Part XIV: Computational Foundations and Encoding

Chapter 25: Optimality of Physical Computation

In Part XIII, we established category-theoretic logical foundations of physics, proving that physical processes follow linear logic and resource conservation laws at bottom level. This chapter turns to thermodynamics and coding theory of computation. Since physical reality is essentially computational processes (QCA), physical laws must contain limitations on “computational cost” and “information transmission efficiency.”

We will first explore thermodynamic cost of computation (Landauer principle), then prove how natural encoding methods (such as Fibonacci coding) achieve optimal information transmission in causal networks. This chapter will reveal that so-called “physical constants” and “conservation laws” are often eigenvalues of optimal information processing systems.

25.1 Landauer’s Principle and Thermodynamic Cost of Irreversible Computation

In QCA discrete ontology, microscopic dynamics $U$ is strictly unitary (reversible). This means information never loses at microscopic level. However, computation in macroscopic world is often logically irreversible. Most typical example is “erase” operation (RESET): regardless of input being 0 or 1, output is forced to 0.

There is profound conflict between logical information loss (many-to-one mapping) and physical microscopic reversibility (one-to-one mapping). Rolf Landauer resolved this conflict in 1961, pointing out: Logical irreversibility must be accompanied by physical heat dissipation. This section will strictly derive Landauer principle from phase space volume conservation of QCA, and establish ontological status of “information as physics.”

25.1.1 Logical Irreversibility and Phase Space Compression

Consider a physical bit (such as spin or QCA lattice point), its logical state space is $\mathcal{S}=\{0,1\}$.

• Logically Reversible Operation (such as NOT): $0\to 1,1\to 0$. This is bijection, entropy unchanged.

• Logically Irreversible Operation (such as ERASE): $0\to 0,1\to 0$. This is compression mapping, entropy decreases.

In QCA framework, microscopic states are described by state vectors $|\psi ⟩$ in Hilbert space.

Quantum Version of Liouville’s Theorem states: unitary evolution $U$ preserves volume (or von Neumann entropy) of state space.

$$S({\rho }_{final})=S(U{\rho }_{initial}{U}^{†})=S({\rho }_{initial})$$

If we want to perform logical erasure (transforming unknown initial state ${\rho }_{mix}=\frac{1}{2}(|0⟩⟨0|+|1⟩⟨1|)$ to definite ground state $|0⟩⟨0|$), system entropy must change from $S=k_B\ln 2$ to $S=0$.

Due to total entropy conservation, this reduced $k_B\ln 2$ entropy must be transferred to Non-computational Degrees of Freedom outside the system, i.e., environmental heat bath.

25.1.2 Strict Derivation of Landauer Bound

Theorem 25.1.1 (Landauer’s Principle)

In any thermal environment at temperature $T$, minimum energy cost $\Delta E$ required to erase 1 bit of information (i.e., resetting physical system with Shannon entropy 1 bit to pure state) is:

$$\Delta E\ge k_{B}T\ln 2$$

Or expressed as minimum heat produced $Q$:

$$Q\ge k_{B}T\ln 2$$

Proof:

Let total system be $\text{System}\otimes \text{Bath}$.

Initial state: System in maximum mixed state ${\rho }_S=\mathbb{I}/2$, heat bath in thermal equilibrium ${\rho }_B=e^{-\beta H_B}/Z$.

Total entropy: $S_{tot}=S({\rho }_S)+S({\rho }_B)=k_B\ln 2+S_B(E_B)$.

Process: Execute unitary operation $U_{SB}$ to achieve erasure.

Final state: System in pure state ${\rho }_S^{\prime }=|0⟩⟨0|$, heat bath in new state ${\rho }_B^{\prime }$.

Due to unitarity, total entropy conserved:

$$S({\rho }_S^{\prime })+S({\rho }_B^{\prime })\ge S_{tot}(\text{equality if no additional correlations})$$

Substituting $S({\rho }_S^{\prime })=0$, we get:

$$S({\rho }_B^{\prime })\ge k_B\ln 2+S_B(E_B)$$

Entropy increase of heat bath $\Delta S_B\ge k_B\ln 2$.

According to thermodynamic definition $dS=dQ/T$, heat absorbed by heat bath (i.e., energy dissipated by system) is:

$$Q=\int TdS\ge T\Delta S_B\ge k_{B}T\ln 2$$

$\square$

25.1.3 Garbage Bits and Heat Dissipation in QCA

In QCA discrete networks, there is no abstract “heat bath.” Landauer principle manifests as propagation of Garbage Information.

Construction 25.1.2 (Reversible Embedding)

Any irreversible logic gate (such as AND gate, $(a,b)\to a\wedge b$) can be embedded into a reversible gate (such as Toffoli gate) by adding auxiliary bits (Ancilla).

$$\text{Toffoli}(a,b,c)=(a,b,c\oplus (a\wedge b))$$

If we initialize $c$ to $0$, third bit of output is computation result. However, first two bits $(a,b)$ still remain at output, becoming garbage bits.

To perform next computation (reset register), we must “remove” these garbage bits.

In QCA, this manifests as radiating wave packets carrying garbage information to infinity (or into horizon). These radiated wave packets carry energy $E=\hslash \omega$.

For wave packet with frequency $\omega$, entropy it carries is $S$. To carry away 1 bit entropy, wave packet must have sufficient phase space volume. At thermal equilibrium, this corresponds to energy $E\ge k_{B}T\ln 2$.

Physical Corollary:

CPU heating is not due to electron friction (that’s engineering defect), but due to reset of logic gates. Each forced reset of $0\to 0$ or $1\to 0$ microscopically emits a photon (or phonon) into environment, carrying away entropy originally stored in bit.

25.1.4 Physical Reality of Information: Exorcism of Maxwell’s Demon

Landauer principle completely resolves Maxwell’s Demon paradox that plagued physics for a century.

Demon attempts to control gate by acquiring information about molecules (velocity), thereby reducing system entropy without doing work.

Landauer pointed out: Demon’s brain (or memory) is a physical system. To work continuously, demon must constantly erase old memories to store new information.

• Measurement Phase: Entropy transfers from gas to demon’s memory (gas entropy decreases, demon entropy increases).

• Erasure Phase: Demon resets memory, according to Landauer principle, must release heat $Q\ge k_{B}T\ln 2$ to environment.

Entropy increase $\Delta S_{env}\ge k_B\ln 2$ produced by this heat precisely cancels entropy reduced by demon through sorting molecules.

Conclusion 25.1.3 (Information as Physical Resource)

Information is not an abstract concept independent of matter; it is a form of Negentropy.

1. Energy Cost: Processing information (especially erasure) must consume free energy.

2. Mass: If $E=m{c}^2$, then erasing 1 bit information corresponds to increasing environmental mass $\Delta m=\frac{k_{B}T\ln 2}{c^2}$.

3. Gravity: Aggregation of large amounts of information (high entropy states) produces gravitational effects (black holes).

This principle establishes thermodynamic bottom line of computational physics. In the next section 25.2, we will explore how nature utilizes Fibonacci Coding and Zeckendorf Arithmetic to achieve optimal local information transmission without erasing information (reversible computation).