{
  "schema_version": 1,
  "id": "PHYS-2026-07-22-02",
  "canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-22-02/",
  "source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-22-02.md",
  "metadata": {
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    "id": "PHYS-2026-07-22-02",
    "date": "2026-07-22",
    "updated_at": "2026-07-22",
    "title": "Entropy increase from unstable representations",
    "summary": "How reversible microscopic dynamics yields an irreversible effective description through coarse-graining, correlations, and low-entropy boundary data.",
    "language": "en",
    "entry_kind": "supplement",
    "status": "published",
    "level": "graduate-advanced",
    "user_difficulty": "unrated",
    "domains": [
      "statistical-mechanics",
      "classical-mechanics",
      "quantum-information"
    ],
    "estimated_minutes": 40
  },
  "content_markdown": "\n## Key claim\n\n**The comfortable belief under pressure:** “Irreversibility must come from a microscopic arrow of time.”\n\n## Theme\n\n**Entropy increase as a statement about unstable representations, not trajectories.**\n\n## Guiding question\n\nIf microscopic dynamics is reversible, why does the same mathematical flow produce irreversible thermodynamics?\n\nWhere exactly does the arrow enter when moving between Hamiltonian mechanics, stochastic processes, and information theory?\n\n## Setup\n\nConsider a classical Hamiltonian system with phase-space point $\\Gamma=(q,p)$, evolving by\n\n$$\n\\frac{\\dd\\Gamma}{\\dd t}=\\{\\Gamma,H\\}.\n$$\n\nIts probability density obeys the Liouville equation\n\n$$\n\\frac{\\partial \\rho}{\\partial t}+\\{\\rho,H\\}=0,\n$$\n\nor equivalently $\\partial_t\\rho=\\{H,\\rho\\}$.\n\nThe fine-grained Gibbs entropy is\n\n$$\nS_{\\mathrm{fine}}\n=-k_B\\int \\dd\\Gamma\\,\\rho(\\Gamma,t)\\ln\\rho(\\Gamma,t),\n$$\n\nand is exactly conserved:\n\n$$\n\\frac{\\dd S_{\\mathrm{fine}}}{\\dd t}=0.\n$$\n\nNow introduce a coarse-graining map $\\mathcal{C}$, which replaces the microscopic distribution by a lower-resolution description:\n\n$$\n\\rho_c=\\mathcal{C}[\\rho].\n$$\n\nThe central question is not merely “why does entropy increase?” but:\n\n**Why does a loss of description behave like a physical law?**\n\nThis is the minimal arena because it contains the entire tension: reversible microscopic flow, irreversible effective evolution, and information loss.\n\n## Analysis\n\n### Derivation — dynamical systems view\n\nThe clean derivation begins with Liouville's theorem:\n\n$$\n\\nabla_\\Gamma\\cdot\\dot{\\Gamma}=0.\n$$\n\nHamiltonian flow preserves phase-space volume. Microscopic trajectories therefore never compress into equilibrium.\n\nIn a mixing chaotic system, however, an initially localized distribution stretches and folds:\n\n$$\n\\text{large-scale structure}\\longrightarrow\\text{fine filaments}.\n$$\n\nThe information is not destroyed; it moves into increasingly fine phase-space structure.\n\nA coarse observer cannot resolve those filaments. Under suitable mixing and coarse-graining assumptions, the effective entropy grows because the reduced description projects away those correlations.\n\n### Scope and assumptions — statistical mechanics view\n\nThe hidden assumption is that coarse-graining is innocent.\n\nIt is not.\n\nIn the Boltzmann description, molecular chaos factorizes the distribution of incoming collision partners:\n\n$$\nf_2(z_1,z_2,t)\\approx f_1(z_1,t)f_1(z_2,t).\n$$\n\nThis closure discards correlations from the autonomous one-particle description. The $H$-theorem then gives\n\n$$\n\\frac{\\dd H}{\\dd t}\\leq 0,\n\\qquad\nH[f]=\\int f\\ln f\\,\\dd^3x\\,\\dd^3p.\n$$\n\nThe exact microscopic two-particle distribution does not remain factorized after collisions. Correlations accumulate.\n\nThe time-asymmetric ingredient is the low-correlation boundary condition imposed on incoming states, schematically\n\n$$\nf_2(t_0)\\approx f_1(t_0)f_1(t_0),\n$$\n\nnot Newton's equations themselves.\n\n### Physical interpretation — information theory and stochastic processes\n\nThe dictionary is:\n\n| Statistical mechanics | Information theory |\n|---|---|\n| microstate $\\Gamma$ | full information state |\n| coarse-graining | compression or marginalization |\n| entropy increase | information unavailable to the reduced description |\n| correlations | hidden memory |\n\nFor a bipartite classical distribution,\n\n$$\nI(A:B)=S(A)+S(B)-S(A,B),\n$$\n\nwhere $I(A:B)$ is the mutual information.\n\nA closed system can keep its fine-grained total entropy constant while subsystem entropies increase because information migrates into correlations.\n\nThis structurally parallels open-system quantum mechanics, where\n\n$$\n\\rho_S(t)=\\Tr_E\\rho_{SE}(t).\n$$\n\nBut the statements are not identical. Thermodynamic coarse-graining is often a chosen family of macroscopic observables, while the quantum partial trace is defined by a selected tensor-product decomposition.\n\nSo we must distinguish:\n\n- **Shared mathematical structure:** reduction by marginalization or partial trace.\n- **Controlled approximation:** an autonomous Markovian master equation.\n- **Analogy:** “the environment stores forgotten information.”\n\n## False claim to diagnose\n\n> Because microscopic equations conserve entropy, the second law is merely an illusion caused by human ignorance and has no objective content.\n\nWhy is this tempting?\n\nThe equation\n\n$$\n\\frac{\\dd S_{\\mathrm{fine}}}{\\dd t}=0\n$$\n\nlooks decisive. If entropy is constant microscopically, thermodynamic entropy seems subjective.\n\nThe failure occurs at the identification\n\n$$\nS_{\\mathrm{fine}}\\equiv S_{\\mathrm{thermo}}.\n$$\n\nThey are different functionals on different state spaces.\n\nAs a counterexample, take a gas initially confined to half a box. Under Hamiltonian evolution, its fine-grained distribution remains an incompressibly deformed region. Yet macroscopic measurements robustly predict expansion toward equilibrium.\n\nThe issue is not mere ignorance. Macroscopic observables are insensitive to the extremely fine correlations produced by mixing, and the low-entropy initial condition is a physical boundary condition.\n\nThe defensible statement is:\n\n> The second law is not a fundamental law of individual trajectories; it is a typical law for macroscopic descriptions conditioned on special low-correlation initial data.\n\n## What follows — and what does not\n\nThe disagreement is not about the microscopic equations of motion. All three lenses can accept reversible microscopic dynamics.\n\nThe conflict comes from the **choice of observables, closure assumptions, and boundary conditions**:\n\n- Hamiltonian mechanics retains the full fine-grained distribution and all correlations.\n- Boltzmann theory closes the dynamics of a one-particle distribution using molecular chaos.\n- Information theory tracks where information excluded from the reduced description can reside.\n\nThe arrow is therefore not inserted into Hamiltonian dynamics. It appears when a reversible theory is mapped onto a restricted state space and supplied with asymmetric boundary data.\n\n## Exercise\n\nConsider the symmetric two-state Markov process\n\n$$\n\\frac{\\dd p_1}{\\dd t}=-kp_1+kp_2,\n$$\n\n$$\n\\frac{\\dd p_2}{\\dd t}=kp_1-kp_2,\n$$\n\nwith $p_1+p_2=1$ and Shannon entropy\n\n$$\nS=-p_1\\ln p_1-p_2\\ln p_2.\n$$\n\n1. Show explicitly that $S(t)$ increases until equilibrium, $p_1=p_2=1/2$.\n2. Identify the mathematical assumption that makes this evolution different from Hamiltonian flow.\n\n<details>\n<summary>Hint 1</summary>\n\nWrite $p_2=1-p_1$ and compute $\\dd S/\\dd t$.\n\n</details>\n\n<details>\n<summary>Hint 2</summary>\n\nUse $x=p_1-p_2$ and solve its evolution first.\n\n</details>\n\n**Oral check 1.** If every microscopic velocity in a gas is reversed, why does entropy decrease only from an extraordinarily special correlated state?\n\n**Oral check 2.** Does tracing out an environment by itself guarantee permanently irreversible decoherence?\n\n<details class=\"solution\">\n<summary>Solution</summary>\n\nFrom the master equations,\n\n$$\n\\frac{\\dd x}{\\dd t}=-2kx,\n$$\n\nso\n\n$$\nx(t)=x(0)\\ee^{-2kt},\n\\qquad\np_{1,2}(t)=\\frac{1}{2}(1\\pm x).\n$$\n\nThe entropy is\n\n$$\nS(x)=-\\frac{1+x}{2}\\ln\\frac{1+x}{2}\n-\\frac{1-x}{2}\\ln\\frac{1-x}{2}.\n$$\n\nIts production rate is\n\n$$\n\\frac{\\dd S}{\\dd t}\n=k x\\ln\\frac{1+x}{1-x}\\geq 0\n$$\n\nfor $|x|<1$, with equality only at $x=0$. Hence $S$ approaches\n\n$$\nS_{\\max}=\\ln 2.\n$$\n\nThe decisive step is the autonomous Markov rule\n\n$$\np(t+\\Delta t)=M_{\\Delta t}p(t).\n$$\n\nIts inverse is generally not a stochastic map. In a coarse-grained derivation, the missing information resides in microscopic correlations and memory that are absent from the state vector $p$.\n\nFor this symmetric doubly stochastic process, $|x|$ contracts and the relative entropy to equilibrium decreases. This is the precise information-space contraction used here.\n\n</details>\n\n## Check your understanding\n\nWhat is the more fundamental source of the thermodynamic arrow: **the dynamics**, **the initial condition**, or **the observer's choice of variables**?\n\nReply with your answer, your attempted solution, or simply say “deeper,” “too easy,” or “too hard.”\n\n## Connections and next step\n\n- New pressure point: fine-grained conservation versus autonomous macroscopic irreversibility.\n- Main domains: classical mechanics, statistical mechanics, and information theory.\n- Toy models: a coarse-grained Hamiltonian gas and a symmetric two-state Markov process.\n- Unresolved: when is entropy increase observer-relative, and when is it structurally robust across macroscopic descriptions?\n- Revisit: equilibrium and typicality in three runs.\n",
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