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 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 Parts I and II we examined equilibrium situations based on an additive scalar (distributive variable) such as ei= pi dot pi/2m in a Maxwell-Boltzmann gas and a vector dynamic quantity such as the momentum vector in a reflection/refraction of light situation or a single particle quantum bound state. We argued that a loss of information in the form P(a1)P(a2)=P(a3) holds in both cases, but in the additive scalar situation this leads to P(a1) = C exp(-a1/T) while in the vector case to exp( -i a1 dot conjugate variable). For a one dimensional quantum bound state a1=p and the conjugate variable is x. In such a case, exp(ipx) is a periodic eigenstate of the translational generator multiplied by -i. Within the bound state, stochastic hits from V(x) the potential lead to an ensemble i..e W(x) = Sum over p a(p)exp(ipx) where a(p)’s are fixed by considering average energy at each x to be the same i.e. { Sum over p a(p) pp/2m exp(ipx) } / { Sum over p a(p)exp(ipx)} + V(x) = En. In this note, we consider the a(p)’s as also being probabilities like exp(ipx) and ask whether they could exhibit the same loss of information as in the vector p if one associate p in a(p) with the additive scalare pp/2m. We argue that such considerations lead to the ground state solution of a quantum oscillator.
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.
Both quantum bound state equilibrium and classical statistical mechanical equilibrium are characterized by d/dt partial density(x,t)=0. For a quantum particle, W(x,t)=exp(-iEt)W(x) and d(x,t)=W*W. Density is linked to fluid flow and arguments of pressure balance such that the overall average fluid flow is zero are made. This pressure-force balance is common in classical statistical mechanics and used to calculate the variation of density with x i.e. obtain a differential equation with d d(x,t) /dt partial. A similar balance equation is used in (1) as the authors try to derive the Schrodinger equation using a fluid dynamical approach. In this note, we try to argue that one may think of the two equilibria in terms of an energy quantity which remains constant throughout space. This energy quantity depends directly on the type of interactions in the system at any point x. For a classical ideal gas, balanced two body time reversal scattering represents the interaction. V(x), a potential, does not change this fundamental manner of interacting at any given point, although it does affect density. As a result the energy related constant at each x is T, related to average kinetic energy. We argue that this already points to the dominant type of interaction creating the statistical interaction i.e. one that depends only on p in interactions which conserve kinetic energy. This “happens” to be equivalent to maximizing Shannon’s entropy, but we argue this is particular to this scenario in the sense that not all interacting systems will have this equivalence. In other words, we argue that one must be careful with regards to the maximization of entropy as a fundamental rule. In particular, we consider the single particle bound quantum state. The interaction exists between the particle and the potential V(x), but one may consider a distribution of free particle momentum probabilities i.e W(x)=wavefunction = Sum over p a(p)exp(ipx). The form exp(ipx) comes from separate arguments as is discussed in the note, but the point is that there are no two body scattering reactions occurring at a given x, neither is there acceleration from on x to another (except for a vrms mathematical quantity). Nevertheless the “free particle” exp(ipx) interacts at each x with V(x). As a result of this interaction, one has free particles exp(ipx) which are no longer associated with pp/2m, but rather with an Eave which is a constant for every p. In other words the identity of exp(ipx) is unusual because it is linked to exp(ipx) and pp/2m, but also to Vk1 a(p-k1), Vk2 a(p-k2) etc where V(x)=Sum over k Vk exp(ikx). Thus each x point is associated with a constant Eave and this defines the equilibrium. (It is also equivalent to conservation of energy at each x.) In such a case, one does not maximize entropy to find a(p) values. In the two cases, the common point seems to be considering the interactions at each x and finding a constant “energy” type quantity which does not vary with x. This we argue appears to define equilibrium for both the classical statistical and quantum bound states. Instead of considering density which does not change in time at each x, there is constant energy which not only does not change in time, but has a constant value at each x which seems to be an alternative rule.
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 previous notes, we suggested that quantum bound states may be analysed in terms of the conditional probability P(p/x)= a(p)exp(ipx)/W(x) (where W(x) is the wavefunction) which allows for the calculation of average kinetic energy at each point x i.e. [Sum over p pp/2m P(p/x)]/ W(x). This added to V(x) yields E (average energy) at each x and yields the time-independent Schrodinger equation). We argued that probability is handled at the level of W(x)=Sum over p a(p) exp(ipx). Measurements (even internal) may knock a particle from one p1 state to another so another set of probabilities (P(x) and P(p)) arise, P(x) being based on: a(p1)exp(i p1x) exp(i p2x). Averages of f(p,x) follow from P(x) f(p,x) P(p/x). Thus, there are two sets of probabilities in quantum mechanics. Entropic dynamics (1) appears to begin with the probability P(x) and considers a transition probability P(x1 |x) with x1-x= an infinitesimal quantity. A product of transition probabilities is then taken to allow for finite x1-x differences. It is stated in (1) that there exists information about x but not p. The maximization of entropy density may be used to determine the form of P(x1 | x). A “drift velocity” is also included (which is later shown to be related to the phase of the wavefunction). Ultimately, the Schrodinger equation is also obtained. We argue one may use P(p/x) to also calculate P(x1 | x) and compare this result with that obtained through the maximization of entropy density. A type of drift velocity appears naturally in this approach. We argue that the form a(p)exp(ipx) already incorporates statistical ideas (i.e. maximum entropy) in that the modulus of exp(ipx) is identical for all x, endowing all x points with a “kind” of equivalent status. exp(ipx) is a function of p and x because one must calculate average kinetic energy at each x and this must change with x i.e. KEave(x) + V(x) = E. Thus, we think there are strong similarities between entropic dynamics and the approach of using P(p/x). There are also some differences in that the coefficient A of delta(x)delta(x) in P(x1 | x) = 1/Z exp{ A delta(x)delta(x) + B d Phi/dx delta(x)} is treated as independent of x in entropic dynamics, but is a specific function of x (related to d ln(W)/dx d ln(W)/dx + 1/W d/dx d/dx W) in the P(p/x) approach.
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.