Reinventing Time
Introduction
Perhaps the most fundamental problem of physics is the nature of time. There is a serious tension between the empirical randomness and irreversibility of quantum observations and the fundamental laws of physics, which are blind to the direction of time. This conflict underlies quantum mechanics’ conceptual difficulties, and it drives the need to reinvent time.
In another post, Is Quantum Randomness Fundamental?, I describe two distinct times: mechanical time and thermodynamic time. Mechanical time is the time of classical, quantum, and relativistic mechanics. It describes change as deterministic and reversible. Thermodynamic time, in contrast, describes the irreversible production of entropy. Insofar as physics does not recognize entropy as a fundamental property, physics regards entropy and the arrow of time as emergent properties or as properties of our perception only.
In the same post, I introduced a contextual model of physical reality, in which entropy and the arrow of time are fundamental properties. A fundamental assumption of the Dissipative Conceptual Model (DCM) is that perfect reversible measurement from the system’s ambient surroundings completely defines a system’s physical state. The DCM contextually defines entropy as a fundamental property of state by:
where dq is an increment of heat added to the system at temperature T and increments are summed from the ambient temperature, Tₐ, to the system temperature, Tₛ. The DCM entropy is a generalization of statistical mechanical entropy, defined with respect to absolute zero. Absolute zero, however, is unattainable. Statistical mechanics is an idealized special case which sets the ambient temperature to absolute zero and eliminates thermal randomness.
Complex System Time
The DCM recognizes both mechanical time and thermodynamic time as physically real. It defines system time as a complex property of state, given by:
where i is the square root of negative one, tₜ is the real thermodynamic time and itₘ is imaginary mechanical time. Mechanics defines mechanical time as real-valued, but this is simply a matter of convention. The DCM adapts a mathematically identical convention but a different interpretation, by replacing real-valued mechanical time t with -i(itₘ). This convention adheres to the geometric interpretation of spacetime, in which the time axis is an imaginary coordinate, and it leaves the equations of mechanics unchanged.
DCM System time is represented by a point on the complex system-time plane (Figure 1A). The horizontal axis spans real-valued thermodynamic time and the vertical axis spans the imaginary and reversible mechanical time component.
A non-equilibrium thermodynamic system generally produces entropy continuously. For example, if a drop of ink is dropped into a glass of water, entropy is produced as the ink disperses. The system evolves irreversibly along the thermodynamic time axis until the ink is uniformly dispersed. At that point, entropy production ceases. The system reaches an equilibrium state and a static point on the thermodynamic time axis. No further change is possible.
Mechanical time describes the reversible changes in state for a friction-free mechanical system. The system produces no entropy, so its evolution is confined to a vertical line at a fixed instant of thermodynamic time. For a system comprising a single particle, we could describe the particle’s position as a function of mechanical time. If the particle is confined to a finite line interval, we could describe its position over a finite interval of time. When the particle bounces off its boundary and reverses its trajectory, its mechanical time would then also reverse.
System time, whether it proceeds reversibly or irreversibly, is empirically measured by the irreversible advance of reference time, as recorded by an observer’s reference clock (Figure 1B). The DCM decouples system time and reference time. Reference time marks the continuous and irreversible “flow” of time, and it provides the time scale across which empirical observations are recorded.
The deterministic laws of physics do not accommodate irreversible change. They describe states confined to a vertical slice of system time. When we observe irreversible change, such as a random quantum measurement result, physics has to resort to metaphysical gyrations. Does observation somehow instantiate random physical change, outside the scope of physical laws? Is reality only a matter of what we perceive? Are all possible outcomes manifested in separate branches of an exponentially branching universe? These ideas have all been considered to address the conflict between physical determinism and the empirical evidence for irreversible and random change.
The DCM provides a much simpler interpretation. An unstable quantum particle can exist temporarily as a metastable state between transitions, during which it can reversibly evolve along a vertical line over mechanical time. At some point, however, it randomly transition to a more stable state. During transition, entropy is produced and thermodynamic time advances. During the transition, the system is irreversibly changing. It is not static, it is not reversibly measurable, and it therefore does not exist as a state. Following the transition, it exists as a new metastable state, reversibly changing along a new vertical line at a later instant of thermodynamic time.
The Two Components of DCM Entropy
The advance of thermodynamic time is defined by the irreversible production of entropy, as expressed by the Second Law of thermodynamics. The Second Law is commonly described in two different ways: 1) heat irreversibly flows from hot to cold, and 2) exergy, which is a system’s potential to do work on its ambient surroundings, is irreversibly dissipated to ambient heat. The DCM recognizes each description as the production of distinct components of the total DCM entropy.
To illustrate these two components of entropy, we consider a fixed volume of an ideal gas at 600 kelvins absolute temperature and 10 atmospheres pressure (State 1 in Figure 2). The system is contextually defined with respect to its ambient reference state at 300 kelvins and one atmosphere (State 3 in Figure 2). The DCM resolves the total entropy into two components: ambient entropy (Sₐ) and the entropy of thermal refinement (Sₜᵣ) (Figure 2).
We first consider the entropy of thermal refinement. As the gas cools from State 1 at 600K to State 2 at 300K, heat is lost to the ambient surroundings. The incremental change in the gas’s entropy of thermal refinement is dSₜᵣ =dq/T, where dq is the increment of heat lost and T is the gas temperature as it cools. The change in entropy is the sum of these increments from 300K to 600K:
where Cᵥ is the heat capacity, defined by dq/dT at constant volume.
The gas at 600K has a positive entropy relative to the gas at 300K, and as it cools, its entropy declines. The entropy of the surroundings at fixed temperature Tₐ, however, increases by dSₛᵤᵣ=dq/Tₐ for each increment of heat exchanged. Since Tₐ is always less than or equal to the gas temperature, the flow of heat from the hotter gas to the cooler surroundings results in a net production of entropy, in accordance with the Second Law. As heat flows from hot to cold, it produces entropy of thermal refinement. As the gas cools, its pressure also declines, from ten atmospheres to five.
We next consider the change in the ambient entropy from State 2 to State 3. We open a valve and let the gas expand into a volume five times its original size. There is no exchange of energy between the gas and its surroundings, so the gas’s energy and temperature do not change. The pressure drops from 5 atm to 1 atm, in accordance with the ideal gas law.
Statistical mechanics describes the increase in entropy as the increase in probability of the gas expanding into the vacuum. The DCM describes the increase in entropy by the dissipation of exergy. The gas in State 2 has five atmospheres of pressure and a positive potential for work relative to the ambient reference. As the gas expands from State 2 to State 3, its exergy is dissipated and its ambient entropy increases, in accordance with the Second Law.
The Two Paths of Irreversible Change
Both dissipation and the flow of heat from hot to cold produce entropy and advances of irreversible time. Both processes illustrate the path of dissipation, whether it is the dissipation of ambient exergy or the dissipation of thermal exergy. Dissipation defines one path of irreversible change. Once exergy is fully dissipated, the system reaches equilibrium with its ambient surroundings. Zero exergy defines the ambient equilibrium state, from which there is no potential for further change.
Refinement defines a second path of irreversible change. Figure 2 and equation 3 show that if the ambient temperature declines, the system’s entropy of thermal refinement increases. To illustrate the path of refinement, we consider a gas initially in equilibrium with its ambient surroundings. While the gas is in equilibrium with its ambient surroundings, it has zero exergy and zero potential to change.
The gas particles’ positions are a contextual property defined by perfect measurement from the ambient surroundings. Perfect measurement cannot resolve the equilibrium gas particles’ positions; it can only define their positions as a probability distribution function spread uniformly across the gas’s configuration space spanning the position coordinates for the gas’s particles. The probability distribution function is a complete specification of the gas’s contextual property for the particle’s positions. The ambient gas’s temperature, pressure, volume, and the probability distribution functions for the particles’ positions and other contextual properties completely define the ambient gas’s physical state. The ambient equilibrium state is the only state consistent with its boundary constraints. There is no uncertainty in the system’s physical state and its entropy is therefore zero.
If the ambient temperature declines while the gas maintains its initial temperature, the gas’s exergy increases with respect to its cooling ambient surroundings. A positive exergy creates metastability and a drive to transition to a new ambient equilibrium state.
As the ambient temperature declines, perfect ambient measurement becomes more refined and fine grained, and the probability distributions defining the gas’s contextual properties become increasingly particle-like. In the limit of absolute zero ambient temperature, each potential ambient state describes a distinct configuration of precisely defined positions and motions. Refinement and increasing exergy create a drive to transition to a new potential ambient equilibrium state. The entropy of refinement measures the uncertainty in a system’s path of actualizing the present by the random selection of increasingly fine-grained and detailed potentialities.
Refinement and Cosmological Evolution
A common interpretation for the arrow of time is that it stems not from physical laws, but from a highly improbable initial state of the universe. According to the Past Hypothesis, the universe (or at least this particular universe) evolved deterministically from a finely tuned and highly improbable initial state to its current improbable state, in which we are here to ponder its origin. At the same time, it has evolved from an initial improbable state of low entropy to its current state of higher entropy, and, paradoxically, higher probability and randomness.
The DCM has a different take on the past hypothesis. The universe did start in a state of near-zero entropy, but it was a high-energy equilibrium state. As an equilibrium state, it existed in the only state available to it. No fine tuning was required. Expansion and cooling of the ambient cosmic microwave background then led to fine graining, and to the unfolding of the future by random actualizations of newly created potentialities.
The universe has evolved from its initial state of equilibrium and zero exergy to its current state of high exergy with respect to its cold ambient microwave background. Its positive exergy drives irreversible dissipative processes. As described in The Arrow of Functional Complexity, preferential selection of dissipative processes with higher utilization of exergy leads to the spontaneous organization of dissipative structures. As long as the universe expands and the cosmic microwave background temperature continues to fall, it will continue to evolve, without ever reaching an equilibrium state of zero-exergy heat death.
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