Probability allows predicting the most and least probable outcomes. However, the probability of an outcome is affected by the physical quantities that describe the universe. The certainty of a single outcome and uncertainties of many outcomes are determined by how uniformly a scalar field (i.e. Potential field, entropy, mass) is distributed over an entity. It is known that an increase in entropy increases the likelihood. In this paper, this knowledge is taken one step further to understand the likelihood of the possible outcomes within an entity which have either a uniform or non-uniform scalar field. Uniform scalar fields over an entity have net-zero scalar field value. An example of Uniform scalar fields over an entity is rolling an unloaded dice where every individual six outcomes have an equal likelihood. Uniform scalar fields over an entity are where most uncertainty occurs as all outcomes have an equal likelihood. The non-uniform scalar field over an entity is where there is most certainty towards a single outcome. For example, a loaded dice has the highest probability for a single outcome. The theoretical model created in this paper is based on two square six by 6 cm dice, where one die is loaded non-uniformly with different chemical molecules of different entropy and mass value and represented with contour lines in a contour map. Another dice is loaded uniformly with the same chemical molecule of the same entropy and mass all over the dice. As the distribution of the chemical molecule is uniform, this configuration represents an unloaded die. Neither entropy nor mass scalar fields alone are capable of determining the outcome of the dice alone. The outcome is also determined by the type of external force, energy acting on the entity (i.e. dice), and the definition of probability. All in all, the Important result is that regardless of the definition of probability, type of external force, energy, or internal scalar field within the entity, the most probable outcome, and the least probable outcome are determined and connected by the gradient of the scalar field (i.e. Gradient of Entropy, ∇ S ) within the entity.
Probability allows predicting the most and least probable outcomes. However, the probability of an outcome is affected by the physical quantities that describe the universe. The certainty of a single outcome and uncertainties of many outcomes are determined by how uniformly a scalar field (i.e. Potential field, entropy, mass) is distributed over an entity. It is known that an increase in entropy increases the likelihood. In this paper, this knowledge is taken one step further to understand the likelihood of the possible outcomes within an entity which have either a uniform or non-uniform scalar field. Uniform scalar fields over an entity have net-zero scalar field value. An example of Uniform scalar fields over an entity is rolling an unloaded dice where every individual six outcomes have an equal likelihood. Uniform scalar fields over an entity are where most uncertainty occurs as all outcomes have an equal likelihood. The non-uniform scalar field over an entity is where there is most certainty towards a single outcome. For example, a loaded dice has the highest probability for a single outcome. The theoretical model created in this paper is based on two square six by 6 cm dice, where one die is loaded non-uniformly with different chemical molecules of different entropy and mass value and represented with contour lines in a contour map. Another dice is loaded uniformly with the same chemical molecule of the same entropy and mass all over the dice. As the distribution of the chemical molecule is uniform, this configuration represents an unloaded die. Neither entropy nor mass scalar fields alone are capable of determining the outcome of the dice alone. The outcome is also determined by the type of external force, energy acting on the entity (i.e. dice), and the definition of probability. All in all, the Important result is that regardless of the definition of probability, type of external force, energy, or internal scalar field within the entity, the most probable outcome, and the least probable outcome are determined and connected by the gradient of the scalar field (i.e. Gradient of Entropy, ∇ S ) within the entity.
Definitions The concept of probability is of considerable importance in optics, as in any situation in which the outcome of a given trial or measurement is uncertain. Under these conditions it is desirable to be able to associate a measure with the likelihood of the outcome or the event in question; such a measure is called the probability of the event. Several different definitions of probability have been adopted at various times in the past. The classical definition is based on an exhaustive enumeration of the possible outcomes of an experiment or trial. If the trial has N distinguishable, mutually exclusive outcomes, which are equally likely to occur, and if n out of these N possible outcomes have an attribute or characteristic that we call ‘success’, then the probability of success in any one trial is given by the ratio n/N . For example, if we roll a die, and if each of the six digits is equally likely to be on top when the die comes to rest, there are N = 6 distinguishable outcomes. If we identify success with an even number, for example, then since there are three different ways in which success can be achieved, it follows that the probability of success when the die is rolled is given by 3/6 = 1/2. Unfortunately, an exhaustive enumeration of all possibilities is not always feasible. Another common definition of probability is based on the notion of relative frequency of success.
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 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).
Local mathematics assumes the existence of number structures of different types, vector spaces, etc. localized at each space time point. Relations between number structures at different locations are based on two aspects: distinction between two so far conflated concepts, number and number value and the "No information at a distance" principle. This principle forbids the choice of the value of a number at one location to determine the value of the same number at another location. Value changing connections, related to a real valued field, g, move numbers between structures at different locations. The effect of the g field, or its exponential equivalent, g(y)=eα(y), on numbers extends to other mathematical structures, vector spaces, etc. The presence of α affects theoretical descriptions of quantities in physics and geometry. Two examples are described, the effect on the Dirac Lagrangian in gauge theory, and the effect on path lengths and distances in geometry. The gradient field of α, A, appears in the Lagrangian as a spin 0, real scalar field that couples to the fermion field. Any value for the mass of A is possible. The lack of direct experimental evidence for the presence of the g or α field means that the field must be essentially constant within a local region of the cosmological universe. Outside the local region there are no restrictions on the field. Possible physical candidates, (inflaton, dark matter, dark energy) for α are noted.
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.
We have argued in previous notes that C(T) exp(-ei/T) (the Maxwell-Boltzmann probability) and exp(-iEt) exp(ipx) (the free particle quantum probability called a wavefunction) have the same informational structure in that product probabilities for a given E1=Ei+Ej for the MB case and Ei,pi and Ej, pj for the quantum case lead to uniform distributions, i.e. ones with minimal information if energy and momentum are conserved. We suggest here that the root for this may be same. We have argued in previous notes that dp/dt = Force cannot have dt be infinitesimally small. This means that there exists a scale at which dt is not 0 or even tiny. We also suggested that the force (even though obeying action-reaction in collisions) may strike at some probabilistic time within dt. We then used special relativity to find that dt =hbar/E and dp=hbar/p. A probability which demonstrates these gradations is exp(-iEt+ipx) (and also Lorentz invariant). This is akin to the notion that p(e1)p(e2)=p(e3)p(e4) and p(p1)p(p2) = p(p3)p(p4) (one dimension for simplicity). In other words, the stochasticity introduced is one of a uniform distribution for product probabilities. One could obtain the same result by starting with this assumption and then suggesting Lorentz invariance. We suggest that this quantum uniform distribution product probability also holds for the Maxwell-Boltzmann case. There, the probability is not complex because p(ei) must be a real number linked to the number of particles with ei. Nevertheless, in an elastic collision of ei and ej, we suggest that the quantum result (and even dp/dt=F) suggest that there is stochasticity in x and t and that one cannot deterministically predict the outcome. Any outcome which conserves energy and momentum has the same likelihood. We suggest that it is the quantum mechanical assumption which is identical to the ideal gas case (which uses quantum mechanical gas molecules albeit at high temperatures). The high temperatures mean that one can only use the conservation laws and the associated stochasticity. We suggest that the stochasticity of interaction inherent in quantum mechanics carries over to collisions in the classical gas. It is not simply that one is approximating the situation in a gas by throwing away information. If one were to try to be as deterministic as possible at some scale (the quantum one) stochasticity arises and there is no full determinism in a collision. The best one can do is use conservation of momentum and sometimes conservation of energy in certain situations like high temperature. If this is the case, one may ask: Why is it possible to perform deterministic classical calculations such as particles bouncing off walls or mirrors or interacting with a potential V(x) in a completely deterministic manner. We suggest that these involve constrained interactions, i.e. the wall or V(x) is associated with a huge mass compared to the particle, unlike in a collision of two identical particles. Action-reaction in the former case is associated with the large mass and so always the same approximately 0 kinetic energy and hence no p(ei)p(ej) =p(ek)p(el), but in the latter, with a mass the same size and so the stochasticity may be seen clearly.