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Modeling the Aggregate Structure of Configuration Space

I. Introducing Absolute Zero into the Aggregate Set of All Possible States

Devin Harris
July 14, 2003

1. The Ultimate Singularity

In past attempts to track the course of time cosmologists knew only that after the big bang the Hubble expansion established a decreasing momentum near enough to a declination toward absolute zero that the critical density Ω appeared to equal one. With the course of time indeterminately riding the dividing line between a big crunch future and an endless dissipation of heat there was no pronounced indication of an underlying intent to the evolution of the universe. Since 1998, studies of distant type 1a supernovae [1][2] combined with studies of the temperature variations in the cosmic microwave background (CMB) [3][4][5][6][7][8] have revealed that neither scenario accurately depicts our future. After decelerating for nearly eight billion years, it is now apparent that a dark energy force emerged approximately six billion years ago, beginning a unique period of cosmological evolution, where expansion accelerates rather than decelerates.

Presently considering a more fully exposed momentum of cosmological time, and measurable certainty that space-time will end in either finite or infinite time at the state of absolute zero, my proposal here is that absolute zero be recognized as a boundary state existing in aggregate state space, located beyond the bulk of all states of greater disorder [9][10][11]. Although the common point of absolute zero for all measures in physics, including mass, energy, density, gravity, and temperature, has not previously played a major role in cosmological theory, there can no longer be any question as to the intent of cosmological evolution. The role of zero in cosmology now appears equal to that of the big bang's origin at an infinitely dense singularity. Time distinctly appears to begin at one extreme of nature and now appears to end at the other.

My position will be that the physical equivalent of absolute zero is a four dimensional flat space singularity referred to here as a state of absolute flat space (AFS), with properties to be presented that have not been previously considered. I suggest that cosmic acceleration plainly indicates AFS is the primary attractor in aggregate configuration space. I shall explain that the crossover point between decelerating and accelerating expansion can be recognized as the beginning of a forced convergence toward AFS, and will conclude by explaining that the four dimensional properties of an AFS are responsible for accelerating expansion.

The first step is to generally consider the state space or configuration space boundary issues of this late time acceleration in the distant future, comparing them to similar state space boundary issues long recognized in our past. The deceleration of the big bang in reverse time accelerates toward a single high energy extreme state where all of space-time collapses into a common point. Considered in reverse, all possible paths converge in state space toward that extreme. This late time acceleration period in our future is nearly identical in that it is riding a critical density toward becoming the opposite extreme, where space is perfectly flat and smooth. Similar to the decreasing measure of states surrounding the singularity at the beginning of time, the position here is that configuration space is also shaped by a decreasing measure of states surrounding zero, and thus paths are forced to converge in state space in any declination toward zero.

In reference to all states in an aggregate configuration space, the collapse of space-time volume at or near the beginning of time, sometimes referred to as the alpha state, is generally assumed to be the highest state of order, if only because it is the extreme of possibility in physics in the direction of increasing order and low entropy. Once all known matter and energy is collapsed to a point of zero volume, physical descriptions end, and thus beyond an infinitely dense singularity [12][13] it is assumed here that no other possibilities exist. Likewise, here agreed to be the more ultimate singularity, an absolute zero temperature and density is the end of all scale, and also is an ultimate boundary limit in aggregate state space, beyond which no other possibilities exist, with the only exception to follow.

Ludwig Boltzmann was the first to model the realm of all possibilities. In developing his statistical approach to the second law [14] Boltzmann introduced probability into physics by utilizing an assumed axis between order and disorder . Explaining entropy and the most general evolution of patterns in nature, Boltzmann determined that the disorder of a closed system increases due to a greater measure of disordered states compared to ordered states. It follows logically from Boltzmann's approach that along an axis the number of ordered states decreases toward an ever fewer measure of ordered states, while in the opposite direction the measure of increasingly disordered states increases. Referred to here as the wedge model, the large-scale structure of states has been reservedly portrayed as closing at the end of highest possible order at a single extreme state, while in the direction of increasing disorder the general assumption is of an endless and indefinite expansion of states without end. A wedge shaped order gradient has become a general description of reality in physics which describes the logical pattern evolution aspect of thermodynamics [15], as well as the universal arrow of time [16][17][18]. Sir Arthur Eddington first proposed a relationship between Boltzmann's applications to closed frames of reference and postulated an aggregate influence to the general direction of time [19], concluding that the second law designates a heat death ending to the universe.

The second law implies that disorder will increase only if the system originates in an initial condition of high order. If we were to treat absolute zero as the lowest possible order, the question arises as to whether a system near zero existing in a condition of extremely low order should be expected to probabilistically gravitate toward greater order. What I mean this question to highlight is that if there is both a high order boundary and a low order boundary in an order gradient there would logically exist a balance somewhere in state space between the two extremes of alpha and zero, meaning a point where a set of states of greater order is equal to a set of states of lesser order. The quantity of disordered states is then not necessarily always greater and the influence of the body of disordered states should decrease as a system approaches the ultimate balance between all states. We can therefore recognize a center position within the order-disorder axis which logically should act as a universal basin of attraction for all dynamic systems.

Yet if we consider the evolution of space-time, there is no evidence to suggest the general direction of time is being drawn toward any such basin of attraction balanced between alpha and zero, or between the highest and lowest states of order. Instead indications are becoming ever more pronounced that the arrow of time is being directed precisely at zero, so that we must be highly curious about the attractive properties of AFS in the set of all possible states (SOAPS). While maintaining Boltzmann's general approach, we must speculate on how the SOAPS might be expanded so that the momentum of time toward AFS, as well as decreasing order and increasing entropy, can still be understood to be attributed to state space.

The most simple explanation for an arrow of time directed precisely at zero would be that a reciprocal negative set of states extends beyond zero. The overall structure of states would thus include an inverse set of patterns, similar to the extension of negative numbers beyond zero in the mathematical plane of real numbers, which are identical yet opposite. My central proposal is that absolute zero is the ultimate balance in aggregate state space and thus the primary attractor in the set of all possible states for all temporal systems, and that an inverse set of states verifiably explains the arrow and predictable course of time. This solution might feel intuitively satisfying since it eliminates the previous asymmetric version of state space. It also carries with it many implications so it will easily be evidenced or disproved. Before I more fully introduce the influence of an inverse set of patterns, it will benefit my argument to first point out that there also exists extremes adjacent to the axis between the alpha state and zero.

2. Adjacent Extremes

In this third modification it is necessary to suspend the use of an order-disorder axis and instead utilize a gradient based on the average density of a state so that we can consider the groups of states which exist adjacent to a more fundamental and imaginable axis of average density. At any point along an average density gradient (ADG) excepting the alpha state and AFS there exists a quantity of states greater than one and necessarily infinite. However, this adjacent axis is also bounded by extremes which have natures related to the alpha and zero states. What I mean here is that there are extreme possibilities of perfectly smooth versus what I can initially only refer to as radical lumpiness. These extremes relate to the ordinary concept of contrast, where color tones are either blended into a single averaged color (low contrast) or the color tones of the image blend into two opposing shades of light and dark (high contrast). We can therefore identify a single extreme of smoothness and using the contrast gradient further hypothesis a single extreme of lumpiness even if such a state is initially difficult to envision. It is suggested here that no possibilities beyond the smooth and lumpy extremes of the contrast gradient are describable by physics or even imaginable.

A top-down perspective of physical reality has been advocated by Piet Hut [20] and recently by Stephen Hawking [21]. The proposal here is that boundary states exist in all directions of state space making the SOAPS definitive as opposed to indefinite. It follows that the realm of all possibilities is not open, as the image below represents.

We can apply this gradient of low to high contrast to the question of why the early universe did not remain smooth. With similar reasoning as Boltzmann's original postulate, we can conclude that the maintenance of a smooth universe during expansion is statistically near impossibility. The universe remaining perfectly smooth is one possibility among many other possible states where space does not remain smooth. The suggestion here is that the path of a dynamic system will follow the basin of attraction balanced between the two contrast extremes, and that the measure of density variation detected since the big bang, and the present distribution of galactic matter, is congruent with this phase space in the contrast gradient which expands and contracts in respect to our position along the density gradient.

If the second law effectively explains why an ordered system decays toward disordered states, why then do ordered systems emerge and why are they maintained? As portrayed, such a model of all states indicates the forces of nature shape the universe and maintain or increase order against the probabilistic trend toward disorder. Given that a study of macro-state structure was successful in providing what is perhaps the only major solution to why the universe evolves as it does, and given that the general wedge model is arguably very vague and rarely scrutinized, perhaps there should be far greater focus placed on other possible applications. We might consider even that a fully accurate model of all states would indicate why there are laws and forces of nature responsible for order and systemization. That is actually the intent of this paper, to indicate that we are set to answer the why questions, even to discover that all of physics is accountable to the probabilities of configuration space.

3. General Probabilities

We can now begin a study of how the general probabilistic features of the proposed model would influence space-time, utilizing the same rational as it has been held that a greater number of disordered states influences a system. The diagram below represents the position of a partially evolved system in respect to the model of aggregate state space proposed. Set B includes all states which are less positive than the present state of the system, which we can refer to as the Beta set. Note that when located at the positive extreme, set A includes only that one single state while set B included the total of all other states. Elsewhere set A includes all states which are more positive than the present state of the system. Consequently set A is always the lesser and set B dominates throughout the course of time over the alpha set, until the two sets become equal as the system approaches zero. Although less forceful, the influence of set A, which is not considered in Boltzmann's approach, is here recognized as having considerable influence upon all systems.



The more obvious probabilities in this model indicate that as space-time enters its state space, all probability is for time to move away from the positive extreme. An unconflicted probability without needing to escalate, would drive the system into its state space toward the negative side, in effect by neutralizing the extremity of the original positive state with an isotropic negative influx. As the system rapidly changes its location in configuration space the momentum toward zero immediately decelerates, slowed by the growing gravitational pull of set A, and also moderately slowed by a growing measure of possible states adjacent to the contrast gradient's basin of attraction. The pathways through the first quarter of state space are recognized as the Period of Divergence. Then as any space-time system crosses the mid point of its native body (positive group) of states it then enters a Period of Convergence. In this volume of state space, the adjacent possible begins to decrease and all previously diverging paths of time are aligned into radial trajectories toward the single omega state of zero.

Boltzmann�s logic indicates a system selecting from static patterns which themselves are not evolving, and in that the process requires time to begin in a state of high order, indicates the early universe as existing in a highly improbable state. This modified state space model alternatively was developed by assuming a block universe view [9][10][11] in which case a system selects from a set of possible patterns quiescently existing in physical form. As Boltzmann recognized, each pattern, each unique arrangement or distribution of matter is equally probable. However, unique from Boltzmann's approach, the logic here actually expects that observers positioned within a universe temporally structured by probability would detect that time originates from an ever more extreme imbalance. Looking into the past an observer will observe an evolution related to an increased measure of states in set B, and ever fewer states in set A. Rather than a logic that indicates the universe began in an improbable state of existence, observers equipped with this model of states even expect that a temporal evolution would most naturally originate with the universe accelerating at the most extreme rate allowed by nature away from the most improbable state, directly toward the most probable, or whatever balance dominates state space, slowing at an ever decreasing rate, to eventually come to rest in the single most probable state.

The modifications so far include introducing zero, the inverse group of states, the contrast gradient, and the converging structure of state space near zero. Before continuing I want to make note of how the model thus far resolves major inconsistencies between the most basic physical properties known in physics and how we presently model all possible states based upon Boltzmann's vision of an order-disorder axis. A model of all states should exactly reflect physics, and although Boltzmann's approach is based conceptually upon order and disorder, all potential states should be at least generally locatable if the model is genuinely effective in describing all states. Absolute zero is unquestionably one of the most basic attributes of physical existence, and the absence of any established location within aggregate state space of absolute zero, or a gradation to zero, in the wedge model, parallels with its most irreconcilable feature, the endless indefinite extension of increasingly disordered states. This point can turn our focus now on how to re-integrate order and disorder into this evolved model, which is far more challenging that one might expect, since although zero is clearly the highest possible state of entropy, the most distinct property of an absolutely flat space is perfect symmetry, which obviously contradicts any designation of zero as the lowest possible order.

To explain the most significant modification brought about by the introduction of absolute zero to the SOAPS I must also propose a modification to how we understand order and disorder. There is a mistake being made in how we conceptualize order highlighted by this advancement in modeling state space, and that misunderstanding is the very reason science is yet unable to understand why the universe is ordered and systematic. In fact what follows is probably the more consequential material of this paper.

II: Defining Two Opposing Types of Order
 

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Copyright © July 14, 2003 by Devin Harris