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.
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 Part I we considered two kinds of statistical processes. One is based on a scalar additive which changes in stochastic processes e.g. kinetic energy in two body elastic collisions or the spatial shell in a 3D random walk problem. We argued that in such a case, one considers a1+a2= a3 (where ai is the distributive scalar) and a product probability scheme i.e P(a1)P(a2) = P(a3) to find that P(ai) = C exp(-ai b) where b is a parameter which may be found if the average value of a is known. Furthermore, we argued that the distributive scalar process is identical to maximizing entropy subject to a constraint Sum over i ai P(ai). The second process is based on a dynamic vector changing in the stochastic process. We used the problem of light reflecting and refracting (which we argued is similar to quantum mechanics for a pure bound state) and noted that the p (momentum) vector changes i.e. an incoming p along the x axis in the positive direction becomes -p on reflection or p2 on refraction in 1 dimension. We suggested that a dynamic probability is needed in such a case, namely an eigenfunction of the generator associated with the dynamic variable, e.g. x for p. The eigenfunction is then periodic. Using exp(ipx)s (eigenfunctions of -id/dx) on may calculate average energy at x and fix weights i.e. {Sum over p pp/2m a(p)exp(ipx)} / { Sum over p a(p)exp(ipx)} + V(x) = En, but there is no maximization of entropy. At first the additive scalar (distributive variable) and dynamic vector approaches seem to be very different (for example, one maximizes entropy and the other does not), but in this note we argue there are a number of similarities. We focus on these similarities and suggest these may perhaps be key features that unite the distributive scalar approach (classical statistical mechanics, level jumping in quantum mechanics) with the dynamical vector approach (statistics of pure quantum bound state). In particular, both types of probabilities are linked with exponential functions although one is real and drops in value exp(-ei/T) and the other is complex and periodic exp(ipx) and P(a1)P(a2) =P(a3) in both cases. Also, both are eigenfunctions of generators. Thus we suggest that the similarities between the scalar additive approach (MB gas) and dynamical vector (quantum bound states) may be the key features of overall equilibrium type statistical mechanics.
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.
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 equilibrium, for a Maxwell-Boltzmann gas for example, is based on localizing the gas to a certain volume and having no average changes in time. In other words, maximization of entropy subject to the average energy constraint sum over i ei P(ei) is linked with these two ideas. Ei (kinetic energy for no V(x)) is a conserved quantity in reactions and overall energy represents a quantity which does not change in time consistent with a center-of-mass which does not change and average values which do not change in time. This approach is associated with a specific kind of statistics, namely that of the constituents of the system. As a result, one obtains the Maxwell-Boltzmann distribution exp(-ei/T) where ei is an extensive property of a particle (or subset of the system) and T an intensive property linked with the whole system. Entire systems, however, may interact. For example, a N particle system with T1 may achieve equilibrium with a N system with T2 yielding a new average (T1+T2)/2. The probabilities exp(-ei/T), however, are for subcomponents of a system. One could argue that momentum is an additive (vector) quantity which is conserved in collisions, but the average momentum of the system is 0 which does not allow one to describe increases in average p and -p which may be described using energy, i.e. through T temperature. This begs the question whether momentum may be associated with maximization of entropy subject to a constraint. In (1), we argued that there were flow probabilities in quantum mechanics linked with exp(-iEt) and exp(ipx), which may be obtained formally through maximization of entropy. Here we argue that while an MB gas describes a localized volume with average values not changing in time, x and t state values change, even if p and pp/2m remain constant for a free particle and it may be possible to have a maximum entropy situation to describe the combination of px and -Et. In other words, we suggest localized versus nonlocalized equilibria. The latter represents a quantum free particle, but one may then try to localize such a particle in a potential. What happens in such a case? One does not switch to a localized maximization of entropy as in the MB gas case, because it is possible to have the potential leave the flow unbroken as argued in (1). In other words, the nonlocalized free particle equilibrium is akin to a particle bouncing back and forth in a box, which in classical mechanics is not considered a statistical process. Here we suggest that the underlying picture is statistical meaning that the classical mechanics approach seems to be an approximation. We consider both a quantum particle trapped in an infinite potential well and the ground state of an oscillator to examine ideas of maximizing entropy beyond exp(ipx). A key common feature of both types of equilibrium is the product rule P(ei)P(ej) = P(ei+ej) or exp(ip1x)exp(ip2x) = exp(i (p1+p2)x). We consider this feature as well.
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.
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)