ENTROPY: DEEP RESEARCH NOTES

Raw material for the Deep Truth Guide. Organized by mechanism area. Every claim sourced.

1. BOLTZMANN’S STATISTICAL MECHANICS

The Equation: S = k ln W

History

What It Actually Means

Microstates vs. macrostates:

Why entropy is about counting, not “disorder”:

The Combinatorial Argument

Boltzmann’s 1877 paper used combinatorics. For N particles distributed among energy levels, the number of arrangements is:

W = N! / (n1! x n2! x n3! x …)

where ni is the number of particles in energy level i. The distribution that maximizes W (subject to constraints of fixed total energy and particle number) is the Maxwell-Boltzmann distribution. This is the equilibrium state. Not because the system “wants” to be there, but because overwhelmingly more microstates correspond to it.

Key Researchers

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2. THE SECOND LAW AND THE ASYMMETRY OF TIME

The Second Law

Clausius formulation (1865): The entropy of the universe tends toward a maximum.

Kelvin-Planck formulation: No cyclic process can convert heat entirely into work.

Statistical formulation: The entropy of an isolated system almost certainly increases, because transitions from lower-W macrostates to higher-W macrostates are overwhelmingly more probable. Not impossible to decrease, just astronomically unlikely.

The Past Hypothesis

The problem: The second law says entropy increases toward the future. But the laws of physics are time-symmetric (with minor exceptions from CP violation). If you run Newton’s equations backward, they work perfectly. So why does entropy have a preferred direction?

The answer (such as it is): The universe started in an extraordinarily low-entropy state. This is a boundary condition, not derivable from the dynamics.

David Albert (Columbia University) named this the “Past Hypothesis” in his 2000 book “Time and Chance.” He argues it should be promoted to a fundamental law of nature, a postulate alongside the dynamical laws and the statistical postulate (equal probability of microstates).

Sean Carroll (Johns Hopkins, formerly Caltech) elaborated on the cosmological implications:

Carroll-Chen proposal (2004): Rather than postulating the Past Hypothesis, Carroll and Jennifer Chen proposed a model where the universe can fluctuate into existence from a maximum-entropy de Sitter state, with baby universes branching off. Each branch starts with low entropy. No need to postulate a special initial condition; the multiverse naturally generates low-entropy beginnings.

Roger Penrose proposed the Weyl Curvature Hypothesis: at the Big Bang, the Weyl curvature tensor (the part of spacetime curvature not determined by local matter) was zero or near-zero. This is a very special gravitational condition. At a Big Crunch or inside black holes, Weyl curvature is enormous. This geometric fact encodes the low gravitational entropy of the early universe.

Quantitative Scale

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3. SHANNON ENTROPY AND INFORMATION THERMODYNAMICS

Shannon’s Definition (1948)

Paper: “A Mathematical Theory of Communication,” Bell System Technical Journal, Vol. 27, pp. 379-423 (July 1948) and pp. 623-656 (October 1948).

The equation:

H = -sum(pi log2 pi)

where pi is the probability of outcome i.

Key insight: Shannon was measuring the same thing Boltzmann measured, but from the information side. Gibbs entropy S = -k sum(pi ln pi) is Shannon entropy multiplied by Boltzmann’s constant k. The mathematical structures are identical.

John von Neumann reportedly told Shannon to call his measure “entropy” because “nobody knows what entropy really is, so in a debate you will always have the advantage.”

Landauer’s Principle (1961)

Rolf Landauer (IBM, 1961) paper: “Irreversibility and Heat Generation in the Computing Process.”

The principle: Erasing one bit of information necessarily dissipates at least kT ln 2 of energy as heat (approximately 2.87 x 10^-21 joules at room temperature, 300K).

Why this matters:

Maxwell’s Demon: Resolution

The paradox (James Clerk Maxwell, 1867): A tiny intelligent being (“demon”) operates a frictionless door between two gas chambers. It lets fast molecules through one way and slow molecules the other way. This sorts the gas into hot and cold without doing work, apparently violating the second law.

History of resolution:

  1. Leo Szilard (1929): Argued the demon must acquire information, and this acquisition costs entropy. Built a one-molecule engine (Szilard engine) to make the argument precise.
  2. Leon Brillouin (1951-1953): Argued measurement requires interaction (e.g., shining light on molecules), which produces entropy. Published “The Negentropy Principle of Information” (1953) in Journal of Applied Physics, Vol. 24, pp. 1152-1163.
    • Source: https://pubs.aip.org/aip/jap/article/24/9/1152/160687/The-Negentropy-Principle-of-Information
  3. Rolf Landauer (1961): Shifted the focus from measurement to erasure.
  4. Charles Bennett (1982): Completed the resolution. The demon can measure reversibly (no entropy cost). But its memory is finite. Eventually it must erase old measurements to make room for new ones. This erasure, by Landauer’s principle, dissipates at least kT ln 2 per bit. The total entropy generated by erasure equals or exceeds the entropy reduction from sorting. The second law holds.

Experimental Verification of Landauer’s Principle (2012)

Paper: Berut, Arakelyan, Petrosyan, Ciliberto, Dillenschneider, Lutz. “Experimental verification of Landauer’s principle linking information and thermodynamics.” Nature, Vol. 483, pp. 187-189 (2012).

Method: Used a single colloidal particle trapped in a modulated double-well potential (created by focused laser beams). This is a physical one-bit memory. By tilting the potential to force the particle into one well (erasing the bit), they measured the heat dissipated.

Result: Confirmed that mean dissipated heat saturates at the Landauer bound (kT ln 2) in the limit of slow (quasi-static) erasure. First direct experimental confirmation of the information-thermodynamics link.

Brillouin’s Negentropy

Leon Brillouin formalized the connection: information is negentropy (negative entropy).

I = k ln(1/W) = -S

Information corresponds to a negative term in total system entropy: S = S0 - I. Gaining information about a system reduces your uncertainty (entropy) about it.

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4. THE FREE ENERGY PRINCIPLE (KARL FRISTON)

The Core Idea

Karl Friston (University College London, 2006-present formulation) proposed that all biological systems, from single cells to brains, can be described by a single imperative: minimize variational free energy.

Variational free energy is an information-theoretic quantity. It is an upper bound on “surprisal” (negative log-probability of sensory observations given an internal model). Minimizing free energy means minimizing surprise, which means maintaining yourself in expected states, which means resisting the second law.

The Formalism

F = E_q[ln q(theta) - ln p(theta, y)]

where:

Minimizing F can be decomposed into:

Connection to Entropy

Under ergodic assumptions (the system visits all accessible states over time), minimizing variational free energy is equivalent to minimizing the Shannon entropy of sensory states. An organism that minimizes free energy places an upper bound on the entropy of the states it occupies. It stays in a restricted set of life-compatible states rather than dispersing into all possible states (which would be death/dissolution).

Active Inference

Organisms don’t just passively update beliefs. They ACT on the world to reduce surprise:

Key Papers

Why It Matters for the Entropy Guide

Friston provides the mathematical framework for understanding HOW biological systems maintain low internal entropy. The answer: by being predictive models of their environment. A bacterium swimming up a sugar gradient is minimizing free energy. A human planning next year’s budget is minimizing free energy. Same principle, different complexity.

The free energy principle connects:

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5. DISSIPATIVE STRUCTURES (PRIGOGINE)

The Discovery

Ilya Prigogine (1917-2003), Russian-Belgian physical chemist. Nobel Prize in Chemistry, 1977, “for his contributions to nonequilibrium thermodynamics, particularly the theory of dissipative structures.”

The Key Insight

Classical thermodynamics (Clausius, Boltzmann) deals with systems near equilibrium. Life, weather, economies, and most interesting phenomena are FAR from equilibrium. Prigogine showed that far from equilibrium, new rules emerge.

Near equilibrium: Fluctuations decay. Systems return to equilibrium. Entropy production is minimized (Prigogine’s minimum entropy production theorem, applicable near equilibrium).

Far from equilibrium: Fluctuations can be AMPLIFIED. Systems can spontaneously self-organize into ordered structures. These structures INCREASE the rate of entropy production (they dissipate energy faster than the unstructured state would). Hence: “dissipative structures.”

The Paradox That Isn’t

Order doesn’t emerge DESPITE entropy production. Order emerges BECAUSE it is a more efficient way to produce entropy. The structure is the entropy production mechanism.

Examples of Dissipative Structures

  1. Benard convection cells: Heat a layer of fluid from below. Below a critical temperature gradient, heat transfers by conduction (no structure). Above the threshold, the fluid spontaneously organizes into hexagonal convection cells. These cells transport heat (produce entropy) far more efficiently than conduction alone.

  2. The Belousov-Zhabotinsky (BZ) reaction: A chemical oscillator. Reactants cycle between states, producing visible color-changing waves and spirals. A self-organizing chemical clock maintained by continuous energy input.

  3. Hurricanes: Organized vortical structures that dissipate thermal energy from warm ocean surfaces far more efficiently than simple radiation or convection would.

  4. Life itself: Organisms are dissipative structures. A tree dissipates solar energy more effectively than bare rock. An ecosystem dissipates more than a desert.

Bifurcation Points

Prigogine showed that dissipative structures emerge at bifurcation points: thresholds where the system’s behavior qualitatively changes. Below the threshold, one stable state. Above it, the system “chooses” between multiple possible organized states. The choice is determined by fluctuations (fundamentally unpredictable). This introduces genuine novelty and irreducible historicity into physics.

Jeremy England’s Extension (2013)

Jeremy England (MIT, now at Georgia Tech/GlaxoSmithKline) extended this thinking:

Paper: “Statistical Physics of Self-Replication.” Journal of Chemical Physics, 139, 121923 (2013).

Key result: Derived a minimum bound on heat production during self-replication: the minimum dissipation is determined by the replicator’s growth rate, internal entropy, and durability. Faster-growing, more-ordered, more-durable replicators must dissipate MORE heat.

Implication: Self-replicating structures are thermodynamically favored in driven systems because they are better at dissipating energy. Life doesn’t fight thermodynamics. Life is thermodynamics finding a faster path to equilibrium.

Follow-up: “Dissipative adaptation in driven self-assembly.” Nature Nanotechnology, 10, 919-923 (2015).

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6. ENTROPY IN PSYCHOLOGY AND NEUROSCIENCE

The Entropic Brain Hypothesis

Robin Carhart-Harris (University of California, San Francisco; formerly Imperial College London).

Original paper (2014): “The entropic brain: a theory of conscious states informed by neuroimaging research with psychedelic drugs.” Frontiers in Human Neuroscience, 8, Article 20 (2014).

Updated paper (2018): “The entropic brain - revisited.” Neuropharmacology, 142, 167-178 (2018).

The Core Claim

The richness of conscious experience can be indexed by the entropy of spontaneous brain activity (measured via EEG, MEG, or fMRI). Higher entropy = richer, less constrained, more fluid conscious states. Lower entropy = more rigid, constrained, repetitive states.

The Entropy Spectrum of Consciousness

Carhart-Harris proposes a spectrum:

LOW ENTROPY (rigid, constrained):

HIGH ENTROPY (fluid, unconstrained):

The Default Mode Network (DMN)

The DMN is a network of brain regions (medial prefrontal cortex, posterior cingulate cortex, angular gyrus, etc.) that is active during rest, self-referential thinking, mind-wandering, and autobiographical memory.

Key finding from psilocybin research: Psychedelics DECREASE functional connectivity within the DMN. The DMN normally acts as a constraint, an entropy-reducing mechanism that imposes top-down predictions and narrative coherence. Psychedelics relax this constraint, increasing neural entropy.

REBUS Model (2019)

Paper: Carhart-Harris and Friston. “REBUS and the Anarchic Brain: Toward a Unified Model of the Brain Action of Psychedelics.” Pharmacological Reviews, 71(3), 316-344 (2019).

REBUS = RElaxed Beliefs Under pSychedelics.

This paper integrates:

The synthesis: Psychedelics relax the precision of high-level priors (top-down beliefs/predictions). This liberates bottom-up sensory information. The brain’s predictive model loosens its grip. Result: increased entropy, reduced constraint, access to states normally suppressed by the DMN’s filtering.

Therapeutic implication: Depression, OCD, and addiction involve pathologically rigid priors (over-weighted beliefs about self, world, future). Psychedelics can “reset” these by temporarily dissolving the constraints, allowing new patterns to form. This is why psychedelic-assisted therapy shows efficacy for treatment-resistant depression.

Quantitative Findings

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7. ENTROPY AND AGING

The Hallmarks Framework

Lopez-Otin et al. (2013). “The Hallmarks of Aging.” Cell, 153(6), 1194-1217.

Lopez-Otin et al. (2023). “Hallmarks of Aging: An Expanding Universe.” Cell, 186(2), 243-278.

The 12 Hallmarks (2023 expanded list)

  1. Genomic instability - Accumulation of DNA damage from endogenous (reactive oxygen species, replication errors) and exogenous (UV, radiation) sources. Approximately 10,000-100,000 DNA lesions per cell per day. Repair mechanisms (base excision repair, nucleotide excision repair, double-strand break repair) gradually become less efficient.

  2. Telomere attrition - Telomeres shorten by approximately 50-200 base pairs per cell division. Human telomeres start at approximately 10,000-15,000 base pairs. When critically short (below approximately 4,000 bp), cells enter senescence or apoptosis.

  3. Epigenetic alterations - DNA methylation patterns drift. Histone modifications change. Chromatin remodeling degrades. The epigenetic landscape becomes noisier. Information encoded in epigenetic marks degrades. (This is directly analogous to information entropy: the epigenome loses signal-to-noise ratio.)

  4. Loss of proteostasis - The proteome accumulates misfolded, aggregated, and damaged proteins. Chaperone systems (HSP70, HSP90) decline. Proteasome activity decreases. Autophagy becomes less efficient. Protein aggregates are the hallmark of Alzheimer’s (amyloid-beta, tau), Parkinson’s (alpha-synuclein), and other neurodegenerative diseases.

  5. Disabled macroautophagy - The cell’s self-cleaning mechanism fails. Damaged organelles and protein aggregates accumulate.

  6. Deregulated nutrient sensing - Insulin/IGF-1 signaling, mTOR, AMPK, and sirtuins lose calibration.

  7. Mitochondrial dysfunction - Mitochondrial DNA (mtDNA) accumulates mutations (it lacks the repair mechanisms of nuclear DNA). Electron transport chain efficiency drops. ROS production increases. A vicious cycle: more damage, more ROS, more damage.

  8. Cellular senescence - Cells that have ceased dividing but refuse to die. They secrete pro-inflammatory cytokines (the SASP: senescence-associated secretory phenotype). Senescent cells accumulate with age and poison their neighbors.

  9. Stem cell exhaustion - The regenerative capacity of tissues declines as stem cell pools shrink and remaining stem cells accumulate damage.

  10. Altered intercellular communication - Inflammatory signaling increases (inflammaging). Endocrine signaling degrades. The systemic coordination of tissues breaks down.

  11. Chronic inflammation - Low-grade, systemic inflammation that increases with age.

  12. Dysbiosis - Gut microbiome composition shifts with age, losing diversity.

The Hayflick Limit

Leonard Hayflick (1961): Discovered that normal human cells can divide approximately 40-60 times before entering replicative senescence (the “Hayflick limit”). This overturned the prior assumption (from Alexis Carrel’s flawed experiments) that cells were immortal in culture.

The mechanism: telomere shortening. Each division loses telomeric DNA. When telomeres become critically short, the DNA damage response is triggered (p53, p21, Rb pathways), permanently arresting the cell cycle.

Entropy Framing

Aging can be understood as the gradual failure of entropy-reducing mechanisms:

Each mechanism requires energy (ATP). As mitochondria degrade, energy supply drops. As damage accumulates, repair demand increases. The system crosses a threshold where repair capacity < damage rate. This is the thermodynamic interpretation of aging: the organism can no longer export enough entropy to maintain its low-entropy internal state.

Aging Clocks and Entropy

Paper: Gladyshev (2022). “Aging clocks, entropy, and the limits of age-reversal.” bioRxiv.

This paper explicitly connects epigenetic clocks (Horvath clock, etc.) to thermodynamic entropy, arguing that the drift in methylation patterns is a form of entropy increase in the epigenomic information channel.

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8. MAXIMUM ENTROPY PRODUCTION PRINCIPLE (MEPP)

The Principle

Rod Swenson (1988-1989): First recognized and articulated the Law of Maximum Entropy Production (LMEP).

Statement: A system will select the pathway or assembly of pathways that minimizes the potential (or maximizes the entropy) at the fastest rate given the constraints.

Key paper: Swenson, R. (1989). “Emergent attractors and the law of maximum entropy production: Foundations to a theory of general evolution.” Systems Research, 6(3), 187-197.

What It Means

The second law says entropy increases. MEPP says: among all the ways entropy COULD increase, nature selects the ways that increase it FASTEST.

This is not universally accepted. It is controversial. But it has explanatory power for pattern formation:

Examples

  1. Why rivers branch: A dendritic river network dissipates gravitational potential energy faster than a single straight channel would. The branching pattern maximizes entropy production rate.

  2. Why ecosystems diversify: A complex ecosystem with many trophic levels and nutrient cycles dissipates solar energy more efficiently than a simple one. Biodiversity as entropy optimization.

  3. Why convection cells form: (Connects to Prigogine.) The structured convection pattern produces entropy faster than conduction alone.

  4. Why life exists: Life catalyzes entropy production. A forest-covered landscape radiates at a different (lower) effective temperature than bare rock, dissipating more solar energy into low-grade heat. Life accelerates the universe’s approach to equilibrium.

Theoretical Framework

Dewar (2003, 2005) attempted to derive MEPP from information theory using Jaynes’ maximum entropy formalism (MaxEnt). The idea: if you know only the constraints on a system and apply maximum ignorance (MaxEnt) to its dynamics, MEPP emerges as the most probable macroscopic behavior.

Paper: Martyushev, L.M. & Seleznev, V.D. (2006). “Maximum entropy production principle in physics, chemistry and biology.” Physics Reports, 426(1), 1-45.

Criticisms

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9. ENTROPY IN ECONOMICS AND ORGANIZATIONS

The Analogy and Its Limits

Organizations are open systems far from equilibrium. They maintain internal order (structure, processes, knowledge) by importing energy (money, talent, information) and exporting entropy (waste, turnover, depreciation). When energy input drops below the maintenance threshold, the organization decays.

Nicholas Georgescu-Roegen (1971)

Book: “The Entropy Law and the Economic Process” (Harvard University Press, 1971).

Key arguments:

Organizational Entropy

Manifestations:

  1. Bureaucracy growth: Rules accumulate but rarely get removed. Each rule is a response to a specific failure. The total rule-set grows monotonically (like a system approaching equilibrium).

  2. Process rigidity: Processes optimized for past conditions become constraints in new conditions. The organization’s “microstate” (actual behavior) becomes increasingly mismatched with the “macrostate” (what the environment demands).

  3. Knowledge decay: Institutional knowledge lives in people. People leave. Documentation rots. Codebases accumulate technical debt. The information content of organizational memory degrades (literally: entropy of the information channel increases).

  4. Coordination costs: As organizations grow, the number of communication links grows as O(n^2) where n is the number of people. More links = more noise = more entropy in the communication system. This is why adding people to a late project makes it later (Brooks’ Law).

  5. Technical debt: Every shortcut in code is a local entropy increase. The codebase’s macrostate (“it works”) is maintained by an increasingly improbable arrangement of microstates (“these specific workarounds compensate for these specific design flaws”). Eventually, small changes cause cascading failures because the system is in a fragile, low-probability configuration.

Maintenance Costs Always Increase

This follows directly from the second law applied to complex systems:

Gall’s Law (John Gall, 1975)

“A complex system that works is invariably found to have evolved from a simple system that worked.” The reverse is not true: complex systems designed from scratch never work. This is an entropy argument: the probability of a random complex configuration being functional is astronomically low (low W for the “working” macrostate at high complexity).

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10. THE ARROW OF TIME

The Three Arrows (Hawking)

Stephen Hawking identified three arrows of time in “A Brief History of Time” (1988):

1. Thermodynamic Arrow: Time flows in the direction entropy increases. This is the most fundamental arrow and arguably explains the other two.

2. Psychological Arrow: We remember the past but not the future. Hawking’s argument: forming a memory is a computation. Computation dissipates energy (Landauer’s principle). Dissipation increases entropy. Therefore, memories can only point toward the low-entropy direction (the past). You cannot remember the future because remembering the future would require reducing entropy, which is overwhelmingly improbable.

More precisely: a memory is a correlation between your brain state and a past event. Creating this correlation (recording information) increases entropy elsewhere. The direction of “recording” is set by the entropy gradient.

3. Cosmological Arrow: The universe is expanding. This is connected to the thermodynamic arrow through the Past Hypothesis: the Big Bang was the low-entropy boundary condition, and expansion is the macroscopic expression of the entropy gradient.

Hawking originally (1985) argued these three arrows must point the same way. He briefly considered a collapsing universe might reverse the thermodynamic arrow but retracted this (“my greatest mistake”).

Penrose’s Contribution

Roger Penrose, “The Emperor’s New Mind” (1989) and “The Road to Reality” (2004):

Why We Remember the Past

The deepest explanation: recording a memory is an irreversible process that increases total entropy. The direction of “recording” is defined by the entropy gradient. Since entropy increases toward the future, all records, traces, memories, and fossils point toward the past (the low-entropy direction).

This means: the psychological arrow is not independent. It is a consequence of the thermodynamic arrow. And the thermodynamic arrow is a consequence of the boundary condition (Past Hypothesis).

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11. NEGENTROPY (SCHRODINGER’S “WHAT IS LIFE?”)

The Book

Erwin Schrodinger. “What is Life?” (1944). Based on lectures delivered at Trinity College Dublin in February 1943.

This book is one of the most influential scientific works of the 20th century. It directly inspired Watson and Crick’s pursuit of DNA structure, and Crick explicitly cited it.

The Central Argument

Chapter 6: “Order, Disorder and Entropy”

Schrodinger’s question: How does a living organism maintain its organization? The second law says isolated systems decay toward maximum entropy. But organisms are not isolated. They are open systems.

Original formulation (1944): “What an organism feeds upon is negative entropy.” Life “sucks orderliness from its environment.”

Correction (later editions): Schrodinger acknowledged that the technically correct term is “free energy” (Gibbs free energy, G = H - TS), not “negative entropy.” He wrote in a footnote: “If I had been catering for [physicists] alone I should have let the discussion turn on free energy instead.”

The Thermodynamics

Gibbs Free Energy: G = H - TS

A process is spontaneous when delta-G < 0 (free energy decreases). This happens when either:

Life maintains low internal entropy by increasing entropy OUTSIDE itself:

Quantitative Example

Photosynthesis: A plant absorbs one high-energy photon (low entropy: one quantum in a specific state) and re-emits approximately 20 infrared photons (high entropy: many quanta in many states). The plant keeps the free energy difference to build glucose. Total entropy increases. The plant becomes more ordered.

Brillouin’s Formalization

Leon Brillouin (1953) formalized Schrodinger’s intuition:

The Deep Connection

Schrodinger identified the core paradox that every subsequent entropy theory addresses:

  1. Boltzmann showed entropy increases (Section 1)
  2. The Past Hypothesis explains why it started low (Section 2)
  3. Shannon showed entropy = missing information (Section 3)
  4. Friston showed organisms minimize information-theoretic entropy (Section 4)
  5. Prigogine showed order emerges FROM entropy production (Section 5)
  6. Schrodinger connected them: life feeds on free energy to maintain local order while increasing global entropy

The organism is a temporary eddy in the entropy current. It persists by channeling the flow, not by opposing it.

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CROSS-CUTTING EQUATIONS REFERENCE

Equation Domain Meaning
S = k ln W Statistical mechanics (Boltzmann/Planck) Entropy = k times log of number of microstates
S = -k sum(pi ln pi) Statistical mechanics (Gibbs) Entropy for non-uniform probability distributions
dS >= 0 Thermodynamics (Clausius) Entropy of isolated system never decreases
dS = dQ/T Thermodynamics (Clausius) Entropy change = heat transfer / temperature
H = -sum(pi log2 pi) Information theory (Shannon) Information entropy in bits
E_erase >= kT ln 2 Information physics (Landauer) Minimum energy to erase one bit
G = H - TS Thermodynamics (Gibbs) Free energy = enthalpy minus temperature times entropy
F = E_q[ln q - ln p] Neuroscience/Biology (Friston) Variational free energy

KEY TIMELINE