In information theory, ln(Probability) is called information. We argue that in various physical problems this information is linked with a conserved quantity and a variable which changes. For example, in the Maxwell-Boltzmann case the conserved quantity is 1/Temperature and the variable e+V(x) = energy. For a quantum free particle, the conserved quantity is momentum and the variable x. “e” may change subject to the conserved quantity 1/T and x may change subject to the conserved quantity p. Given that probability may be unnormalized, changes may be expressed in relative terms i.e. in terms of a derivative of ln(P) i.e. information. We argue that the derivative of ln(P) with respect to the variable which changes yields the value of the conserved quantity in special cases for which the conserved quantity is the only information governing changes in the probability distribution. Thus one has an eigenvalue equation: d/dvariable ln(P) = conserved quantity. This implies that ln(P) = variable * conserved quantity. This may lead to two different kinds of conservation equations. In the MB case, e is the variable and 1/T the conserved quantity, but ln(P(e1)) + ln(P(e2)) = 1/T (e1+e2) = ln(P(e1+e2)). An eigenvalue equation allows for this conservation. In the quantum free particle case: ln(P(k1,x)) + ln(P(k2,x)) where P=wavefunction = exp(ipx) (i.e. the square-root type probability is used), one has k1+k2=k3 with x the variable canceling out. (In the MB case, it was 1/T the conserved quantity canceling). Again, the eigenvalue equation allows for this conservation, this time of the conserved quantity. We note that the eigenvalue equation is a special case and not only probability distributions are associated with it. For example, a power law distribution’s changes in its variable do not depend only on the conserved quantity 1/T in P(e/T)=power law.
In information theory, ln(Probability) is called information. We argue that in various physical problems this information is linked with a conserved quantity and a variable which changes. For example, in the Maxwell-Boltzmann case the conserved quantity is 1/Temperature and the variable e+V(x) = energy. For a quantum free particle, the conserved quantity is momentum and the variable x. “e” may change subject to the conserved quantity 1/T and x may change subject to the conserved quantity p. Given that probability may be unnormalized, changes may be expressed in relative terms i.e. in terms of a derivative of ln(P) i.e. information. We argue that the derivative of ln(P) with respect to the variable which changes yields the value of the conserved quantity in special cases for which the conserved quantity is the only information governing changes in the probability distribution. Thus one has an eigenvalue equation: d/dvariable ln(P) = conserved quantity. This implies that ln(P) = variable * conserved quantity. This may lead to two different kinds of conservation equations. In the MB case, e is the variable and 1/T the conserved quantity, but ln(P(e1)) + ln(P(e2)) = 1/T (e1+e2) = ln(P(e1+e2)). An eigenvalue equation allows for this conservation. In the quantum free particle case: ln(P(k1,x)) + ln(P(k2,x)) where P=wavefunction = exp(ipx) (i.e. the square-root type probability is used), one has k1+k2=k3 with x the variable canceling out. (In the MB case, it was 1/T the conserved quantity canceling). Again, the eigenvalue equation allows for this conservation, this time of the conserved quantity. We note that the eigenvalue equation is a special case and not only probability distributions are associated with it. For example, a power law distribution’s changes in its variable do not depend only on the conserved quantity 1/T in P(e/T)=power law.
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.
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.
Information theory, which applies to compressing messages, suggests that ln(probability) is information. Shannon’s entropy is -1 times the average information i.e. -Sum over i P(i)ln(P(i)). For a classical gas in the canonical approximation, this is of the form of an average conserved quantity e.g Eave= Sum over i ei P(ei). In deterministic classical mechanics one may characterize an elastic two body collision by e1+e2 = e3 + e4. In a Maxwell-Boltzmann (MB) gas these same deterministic reactions occur, but there is also a static probability P(ei). The question seems to be whether P(ei) brings any new “physical” information into a scattering reaction. One already knows that a particle contributes energy ei. Is there something else brought in? If the answer is no, then using the product of probabilities P(e1)P(e2) and taking ln leads to ln(P(e1)) + ln(P(e2)) suggests that ln(P(e1))= A (e1-B) where A and B are constant. Thus information considerations link probability P(ei) to a deterministic quantity ei. One does not have to consider reaction balance or the maximization of entropy subject to a constraint. One simply finds the appropriate probability Pappropriate(e1) which describes an overall change to the state e1. For the MB case, P(appropriate)(ei) = P(ei) (MB) which gives a static description of the number of particles with ei. In other cases, however, there may be bias. Consider quantum scattering of fermions or bosons. If the collisions are elastic, one again has the classical deterministic e1+e2=e3+e4 expression. There is also a static P(ei) (Fermi-Dirac) for fermions and P(ei) Bose-Einstein for bosons. One is, however, interested in collisions which create and maintain the equilibrium situation. If there is bias in these collisions, then Pappropriate(ei) may differ from P(ei). The information e1 is the full contribution of particle 1 to the collision. Thus one also needs a full probability describing probability movement out of state e1. P(e1) is proportional only to the probability flow out of e1 due to scattering, but there is a “bias” factor namely 1-P(e1) which prevents another particle from entering state e1. This biases a reaction. As a result Pappropriate(e1) = P(e1)/ (1-P(e1)) for a fermion. This may be seen from a reaction balance equation:P(e1)(1-P(e3)) P(e2)(1-P(e4)) = P(e3)(1-P(e1))P(e4)(1-P(e2)) for fermions. For bosons there is an enhancement. One may obtain product probabilities by forming P(ei)/(1-P(ei)). This is the full contribution in terms of probabilities of particle i to a collision and so its ln should equal A(e-B)i which is its full information contribution to the reaction. If the two are not equal, it means there is more information (excluding parameters such as T temperature and u the chemical potential) than ei in the problem.
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.
In part I we argued that Newton’s first law implies that constant momentum (i.e. momentum conservation) is associated with spatial invariance i.e. a lack of information in a certain direction. This is equivalent to stating there exists no force in that direction. We argued that one may apply the generator for translation in the spatial direction i.e. d/dx for x and apply it to a function f(x) such that df(x)/dx = (px). The simplest form is f(x)= (px)x. We also argued that it is precisely this function which appears in Fermat’s least time principle because time = distance/speed and one may multiply time by |p| = momentum magnitude. Given that momentum and displacement are parallel vectors for a light ray, p dot r = p distance = (px)x + (py)y for a two dimensional problem (e.g. reflection or refraction). “Distance d” is the key function to use because dd= xx + yy implies that for fixed y, d d/dx = x/d = sin(theta1) (angle measured from y axis) so p d d/dx = (px) which is a conserved quantity if x is invariant. Thus conservation of momentum for two rays is equivalent to extremization. Extremization, we argue, suggests there may be a built-in physical mechanism which allows a free particle to correct itself if perturbed slightly just like a Maxwell-Boltzmann gas which extremizes entropy subject to constant average energy does. This built-in mechanism should be linked to motion in space. Writing (px)x as ln(exp(ipx), one not only obtains an eigenvalue equation i.e. -d/dx exp(i (px) x) = (px) exp(i px x), but the notion of a relative change arises if exp(i px x) is interpreted as a kind of probability. A relative change is natural because an absolute one ignores the idea that probability may differ with x even though (px) is constant. Thus from the idea of invariance in space found in Newton’s first law, which is equivalent to conservation of momentum in that direction, the equation d/dx f(x) = (px) leads directly to -id/dx exp(i px x) = (px) exp(ipx x) where exp(i px x) represents a physical probability type of wave which stablilizes the constant (px) motion because this motion is equivalent to extremizing time or |p| time or p dot r / speed of light in Fermat’s principle. In other words, the quantum mechanics of a free particle is associated with stabilizing wave behaviour exp(ipx x) which was not seen experimentally at the time of Newton or Fermat, but seems to be part of their formalisms, with Fermat’s principle derivable from Newton’s first law.
For the case of physical quantities with units, such as mass, time, length etc, a change in the value of the quantity makes sense by itself in addition to a relative change. In other words, one does not need to know the mass of an object to understand what a change of 1 kg is. Probability, on the other hand, is a strictly relative quantity and so needs to be normalized. If it is not normalized, one needs to know relative probabilities. To say that an unnormalized probability changes by a value of 10 does not mean anything. Given that equations often solve for an unnormalized probability function, we argue that the notion of the relative probability should appear in equations. Even in the case of ln(P(y)), called information in information theory, for which there is no derivative, taking a derivative, say d/dy, immediately creates a relative probability derivative. Thus the form ln(P) is in keeping with the notion of a relative derivative. Here we examine the idea of a relative derivative and its association with information ln(P), Fisher information, quantum mechanics and the Fokker-Planck equation as well as classical statistical balance in the presence of a potential.