Irreversibility, probability and laws of nature

I. Prigogine

Barbara Drossel

It is often claimed that the fundamental laws of physics are deterministic and time-symmetric and that therefore our experience of the passage of time is an illusion. This paper will critically discuss these claims and show that they are based on the misconception that the laws of physics are an exact and complete description of nature. I will argue that all supposedly fundamental deterministic and time-symmetric laws have their limitations and are supplemented by stochastic and irreversible elements. In fact, a deterministic description of a system is valid only as long as interactions with the rest of the world can be ignored. The most famous example is the quantum measurement process that occurs when a quantum system interacts with a macroscopic environment such as a measurement apparatus. This environment determines in a top-down way the possible outcomes of the measurement and their probabilities. I will argue that more generally the possible events that can occur in a system and their probabilities are the result of top-down influences from the wider context. In this way the microscopic level of a system is causally open to influences from the macroscopic environment. In conclusion, indeterminism and irreversibility are the result of a system being embedded in a wider context.

Danker L. N. Vink

Richard Schlegel

Harold D. Kirk

The Born rule — the statement that the probability of obtaining outcome |ϕi⟩ from quantum state |ψ⟩ equals |⟨ϕi|ψ⟩|² — is normally taken as a fundamental postulate of quantum mechanics. This paper derives the Born rule from the bridge law connecting the the Fubini–Study geometry of quantum state space to physical time through irreversible contraction of the accessible future state space Ω. The bridge law, d(ln Ω)/dτ = −κ Γ(τ), specifies that each recording event depletes generative capacity at a rate set by the structural efficiency constant κ = π — the Fubini–Study volume of ℂP1. This identifies the Fubini–Study metric as the natural measure of state-space contraction. Three physical requirements — unitary invariance, projective consistency, and smooth locality — uniquely select this metric, leaving no freedom in the probability assignment. The probability of each outcome is the fraction of the local Fubini–Study contraction associated with that outcome, yielding pi = |⟨ϕi|ψ⟩|². The framework further predicts finite-time corrections to the Born rule. Because records form through irreversible thermodynamic processes with characteristic timescale τdet, the effective probability of registering independent events at short delay τ is modulated by the record-completion function R(τ) = 1 − e−|τ|/τdet. This produces a distinctive suppression of photon coincidences in the second-order correlation function g(2)(τ) at picosecond timescales in Hanbury Brown–Twiss measurements using superconducting nanowire single-photon detectors. The effect depends on detector material properties through the detector transition time τdet and is experimentally testable with existing instrumentation. The result therefore provides both a geometric derivation of the Born rule from the bridge law of irreversible record formation and a concrete experimental prediction linking quantum probability to the thermodynamics of measurement. This work forms part of the author's research programme Emergence as Regime Formation, investigating the origin of time, probability, and physical constants through irreversible record formation.

John Onimisi Obidi

✨On the Significance and Implications of the Obidi Probability Law Derived from the Principles of the Theory of Entropicity (ToE) The Obidi probability law is a central result in John Onimisi Obidi's Theory of Entropicity (ToE), and it marks a radical departure from classical probability theory. In standard Kolmogorov or quantum mechanics, the rule that the sum of all probabilities equals one is taken as an axiom with no dynamical origin. In ToE, Obidi derives this rule from the geometry and dynamics of the universe's Hilbert space, elevating probability from an epistemic bookkeeping rule to an ontological conservation law. The Connection Between Usual Probability and the Obidi Probability Law The simplest and perhaps clearest way to express the relationship between the usual [traditional] probability law and Obidi's entropic probability law is this: usual probability tells you what you might see, while the Obidi probability tells you why you see what you see. Usual probability is a predictive rule; Obidi probability is a structural law. They share a mathematical surface, but they arise from entirely different conceptual foundations. Understanding this distinction is essential for appreciating the depth of the Obidi Probability Law and its implications for physics. Mathematically, the two frameworks look deceptively similar. Usual probability is built on the normalization condition ∑iPi=1, while the Obidi Probability Law expresses a unique conservation law through the relation Po(t)+Pe(t)=1. Both involve quantities between 0 and 1 that sum to unity. This superficial resemblance is precisely why the Obidi formulation feels familiar at first glance: it preserves the algebraic structure of probability. Yet the meaning of these quantities is entirely different. In usual probability, the numbers represent degrees of belief or frequencies of outcomes. In the Obidi probability, they represent the partitioning of physical reality between coherent and entropic sectors of the universe's informational geometry. Usual probability is fundamentally epistemic. It answers the question, "Given what I know, what outcome should I expect?" It depends on the observer's knowledge, uncertainty, and measurement context. It is a rule of inference, not a physical law. The Born Rule, Bayesian updating, and the Kolmogorov axioms all fall into this category: they describe how observers assign probabilities, not how the universe itself behaves. In this sense, usual probability is a tool for reasoning about outcomes, not a description of the underlying structure of reality. Obidi probability is ontological. It answers a different question entirely: "How is reality partitioned between coherence and entropy?" It is observer‑independent, conserved, and rooted in the entropic geometry of the universe. It does not describe ignorance or uncertainty; it describes the physical distribution of amplitude between the coherent sector Po and the entropic sector Pe. It is not optional, not interpretive, and not dependent on measurement. It is a law of nature, not a rule of reasoning. In this sense, Obidi probability is a structural invariant of the universe's informational architecture. The connection between the two emerges when the entropic sector becomes negligible. When Pe(t)≈0, the coherent sector satisfies Po(t)≈1, and the usual Born‑rule probability appears as the projection of the coherent sector onto the observer's measurement basis. In this limit, usual probability is recovered as a special case of the Obidi Probability Law. This relationship mirrors other major correspondences in physics: Newtonian mechanics as a limit of relativity, classical thermodynamics as a limit of statistical mechanics, and classical geometry as a limit of quantum geometry. In each case, the familiar theory is not wrong - it is simply the low‑entropy, low‑curvature, or low‑energy approximation of a deeper structure. Usual probability plays the same role relative to Obidi probability. Measurement provides the final bridge between the two frameworks. In standard quantum mechanics, measurement is modeled as a discontinuous collapse of the wavefunction, with probabilities jumping abruptly according to the Born Rule. In Obidi's Theory of Entropicity (ToE), measurement is reinterpreted as an irreversible flow of amplitude from the coherent sector Po into the entropic sector Pe. This entropic transfer supplies a physical mechanism for what usual probability merely describes. The Born Rule becomes the visible cast or projection of a deeper entropic process unfolding beneath the surface of quantum mechanics. In summary, usual probability is what the observer computes, while the Obidi probability is what the universe is doing underneath. How the Obidi Probability Law Is Formulated In the Theory of Entropicity (ToE), the total quantum state of the universe is written as: Ψ(t) = ψ_o(t) + ψ_e(t) where: ψ_o(t) = the component of the state in the observer (coherent) sector ψ_e(t) = the component of the state in the entropic (irreversible) sector ψ_o(t) is orthogonal to ψ_e(t) This orthogonality is written as: ψ_o ⟂ ψ_e The total state always has unit norm: ‖Ψ(t)‖² = 1 If you decompose a vector into orthogonal components: Ψ = A + B then automatically: ‖Ψ‖² = ‖A‖² + ‖B‖² This is not the Born rule. This is Pythagoras in Hilbert space. Here, we are not assuming probability. We are deriving a conservation identity. From this, the Obidi Probability Law follows immediately: P_o(t) + P_e(t) = 1 where: P_o(t) = ‖ψ_o(t)‖² P_e(t) = ‖ψ_e(t)‖² This means that: P_o(t) is the probability associated with the coherent observer sector P_e(t) is the probability associated with the entropic sector Their sum is always 1, because they partition the entire Hilbert space of reality This is the entropic probability conservation law in ToE. In other words: The Theory of Entropicity (ToE) begins by decomposing the universe's total Hilbert space into two orthogonal sectors: Hₒ - The Coherent Observer Sector This is where quantum amplitudes evolve unitarily. It contains all accessible information. It is the sector in which observers, measurements, and coherent quantum states live. 2. Hₒ - The Entropic Sector This is the domain into which amplitude flows irreversibly. It represents informationally inaccessible states. It is responsible for decoherence, irreversibility, and the arrow of time. The total state of the universe is written as: Ψ(t) = ψₒ(t) + ψₑ(t) with the orthogonality condition: ψₒ ⟂ ψₑ We then write the probabilities as: Pₒ(t) = |ψₒ(t)|² Pₑ(t) = |ψₑ(t)|² In the Theory of Entropicity (ToE), the above [Pₒ(t) = |ψₒ(t)|² and Pₑ(t) = |ψₑ(t)|²] is NOT the Born rule - it is a sector‑weight derived from Hilbert‑space geometry. With the norm conservation of the full state which demands that: |Ψ(t)|² = 1, this immediately yields the Obidi Probability Law: Pₒ(t) + Pₑ(t) = 1 This is the entropic probability conservation law in ToE. Explore: Entropic Field Significance of the Obidi Entropic Probability Law From axiom to conservation principle Unlike classical probability, which partitions events, the Obidi law partitions sectors of physical reality. The sum is not imposed but follows from the Hilbert‑space structure and the combined unitary–entropic evolution . Ontological status of probability Probability becomes a structural invariant of the universe's geometry and dynamics, not just a measure of ignorance. This reframes probability as a fundamental physical law rather than a statistical convention. Mechanism for irreversibility and decoherence The entropic operator transfers amplitude from the observer sector to the entropic sector, generating irreversibility, decoherence, and the arrow of time while keeping total probability constant. Unified explanation of physical phenomena In ToE, this law underpins explanations of wavefunction collapse, classical irreversibility, and even the black hole information paradox: measurement outcomes correspond to irreversible amplitude transfer into the entropic sector. Conceptual shift in physics By embedding probability conservation in the geometry of reality rather than in human observation, the Obidi law challenges the Copenhagen interpretation and suggests a deeper, sector‑based ontology of quantum processes. In short: The Obidi probability law is significant because it redefines probability as a derived, conserved quantity rooted in the universe's Hilbert‑space structure, linking quantum coherence, entropy, and the arrow of time in a single, unified framework. The Obidi Probability Law is precisely a theory of the visible and invisible sectors of reality. This is one of its deepest conceptual achievements, and it's why it feels so different from anything in standard physics. In the following sections, we lay it out clearly and technically for the reader, so as not to lose sight of the elegance of the idea. The Obidi Probability Law teaches us about the visible and invisible sectors of reality The Obidi Probability Law divides the universe into two complementary informational sectors: the coherent sector Po - the visible, reversible, interference‑preserving part of reality the entropic sector Pe - the invisible, irreversible, information‑absorbing part of reality And the law states: Po(t)+Pe(t)=1 This is not a bookkeeping identity. It is a conservation law describing how reality is partitioned between what can be observed and what cannot [what we can reasonably access and what we cannot; and this has nothing to do with the limits of our laboratory or experimental apparatus or devices, but the inherent constraints of nature itself, independent of how accurate or efficient our tools are]. 1. The "visible sector": the coherent part of reality The coherent sector Po corresponds to: superposition interference unitary evolution reversible dynamics the part of the wavefunction that can produce observable outcomes This is the sector that standard quantum mechanics foc