medicine-and-health physics psychology

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Half-life

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Description

Half-life names a specific shape of decline: exponential decay, in which a constant fraction of whatever remains is lost per unit time. The signature property is scale-invariance — because the loss is proportional to the amount present, the quantity takes the same fixed interval to fall by half regardless of how much there was to begin with, and regardless of when the clock starts. That fixed interval is the half-life. A thousand grams and a single gram of the same isotope both shed half their mass over the same span; a drug at peak concentration and the same drug hours later both halve over the same window. The machinery is the differential equation dx/dt = −λx, whose solution x(t) = x₀·e^(−λt) falls by a factor of two every t½ = ln2 / λ. The decay constant λ (the fraction lost per unit time), the half-life t½ (the time to halve), and the e-folding time or mean lifetime τ = 1/λ (the time to fall to 1/e ≈ 37%) are three parameterizations of one curve. What makes the frame portable is that none of this is specific to physics: any process whose rate of loss is proportional to the amount remaining has a half-life. The cross-domain reach is wide. Radioactive nuclei decay with half-lives from microseconds to billions of years — the original and cleanest instance. A drug’s plasma concentration falls with a pharmacokinetic half-life that sets the dosing interval (steady state on repeated dosing arrives in about four to five half-lives, and after the last dose the drug is nearly gone over the same span). Memory retention falls along Ebbinghaus’s forgetting curve with a characteristic time constant; customer cohorts churn at a roughly constant fractional rate; caches expire on a TTL; unmaintained knowledge and skills decay. In each the diagnostic is one question: is a constant fraction lost per unit time (exponential — has a half-life), or a constant amount (linear — does not)?

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Aliases

“Half-life” is one of several names for parameters of the same exponential-decay curve. The decay constant λ is the fraction lost per unit time; the e-folding time (or mean lifetime) τ = 1/λ is the time to fall to 1/e; the half-life t½ = ln2/λ ≈ 0.693/λ is the time to fall by half. They are interconvertible, and which one is quoted is a domain convention — physics favors half-life and mean lifetime, engineering favors time constants, finance favors decay factors. The term itself is Ernest Rutherford’s: he coined “half-life period” in 1907 for the time in which half of a radioactive sample transforms, and it was shortened to “half-life” in the early 1950s. It has since spread as a general metaphor for anything that loses relevance at a steady proportional rate — the “half-life of knowledge,” the “half-life of a news story.”

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Triggers

User-initiated: User describes something losing a constant proportion over time, decaying toward zero, or halving over a repeatable interval. Vocabulary cues: “half-life,” “exponential decay,” “decays to half,” “drops off exponentially,” “decay rate,” “TTL,” “churn,” “forgetting.” Agent-initiated: Agent notices a quantity declining at a rate that scales with how much is left — fast at first, then ever slower, approaching but never quite reaching zero. Candidate inference: “this looks like exponential decay; what is the half-life, and is the loss genuinely a constant fraction per unit time or a constant amount?” Situation-shape signals: A decline that is fast early and slow late, flattening toward a floor of zero; a process with a natural “time to halve” independent of the starting level; dosing, expiry, churn, or forgetting phenomena; the temptation to model proportional loss as though it were linear.

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Exclusions

• Constant-rate (linear) decay — a fixed amount lost per unit time (a tank draining at constant flow, straight-line depreciation, a fixed budget burned per day), not a fixed fraction. Scale-invariance is the load-bearing feature: under linear decay there is no single “time to halve” — it depends on where you started. If the loss per step is constant in absolute terms, half-life is the wrong frame.
• Exponential growth / compounding — the same machinery with the sign flipped: a doubling time rather than a halving time. Compound interest, snowball-effect, and viral spread all grow at a rate proportional to current size; half-life is specifically the decay branch, and applying it to a growing quantity inverts the dynamics.
• Saturation / bounded approach to a ceiling — saturation is a concave approach upward to an upper bound (each added unit of input buys less); half-life is proportional decay downward toward zero. They are mirror curves in opposite directions, and the test — approaching a ceiling from below, or depleting toward a floor? — tells them apart.
• Mean-reversion toward a nonzero baseline — mean-reversion has a restoring force pulling a quantity back to a baseline it can sit above or below and oscillate around; half-life is one-directional depletion of a nonnegative quantity toward zero, with no baseline, no overshoot, and no return. If there is a level the system returns to, rather than a floor it decays toward, the frame is mean-reversion.

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Structure

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Relationships

• saturation — mirror image: saturation approaches a ceiling from below (bounded growth, diminishing returns); half-life decays toward a floor of zero (proportional loss). Opposite direction, opposite driver.
• mean-reversion — the same exponential-relaxation math (rate proportional to distance-from-target); mean-reversion’s target is a nonzero baseline it restores toward from either side, half-life’s target is zero and the motion is one-way depletion.
• snowball-effect — the growth branch of the same exponential form: rate of change proportional to current size, but compounding up rather than down.

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Examples