2.1 The Thermodynamics of Parting: The Essence of Time’s Passage Is the Orthogonalization of Microscopic States. Each Second’s Passing Is an Eternal Farewell to the Previous Microscopic State. Entropy Increase Is the Ever-Growing “Never to Return.”

In Chapter 1, we explained inertia as attachment to the “unchanging.” However, no matter how hard we resist inertia, time continues to flow. The deepest experience this flow brings us is often not the joy of “progress,” but the sorrow of “loss.”

Why is time unidirectional? Why do we remember the past but cannot remember the future? Why does every farewell carry an irreversible finality?

This section will re-examine the second law of thermodynamics from the microscopic perspective of QCA. We will prove: The arrow of time is not determined by probability, but driven by the “orthogonalization” of quantum states. This orthogonalization makes every moment a unique, unrepeatable “limited edition.”

2.1.1 The Uniqueness of Microscopic States

In classical mechanics, a state is a point $(p,q)$ in phase space. If we reverse all particles’ velocities, time flows backward, and a broken cup restores itself.

But in the QCA universe, although the underlying evolution operator $\hat{U}$ is unitary (reversible), for a macroscopic observer (Agent), the evolution of microscopic states is orthogonalization.

According to the Margolus-Levitin theorem, the minimum time required for a quantum system to evolve from state $|{\psi }_0⟩$ to an orthogonal state $|{\psi }_{\perp }⟩$ is determined by its average energy $E$:

$$t_{orth}\ge \frac{h}{4E}$$

This means that as time passes, the universe’s wave function $|\Psi (t)⟩$ continuously enters new, unexplored dimensions in Hilbert space.

• The universe at $t=1$ and the universe at $t=0$ are not merely “different”; they are mutually exclusive in some geometric sense (inner product approaches 0).

• Every “now” is an overwrite of the “past.”

2.1.2 Entropy Increase as a Measure of “Forgetting”

What we call “entropy increase” means information loss in information theory.

For a finite observer (with $M_I\ll M_{universe}$), they cannot track all microscopic particles’ trajectories.

When a cup breaks, the information about “cup shape” does not disappear (according to unitarity); it merely diffuses into the environment’s microscopic degrees of freedom (phonons, thermal radiation), becoming entanglement entropy that the observer cannot read.

Definition 2.1 (Physical Definition of Parting):

Parting is the irreversible decline of mutual information $I(A:B)$ between two subsystems.

$$I(A:B)\to 0$$

When mutual information falls below a certain threshold, not only are you separated in physical space, but you are also disconnected in information geometry. You become thermal noise in each other’s universe.

2.1.3 The Geometry of “Never to Return”

Why can’t we return to the past?

Not because physical laws prohibit reversal, but because the paths back are extremely narrow.

Phase space is a high-dimensional manifold. Ordered states (low entropy) occupy only a tiny volume $V_{order}$. Disordered states (high entropy) occupy the vast majority of volume $V_{chaos}$.

$V_{chaos}/V_{order}\approx 10^{10^{23}}$.

Once you step out of $V_{order}$ and enter the ocean of $V_{chaos}$, although theoretically a path exists to take you back (Poincaré recurrence), the probability of randomly finding this path in your lifetime is zero.

This is the physical source of sorrow:

We know that beautiful low-entropy state (childhood, first love, unbroken times) still exists in phase space, but we also know that geometry itself blocks our path of return.

We are trapped in a high-dimensional maze, holding old maps, but can no longer find the old coordinates.

Conclusion:

The sorrow of time is essentially mourning for information loss.

Every second, we lose part of the “past.” Those smiles, those vows, those touches, as the wave function orthogonalizes, gradually become unrecognizable weak fluctuations in background radiation.

We must learn to say goodbye, because this is the price of evolution.

(Section 2.1 Complete)