Meta Theory of the Zeckendorf-Hilbert Universe

9.3 Thermodynamic Time Arrow: From Density of States $\rho (E)$ to Partition Function and Locking of Entropy Increase Direction

In Chapter 8, we established the microscopic definition of time: time flow rate $\kappa (E)$ equals density of states $\rho (E)$. In Sections 9.1 and 9.2, we applied this microscopic time to the dynamics of expanding universe. However, dynamical equations (whether Schrödinger equation or Einstein equations) are usually time-reversal symmetric at the microscopic level. This leads to one of the most famous paradoxes in physics: if microscopic laws are reversible, why does macroscopic world time always point to the future (entropy increase direction)?

This section will prove that, under the framework of unified time theory, the thermodynamic time arrow is not an additional assumption of statistical mechanics but a natural property of spectral geometry. Since $\kappa (E)=\rho (E)$, the direction of time flow is locked to the direction of density of states growth. Macroscopic time arrow is essentially gradient flow of information capacity.

9.3.1 Thermodynamic Representation of Spectral Geometry: From Trace to Partition Function

First, we need to connect microscopic scattering time ($\text{Tr}\mathsf{Q}$) with macroscopic thermodynamic quantities.

In statistical mechanics, the core object is the partition function $Z(\beta )$ of canonical ensemble, where $\beta =1/k_{B}T$ is inverse temperature.

$$Z(\beta )=\text{Tr}(e^{-\beta H})={\int }_0^{\infty }\rho (E)e^{-\beta E}\mathrm{d}E$$

This shows that partition function is the Laplace transform of density of states $\rho (E)$.

Using unified time identity $\rho (E)=\frac{1}{2\pi \hslash }{\tau }_{tot}(E)$, we can rewrite partition function as integral of time delay:

$$Z(\beta )=\frac{1}{2\pi \hslash }{\int }_0^{\infty }{\tau }_{tot}(E)e^{-\beta E}\mathrm{d}E$$

This formula reveals the dynamical origin of thermodynamics: partition function is a weighted sum of all possible microscopic time delays of the system. The thermodynamic weight of the system at temperature $T$ depends on how long it can “dwell” at each energy layer.

9.3.2 Entropy as Logarithmic Time Density

Thermodynamic entropy $S(\beta )$ is defined by $S=k_B(\ln Z+\beta \bar{E})$. In microcanonical ensemble (fixed energy $E$), Boltzmann entropy is:

$$S(E)=k_B\ln \Omega (E)\approx k_B\ln \rho (E)$$

Substituting unified time identity:

$$S(E)=k_B\ln \left(\frac{{\tau }_{tot}(E)}{2\pi \hslash }\right)$$

Theorem 9.3.1 (Time-Entropy Equivalence Principle)

Microcanonical entropy strictly equals the logarithm of total scattering time delay of the system (plus fundamental constant terms).

$$S(E)\sim \ln {\tau }_{tot}(E)$$

This relationship applies not only to black holes (where $\tau \sim e^{S_{BH}}$) but to any statistical system. It gives an operational definition of entropy: entropy is a measure of system’s “viscosity” or “time impedance.”

• Low Entropy State: ${\tau }_{tot}$ is small, system responds quickly to external perturbations (“transparent”).

• High Entropy State: ${\tau }_{tot}$ is large, system has extremely rich internal states, external perturbations entering will experience long multiple scattering and entanglement (“turbid”).

9.3.3 Spectral Origin of Time Arrow: Hagedorn Growth and Asymmetry

Why does time always flow forward (entropy increase)? This depends on how $\rho (E)$ changes with $E$.

For most physical systems (QCA networks, field theory, black holes), Hilbert space dimension grows exponentially with energy (or volume). For example, Hagedorn spectrum in string theory or QCA:

$$\rho (E)\sim E^{\alpha }{e}^{{\beta }_{H}E}$$

This means ${\tau }_{tot}(E)$ is a strongly increasing function of energy.

Definition 9.3.2 (Time Asymmetry of Spectral Flow)

Examine diffusion of a wave packet in Hilbert space. Due to exponential growth of $\rho (E)$, the high-energy end (or high-entanglement end) of state space is much larger than the low-energy end.

According to Fermi’s golden rule, transition rate $W_{i\to f}\propto |M_{if}{|}^2\rho (E_f)$.

Even if microscopic matrix elements $|M_{if}{|}^2$ are symmetric, transition probabilities are greatly weighted by final state density $\rho (E_f)$.

$$\frac{P(E\to E+\delta E)}{P(E+\delta E\to E)}=\frac{\rho (E+\delta E)}{\rho (E)}\approx e^{{\beta }_H\delta E}\gg 1$$

This is the microscopic mechanism of thermodynamic time arrow: systems tend to evolve toward regions with higher density of states (i.e., larger time delay), purely because there are more “rooms” there.

Corollary 9.3.3 (Time Deceleration Effect)

As the universe evolves, entropy $S$ increases, meaning ${\tau }_{tot}$ increases.

This leads to a counterintuitive but profound conclusion: the intrinsic time flow rate of the universe is gradually slowing down.

Early low-entropy universe, physical processes (interactions) occur extremely fast ($\tau$ small); late high-entropy universe (filled with black holes and radiation), physical processes become extremely slow ($\tau$ large). We feel time flowing “uniformly” because our biological clocks are also made of the same matter, we are synchronously decelerated.

9.3.4 Entropy Increase Locking in Expanding Universe

In Section 9.1, we saw that cosmic expansion introduces non-Hermitian dissipation. How does this dissipation agree with the above entropy increase?

Consider horizon $\partial V$. Horizon is a sink of information. As $H$ decreases (end of inflation or matter-dominated period), horizon radius $R_H=c/H$ increases.

According to holographic principle, the maximum information capacity that horizon can accommodate (i.e., maximum potential entropy of entire universe) is $S_{max}\propto A_H\propto H^{-2}$.

Meanwhile, matter entropy $S_{matter}$ inside the universe is also increasing.

The second law of thermodynamics in cosmology is stated as increase of generalized entropy:

$$\frac{d}{dt}(S_{matter}+S_{horizon})\ge 0$$

Since horizon expansion sweeps over more lattice sites, ${\rho }_{total}(E)$ (effective density of states of entire universe) monotonically increases with time $t$.

This cosmological growth of density of states locks the time arrow: as long as the universe is expanding (or horizon is expanding), the system is always in a non-equilibrium state with “continuously growing phase space volume,” thus driving the system to diffuse into larger phase space.

Conclusion 9.3.4

Time arrow is not an accidental choice of initial conditions (past hypothesis) but an inevitability of QCA geometric evolution:

1. Microscopically: $\tau \sim \rho$.

2. Structurally: $\rho (E)$ grows exponentially with complexity.

3. Macroscopically: System evolution is probability flow toward high $\rho$ (high delay) regions.

Time flows forward because the future contains more microscopic states than the past.

In the next section, we will explore the ultimate corollary of this logic: if vacuum itself has non-zero density of states, it will manifest as a repulsive force—this is the geometric nature of dark energy.