Classical statistical mechanics often utilizes the idea of maximizing entropy subject to a physical constraint For example, for a Maxwell-Boltzmann gas, this constraint is Sum over i ei p(ei) where ei=pp/2m (no V(x)). We have argued previously that there may be many possible constraints, i.e. an average of any power of ei could be a physical constraint. We stated that a conserved quantity associated with interactions present (creating equilibrium) should be used. Here we examine the constraint in terms of mechanical resistance i.e. pressure. We further consider the free particle quantum wavefunction exp(-iEt+ipx) in terms of resistances in time and space. We argue that the form exp(-ipx)exp(ipx) = 1 removes the stochasticity associated with motion due to the resistance (impulse) p Thus 1 or 1/L (L=length) seems to be a smoothed over picture associated with a constant flux and overlooks the inherent stochasticity associated with motion. This stochasticity is crucial when describing impulse hit reactions (n-slits, index of refraction interfaces, interactions with V(x)), but if a particle does not interact in space, then the W*(x)W(x) picture seems to be used to describe the v(average)=x/t scenario. The issue is that for a quantum bound state, there is also an rms velocity based on a wavefunction and one may create an average W(x)W(x) (W(x) is real for a bound state), but like exp(-ipx)exp(ipx), this bypasses some of the stochasticity in the problem, we argue. It only demonstrates stochasticity associated with differences in momentum.
Classical statistical mechanics often utilizes the idea of maximizing entropy subject to a physical constraint For example, for a Maxwell-Boltzmann gas, this constraint is Sum over i ei p(ei) where ei=pp/2m (no V(x)). We have argued previously that there may be many possible constraints, i.e. an average of any power of ei could be a physical constraint. We stated that a conserved quantity associated with interactions present (creating equilibrium) should be used. Here we examine the constraint in terms of mechanical resistance i.e. pressure. We further consider the free particle quantum wavefunction exp(-iEt+ipx) in terms of resistances in time and space. We argue that the form exp(-ipx)exp(ipx) = 1 removes the stochasticity associated with motion due to the resistance (impulse) p Thus 1 or 1/L (L=length) seems to be a smoothed over picture associated with a constant flux and overlooks the inherent stochasticity associated with motion. This stochasticity is crucial when describing impulse hit reactions (n-slits, index of refraction interfaces, interactions with V(x)), but if a particle does not interact in space, then the W*(x)W(x) picture seems to be used to describe the v(average)=x/t scenario. The issue is that for a quantum bound state, there is also an rms velocity based on a wavefunction and one may create an average W(x)W(x) (W(x) is real for a bound state), but like exp(-ipx)exp(ipx), this bypasses some of the stochasticity in the problem, we argue. It only demonstrates stochasticity associated with differences in momentum.
Maximization of entropy subject to a constraint Sum over i ei P(ei) = Eave may be used to determine the Maxwell-Boltzmann factor exp(-ei/T), where T is temperature. In the case of a potential V(x), one may maximize -P(x) ln(P(x)) subject to the constraint TV(x)P(x). Average potential energy is not an a priori known value, but we have argued in the past that maximization of entropy subject to a constraint, followed by a derivative leads to a balance equation. What about a classical mechanical bound problem? This is usually considered nonstatistical . We suggest it may be a statistical problem, but one for which maxent applies to flow, in particular the quantum bound problem. We have argued before that for a free particle, one may maximize Shannon's entropy -P(t)ln(P(t)) subject to a periodic constraint i Et P(t) and -P(x)ln(P(x)) subject to i px P(x) to obtain exp(-iEt) and exp(-ipx). We argue these are associated with flows in time and space, and that there are problems in nature based on such flow and its interference. Classically, a bound system is a flow problem because a particle moves to the right and then left, between turning points. This means the particle is confined or localized,, but exp(ipx) is not. exp(ipx) is two dimensional to account for motion in a certain direction, but both cos(px) and sin(px) exist everywhere in space. A bound system should not have one directional motion,on average, thus either cos(px) or sin(px) applies, i.e. a certain parity. How does one account for a probability value in all of space? We argue that in the flow exp(ipx) picture, one must consider a bound problem in equilibrium with particles trying to enter or escape. By creating an ensemble W(x)=Sumo v over p a(p)exp(ipx) and having W(x) tend to 0 at x= +-/infinite one may create such a flow picture. At first one may consider any a(p) values, but we argue that either a(p)=-a(p) or a(p)=a(-p) for all p in this note, i.e. there is no mixing of parity. We try to show this. We then argue that V(x) does not impede this flow statistics, but only imposes conditions on the average kinetic energy at x. Thus Newtonian mechanics deals only with this restriction and not with the underlying average of maxent flow probabilities exp(ipx). A classical statistical gas is not a flow problem, but a balance problem with or without V(x)..Interparticle collisions change the kinetic energy of a particle and hence break its exp(-iet) flow probability. The stochasticity is then not in the flow, but in the flow-breaking collisions We suggest that this is linked to maximization of entropy subject to a constraint and deals with different probabilities than flow ones (exp(ipx) and exp(-iEt). A Rabi frequency problem on the other hand is a flow in time problem with the magnetic field not inhibiting this flow. Finally, there may be problems which at first appear to be a balance problem, but are really a flow one, like one dimensional photon reflection/refraction.
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.
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 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.
Extremization of entropy, usually subject to an average energy constraint,(Maxwell-Boltzmann gas) is often associated with maximization of the number of microstates N gas particles may represent. The math procedure of extremization (variational calculus) is then associated with the ln of a factorial expression which reduces to n(ei)ln(n(ei)) using Stirling’s approximation. The central limit theorem, on the other hand, states that the distribution of sample means of a random variable approaches a Gaussian, i.e exp(-pp/(2sig sig)). Here sig sig is the standard deviation. For a random variable, the important or characteristic information is the standard deviation associated with the average of the square pp. We suggest here that an alternative way of looking at maximization of entropy (which is a stability statement we argue) other than ln of permutations is to consider the stability of the variation of the variance of pp subject to a change in the probability, with pp written in terms of P(p). Average kinetic energy in a statistical problem represents such a variance and should be invariant under changes of the probability subject to an average kinetic energy constraint, we suggest. How does this idea apply to quantum mechanics? In quantum mechanics, one has a statistical function exp(ipx) and for bound states creates W(x)=wavefunction = Sum over p a(p)exp(ipx). a(p)a(p) is formally the probability for p, but this does not mean that a(p)a(p) is a random variable because one has the constraint: Sum over p a(p)a(p) pp2/m + Integral dx V(x) W(x)W(x) = En. In the case of a harmonic oscillator, however, one knows a priori that the two summands are each equal to En/2. Furthermore there are a set of En values, starting at a ground state and increasing. All En values, except the ground state, are metastable. Thus the ground state is a special state. Given that Sum over p a(p)a(p)pp2/m = Eo/2, a priori, one may consider that all of the information of a(p)a(p) is associated with Eo. This is exactly the case for a Gaussian from the central limit theorem. Alternatively, from information theory, information = ln(probability) which yields the same result. If this is the case, then pp/2m a(p)a(p) written strictly in terms of P(p)=a(p)a(p) should be stationary upon changing P(p) subject to the E/2 constraint. This special situation, however, only holds for the ground state oscillator. What about average potential energy for the ground state oscillator? If the probability for p (momentum) is a Gaussian due to the Central Limit theorem, this means that the standard deviation is the information (and is linked with randomness). Taking a Fourier transform of a(p) cannot add more information to the picture, we argue. W(x)W(x) should also be a Gaussian, as it is. As a result, xxW(x)W(x), written in terms of P(x)=W(x)W(x) subject to the same Eo/2 constraint should represent a maximization or stationary situation, we argue.
For a Maxwell-Boltzmann gas (MB) it is known that T (temperature) is a statistical parameter related to the average energy. It appears in f(ei) = C exp(-ei/T) so T has the same units as ei, a single particle energy. If T increases, ei increases identically if ei/T is treated as constant. Thinking in terms of ei=pp/2m, this means the momentum spread (i.e. variance of the Gaussian increases) and one expects entropy to increase In (1) we argued that exp(-ei/T) for an MB gas and exp(ipx) for a quantum free particle (constant p), follow from the same statistical rule P(a1)P(a2)=P(a1+a2), but ei is a scalar and p a vector. Furthermore exp(ipx) is associated with spatial invariance and the generator of translations d/dx. In this note, we try to argue that a constant p may act as a statistical parameter just as T, although in this case hbar/p has the behaviour of a length i.e. the wavelength. If p increases (just like T is increased), the wavelength decreases meaning there are more peaks in a fixed length or more entropy, we argue. Thus there seems to be a statistical mechanical similarity to increasing T for an MB gas and to increasing p for a single free quantum particle. The association of x and p in exp(ipx) has implications, not only for the resolution of x space, but the increased entropy suggested by more peaks of cos(px). In addition, this means more peak places for impulses to occur. Thus a large p not only represents a large impulse, but is associated with more peak points for this impulse to be delivered and more entropy. In (2) we argued that the humps of cos(px) encode information about the impulse/force used to create p and which may be delivered if p hits an object. Thus there is a somewhat unusual relationship between p and x that one does not find in Newtonian mechanics, but has its roots in special relativity i.e. in the Lorentz invariant -Et+px. We suggest that just like exp(-V(x)/T) in an MB gas means that for a large force=-dV/dx, exp(-V/T) changes rapidly in x, a large p, associated with a large force should also imply that the humps in exp(ipx) rise (and fall) sharply, thus leading to sharp peaks. Thus sharp peaks are indicative of high internal force and increased entropy for a free particle exp(ipx), we argue.