the probability of each is unknown. Thus unlike an MB gas where energy/momentum is being redistributed because the two particles of the outcome exist at the same time, here a probability of 1 must somehow be distributed to the two events.. Given that the incident momentum is p, the reflected -p and the refracted p2, only considering a specific time event i.e. reflection or refraction, does not conserve momentum. In the MB case, a single time event i.e. scattering conserves both momentum and energy. Thus it seems that one does not deal with events in time. There is a distribution of probability in such a way as to ensure conservation of momentum, but does this probability already exist or get created at the interface point? We argued in (3) that if the scenario were deterministic one would have to follow it in time and treat each event separately. Rather we suggest that a distribution of probabilities allows for conservation of momentum, but this means that all three momenta exist at the interface x point as one is not considering an event at a precise point in time. This then allows for each momentum to have its own x distribution based on momentum p so that one may impose conservation of momentum and not have a deterministic time associated with the interface point.. This last idea of conservation of momentum appearing is not discussed in (3) and allows for p to behave like a parameter for x while at the same time being a variable which is conserved. In short, in an MB gas one distributes energy between two colliding particles in order to conserve total energy while for a quantum particle or photon which reflects/refracts one distributes probability to the two scenarios (reflection or refraction) such that momentum is conserved. Thus the nature of what is “distributed” is very different.
In Part I we considered light moving in the positive x direction stiking a medium with a different index of refraction with some reflecting and the rest refracting. We argued one may immediately write a conservation of energy equation, but this is a single equation while there are two unknowns, the intensity of the reflected beam and the refracted, assuming the incident intensity is known. (We also argued that one also has the relation E=pc for a photon, but this is not needed to solve for the two intensities.) Our main point was that instead of seeking another independent equation (containing new information), one may write the conservation of energy equation as a product of two equations A+ B=C ((1a)) and A/c1 -B/c1 = C/c2 ((1b)) where AA/c1 = BB/c1 + CC/c2 is the energy conservation equation. Thus energy conservation contains all of the information needed, but one must move to a square root of probability picture in order to obtain the two equations needed to solve for the reflected and refracted values B,C and then the intensities BB and CC. In Part I, we noted that the reflection/refraction problem may be presented as a steady state one in which the scenario is not changing in time. A Maxwell-Boltzmann gas in equilibrium also does not change in time, but here there is a difference, namely a lack of time reversal invariance. In other words there is a distinct direction of motion i.e. the incident and refracted beams move in the positive direction only and the reflected in the negative x. The problem, however, is also a probabilistic one because given a single photon, it either reflects or refracts with certain probabilities. Thus we suggest there exists in this physical steady state problem a dynamic or directional probability. This probability is related to space and may be positive or negative unlike the usual time probabilities linked to events i.e. classical probability theory. The conservation of energy equation is a classical probability one, but dynamic motion probabilities are present in the problem and should also be manifest in equations. In fact the two dynamic probability equations allow for the solving of the reflected/refracted intensities (probabilities), but are also directly linked to the conservation of energy equation which may be written as: AA/c1 -BB/c1 = CC/c2 which points towards flow in two different directions. Thus we argue that a dynamic directional probability is responsible for the decomposition of the energy conservation equation into two dynamic probability ones.
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 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)
In a previous note (1), we argued that maximization of Shannon’s entropy -P(x)ln(P(x)) subject to the periodic (complex) constraint i p x P(x) yields the free particle two-dimensional probability exp(ipx). Similarly one may obtain exp(-iEt) (with a minus associated with E to allow for -Et+px to remain invariant under t->t+constant/E and x->x+constant/p), also for a free particle. We argued that multiplying the two together exp(-iEt+ipx) yields a phase which should be invariant when seen from frames moving at constant average velocity v=x/t.. This means that not only -Et+px, but -EE+pp= momo (c=1) is a Lorentz invariant. The latter, however, represents conservation of energy for no potential. We argue that conservation of energy for a free particle is linked to both the maximization of Shannon’s entropy for both E and p, i.e. it is an equilibrium in both time and space. We have argued before that in the presence of a potential V(x), one may replace E with E-V(x) in the Klein-Gordon equation. Here we wish to examine the statistical features of this situation in more detail. Finally, we note that the third statistical feature is present, associated with the absorption or emission of a photon. The photon wavefunction exp(-iEt+ipx) is subject to both E and p maximization of Shannon’s entropy and it is associated with an equilibrium E1 or E2 energy level of say an electron in an atom. The difference is that this absorption or emission seems to occur stochastically without being directly linked to the momentum impulse hits between the electron and V(x). In describing photon emission/absorption, one may maximize Shannon’s entropy using only conservation of energy, i..e. use the Maxwell-Boltzmann factor. The x and t notion of maximum entropy are still present as the photon wavefunction is exp(-iEt+px). The photon momentum, however, is conserved with the total momentum of the atom and not the impulse hits of the electron with V(x). The two stochastic features occur together.