Given an unbiased die, the probability is ⅙ for a result. This probability value may be obtained directly by imposing the condition of a lack of bias and noting there are six possible results. Alternatively, one may maximize Shannon’s entropy subject to the constraint Sum over i p(i)=1 where i runs from 1 to 6. Maximization of Shannon’s entropy seems to remove as much bias as possible with the constraint indicating what bias remains. For the die case the constraint has no bias, but for an average energy constraint Sum over i ei p(i) there is, namely ei. In previous notes we have argued that the constraint used in the maximization of entropy is “special”. Here we argue that it represents a conserved quantity if interactions occur. Thus there may be many constraints in a system, but choosing a conserved quantity in an interaction is rarer, but seems to be linked with equilibration. Finally, we suggest that maximization of entropy subject to a conserved quantity constraint removes as much bias as possible and it is this maximized entropy-probability which is subject to the second law of thermodynamics. It is possible, however, to have probabilities and to create entropies which are not based on maximization of entropy. For example, Shannon’s entropy may be applied to the spatial density of various n-slit interactions for a particle with constant momentum p. In this case, even though there is no maximization of entropy, the emerging particle has the same p magnitude (hence same uncertainty within a wavelength), but there is the added uncertainty of its direction. As a result, it seems that entropy increases in accordance with the second law. A free particle with momentum p, however, has a two dimensional probability distribution in space exp(ipx) characterized by a wavelength hbar/p. This may be taken as related to the entropy of the free particle, but a free particle may undergo spontaneous interactions which either increase or decrease p, hence increase or decrease hbar/p and entropy. In other words, the second law does not apply to such an entropy, but this entropy is associated with an unusual constraint. This conclusion seems to hold when one considers two body scattering. Thus we suggest that there are various kinds of entropy. Some are linked with maximization of entropy subject to a conserved quantity constraint (and are associated with equilibration) and others are not. The latter do not necessarily obey the second law of thermodynamics, we argue.
Given an unbiased die, the probability is ⅙ for a result. This probability value may be obtained directly by imposing the condition of a lack of bias and noting there are six possible results. Alternatively, one may maximize Shannon’s entropy subject to the constraint Sum over i p(i)=1 where i runs from 1 to 6. Maximization of Shannon’s entropy seems to remove as much bias as possible with the constraint indicating what bias remains. For the die case the constraint has no bias, but for an average energy constraint Sum over i ei p(i) there is, namely ei. In previous notes we have argued that the constraint used in the maximization of entropy is “special”. Here we argue that it represents a conserved quantity if interactions occur. Thus there may be many constraints in a system, but choosing a conserved quantity in an interaction is rarer, but seems to be linked with equilibration. Finally, we suggest that maximization of entropy subject to a conserved quantity constraint removes as much bias as possible and it is this maximized entropy-probability which is subject to the second law of thermodynamics. It is possible, however, to have probabilities and to create entropies which are not based on maximization of entropy. For example, Shannon’s entropy may be applied to the spatial density of various n-slit interactions for a particle with constant momentum p. In this case, even though there is no maximization of entropy, the emerging particle has the same p magnitude (hence same uncertainty within a wavelength), but there is the added uncertainty of its direction. As a result, it seems that entropy increases in accordance with the second law. A free particle with momentum p, however, has a two dimensional probability distribution in space exp(ipx) characterized by a wavelength hbar/p. This may be taken as related to the entropy of the free particle, but a free particle may undergo spontaneous interactions which either increase or decrease p, hence increase or decrease hbar/p and entropy. In other words, the second law does not apply to such an entropy, but this entropy is associated with an unusual constraint. This conclusion seems to hold when one considers two body scattering. Thus we suggest that there are various kinds of entropy. Some are linked with maximization of entropy subject to a conserved quantity constraint (and are associated with equilibration) and others are not. The latter do not necessarily obey the second law of thermodynamics, we argue.
In information theory, ln(probability(i)) equals information. The approach of maximization of Shannon’s entropy - P(ei) ln(P(ei)) subject to a constraint ei P(ei) yields: ln(P(ei)) = -ei/T where ei is a conserved piece of information associated with an interaction and -1/T, a Lagrange multiplier. In the case of tossing a die or coin, the constraint is P(i), so the Lagrange multiplier is not important, but P(ei) is the conserved quantity and is associated with interactions i.e. trial runs. In the case of a quantum free particle, one may ask: Where is the probability and where are the trial runs or interactions? A free particle, classically, has an energy and momentum as well as a time and x associated with v=constant=x/t. There do not seem to be any reactions so it is not clear what constraint exists, nor is it clear that there is any probability present as E,p and v=x/t are deterministic. In this note, we speculate on the relevant information and the presence of probability for a free particle. Such a particle does not seem to be probabilistic at first glance but, we argue that its intrinsic probability is not defined through interactions (like the classical statistical gas), but rather is inherent due to the various different ways it may have been formed. We argue that it creates other visible probabilities that are not based on maximization of entropy subject to a constraint.
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 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.
Given a coin, one has two complementary pieces of information and a corresponding probability of .5 and .5. Similar considerations apply to a die. As a result, one may get the impression that there exists a unique probability distribution in a probabilistic system and that this single probability scheme is linked to the constrained maximization of Shannon’s entropy or Fisher’s information. We argue however, that there may be different probability schemes in the same problem associated with different pieces of information which are relevant to the physics which occurs in the system. Thus, we argue that it is the physics which selects the probability distribution to be examined. For example, one may loosely think in terms of a detailed probability scheme versus a kind-of coarse grained one in the same system. For example, for a quantum free particle, P(x)=1 i.e there is the same probability for a particle to be at any x point. This, however, is based on velocity, i.e. the relationship between dx and dt. There is another piece of information in the system, namely mo which together with v yields momentum (either relativistic or nonrelativistic) .This need not be measured using a clock (which velocity requires), but by the impact of an impulse on a target. Thus, nonrelativistically p=m1v1=m2v2 give the same impulse, but have different velocities. Quantum mechanics shows that a free particle is governed by exp(ipx) which contains two unnormalized probability distributions cos(px) and sin(px). These are key for impulse interactions (i.e. the physics of interactions such as with a 2-slit apparatus etc), yet exp(ipx)/sqrt(L) maps into the “coarse-grained” P(x)=exp(ipx)exp(-ipx) / L = constant. P(x) is one probability scheme in the problem and exp(ipx) another (for a free particle). Each answers some physical questions, but one needs to know which probability to use (i.e. what information is relevant) to a particular situation. P(x) is relevant to motion in space not involving interactions. exp(ipx) is relevant to interactions (even with a magnetic vector potential causing no magnetic field) or with a 2-slit apparatus. Furthermore, in a quantum bound state, one has W*(x)W(x) = P(x) (W(x)=wavefunction =Sum over p a(p)exp(ipx)) and a*(p)a(p).=P(p). Each probability scheme corresponds to the analysis of a different kind of measurement, but various schemes may also be linked. Thus one may have to start with one to create the others. For example, exp(ipx) is the base for creating W(x)=Sum over p a(p)exp(ipx), which ultimately yields P(x)=W*(x)W(x). As a result, we argue that if probability exists in a system, it is not necessarily the case that there is a unique probability distribution. There may be various probability schemes linked with different information and different measurements/interactions. Thus, physics leads one to focus on a specific probability scheme. As a result, we argue that one must be careful when discussing pure statistical quantities and their extremization, because these quantities, like Shannon’s entropy and Fisher’ s information, contain a classical probability distribution, but they do not indicate which distribution to use in the case that a system has more than one. . As a result, the analysis of a system requires knowledge of the physics (interactions) occurring.We argue against thinking in terms of physics following directly from statistical considerations. The constrained extremization of one statistical quantity may yield a probability distribution related to one piece of information, but not to another. Furthermore, Shannon’s entropy and Fisher’s information pertain to classical probability 0<=P(x)<1 and Integral P(x)dx =1. The wavefunction, exp(ipx)/sqrt(L), is comprised of two probability distributions and ADD and MULTIPLIES statistically (i.e. OR and AND ) situations, but is involved in relative average calculations because it is not normalized. It is also different because it allows for positive and negative values. These are properties which do not follow from statistics using a classical P(x).
The notion of entropy originates historically in classical physics and is intertwined with thermodynamics. Shannon’s entropy: - Sum over i P(i) ln(P(i)), where P(i) is the probability for i, however, seems to apply to any probability, but in quantum mechanics appears to be particularly linked to spatial and momentum entropy (1) i.e. - Integral dx W*(x)W(x) ln[ W*(x)W(x)] and - Sum over p a*(p)a(p) ln[ a*(p)a(p) ] where W(x)= wavefunction = Sum over p a(p) exp(ipx). In this note we speculate that one may create an entropy from the real part of exp(ipx) i.e. cos(px) as this term is associated with probability within a wavelength even though the “center of mass” of the particle may be moving uniformly at v=x/t=constant and W*(x)W(x)=1 or a constant. The latter implies an entropy independent of p but there is uncertainty in a free particle due to its wavelength which is experimentally manifested. We argue there is no reason to limit entropy to x or p considerations. For a bound state, a pattern of crests and troughs of different lengths and heights may be produced and so again there is a pattern of unnormalized probability distributions with different weights. The absolute value may be taken so negative wavefunction values are not an issue. For both the free particle and the bound state two features emerge. First the shape of the distributions is linked to the curvature which in turn is associated with the kinetic energy. Thus the probability distributions which describe the quantum system are governed by kinetic energy. Secondly, a pattern of distributions, both negative and positive, ensures that exp(ipx) is perpendicular to other exp(ip1 x) functions and Wn(x) is perpendicular to other Wn1(x)’s. In other words energy identifies the statistical state which exists throughout space and one cannot confuse the identity of one state with another. Thus they are orthogonal throughout space and probability distributions (linked to entropy) seem to enforce this. The inner product of W*(x) and W(x) is like an inner product of a product probability of a state and its time reversed state. We suggest one should be able to define a Shannon’s entropy expression for both |cos(px)|/C1 from exp(ipx) over a half wavelength where C1 is a normalization factor and |W(x)|/C2 roughly over the length over which it has humps i.e. roughly the classical turning points for an oscillator for example. This entropy is not additive and does not map to classical entropy in the large n (energy level) limit. Furthermore it may not be linked to thermodynamics because it is not linked to x or p. Nevertheless, it might be used to describe uncertainty within the quantity system and equilibrium/stable scenarios which are observed in nature. For example, different Wn(x) states are statistically stable states, but one may knock out the particle and it will be in an exp(ipx) state which is also stable. Furthermore, a photon may excite Wn1(x) to Wn2(x), thus this new entropy seems to be linked to stable states allowed in nature. The “price to pay”, however, seems to be uncertainty in wavelength regions for a free particle and the crest/trough regions for a bound state. We also consider some properties of a similar entropy for bound states.