Derivation of Landauer's Principle from the Obidi Curvature Invariant (OCI) ln 2 of the Theory of Entropicity (ToE)

Landauer's Principle: Overview

Landauer's Principle, proposed by Rolf Landauer in 1961, states that any logically irreversible computation, such as erasing one bit of information, must dissipate a minimum amount of energy as heat into the environment. Specifically, the minimum energy cost to erase one bit is (k_B T ln 2), where (k_B) is Boltzmann's constant, (T) is the absolute temperature of the system, and (ln 2 approx 0.693) is the natural logarithm of 2. This principle links information theory with thermodynamics, showing that information processing has unavoidable physical costs due to the second law of thermodynamics.

The principle applies to irreversible operations (e.g., resetting a bit from an unknown state to 0), but reversible operations can, in theory, be performed with arbitrarily small energy dissipation. It has implications for computing, quantum information, and the limits of energy-efficient devices.

Standard Derivation from Thermodynamics

The derivation relies on the connection between information entropy (from Shannon's theory) and thermodynamic entropy (from statistical mechanics), combined with the second law of thermodynamics, which states that the total entropy of an isolated system cannot decrease. Here's a step-by-step explanation of a simple, classical derivation:

Model the System: Consider a physical system representing one bit of information, such as a particle in a double-well potential (two possible states: "0" and "1", like left or right well). Initially, the bit is in an unknown state, so it has equal probability (1/2) of being in either well. This corresponds to 1 bit of Shannon information entropy: (H = -sum p_i log_2 p_i = -2 times (0.5 \log_2 0.5) = 1) bit.

Link to Thermodynamic Entropy: 

In statistical mechanics, the thermodynamic entropy (S) of a system with (Omega) microstates is (S = k_B ln Omega). For our two-state system (assuming equal probability and thermal equilibrium), (Omega = 2), so the entropy is (S = k_B ln 2). (Note: Shannon entropy in nats is (H = ln 2) for 1 bit, and thermodynamic entropy is ( S = k_B H), bridging the two.)

Erasing the Bit: Erasure means resetting the system to a definite state (e.g., "0"), regardless of its initial state. This is logically irreversible because you can't recover the original bit from the final state. After erasure, there's only one possible state ((Omega = 1 )), so the entropy of the system drops to (S' = k_B ln 1 = 0). The change in system entropy is (∆S_{system} = -k_B ln 2).

Apply the Second Law: The second law requires that the total entropy of the universe (system + environment) must not decrease: (∆S_{total} >= 0). If the system loses entropy, the environment must gain at least that much: (∆S_{environment} >= k_B ln 2).

Heat Dissipation: Assuming the environment is a thermal reservoir at temperature ( T ), the entropy gain of the environment is related to the heat (Q) it absorbs by (∆S_{environment} = Q / T) (for reversible heat transfer; the inequality holds for irreversible). Thus, the minimum heat dissipated to the environment is (Q >= T ∆S_{environment}>= k_B T ln 2). This heat corresponds to the minimum energy cost for erasure.

This derivation assumes classical, equilibrium conditions. More rigorous treatments account for fluctuations or use detailed models like compressing phase space in a bistable system. Experimental verifications have confirmed this limit in systems like colloidal particles or nanomagnetic bits.

Quantum Extensions

In quantum mechanics, the principle can be derived using von Neumann entropy or quantum Shannon entropy. For a qubit in a mixed state (equivalent to 1 bit of classical uncertainty), erasure involves coupling to a thermal bath, leading to similar entropic costs, but with potential modifications due to entanglement or coherence. The minimum free energy cost separates classical and quantum contributions, but the core bound remains ( k_B T ln 2).

Derivation in the Context of the Theory of Entropicity (ToE)

In the audacious Theory of Entropicity (ToE) proposed by John Onimisi Obidi, Landauer's Principle is derived from first principles by treating entropy as a fundamental field, with the Obidi Curvature Invariant (OCI = (ln 2)) as the minimal curvature deformation in this entropic field, and derived from the convexity theorem and Kullback-Leibler (Umegaki) Divergence and Araki Relative Entropy formalisms. Here's a summarized derivation based on ToE's framework:

Entropic Field and OCI: ToE posits entropy (S(x)) as an ontic field, where information manifests as localized curvature. OCI ((ln 2)) is the smallest nonzero curvature representing a binary distinction (e.g., two states), akin to the entropic "quantum" of distinguishability.

Bit Erasure as Field Flattening: Erasing a bit is reinterpreted as "flattening" this curvature from (ln 2) to 0 in the entropic manifold. This irreversible reconfiguration incurs a minimal entropic cost of (ln 2) (in nats), generalized from the binary entropy (S = k_B ln 2).

Energy Cost via Entropic Dynamics: Using ToE's Master Entropic Equation—MEE (derived from the Obidi Action, a variational principle for (S(x))), the entropic change couples to energy via temperature: the cost to flatten the curvature dissipates as heat (k_B T ln 2), ensuring causality and the arrow of time. This emerges as the "Landauer-Bennett cost" for irreversible updates, where (∆N_c = 1) for one bit.

Obidi's Theory of Entropicity (ToE) thus unifies and generalizes Landauer's Principle with broader physics by rooting it in entropic geometry, rather than treating it as a statistical add-on.  

References

1)

https://theoryofentropicity.blogspot.com/2026/01/how-obidi-discovered-ln2-as-universal.html

2)

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3)

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