> ## Documentation Index > Fetch the complete documentation index at: https://agentconcepts.io/llms.txt > Use this file to discover all available pages before exploring further. # half-life > Exponential decay at a scale-invariant rate: a constant fraction of what remains is lost per unit time, so the quantity falls by half over a fixed interval no matter how large it started or when you start the clock. medicine-and-health physics psychology # Half-life ## 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)?* ## 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." ## 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. ## 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](/concepts/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](/concepts/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](/concepts/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. ## Structure Internal structure of half-life: a table of its component slots and the concepts that fill them. ## Relationships Relationship neighborhood of half-life: a graph of the concepts it connects to and the concepts it is a part of. * [saturation](/concepts/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](/concepts/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](/concepts/snowball-effect) — the growth branch of the same exponential form: rate of change proportional to current size, but compounding up rather than down. ## Examples Rutherford observed that a radioactive sample always takes the same amount of time for half of it to transform, no matter how much sample there is, and coined "half-life period" for that interval. It is the archetype of a constant fractional loss rate: each nucleus has a fixed probability of decaying per unit time, so the population halves over a fixed span — from microseconds to billions of years depending on the isotope. The scale-invariance is exact, and it is what makes radiometric dating possible: the ratio of remaining to decayed material reads out elapsed time without any need to know the original quantity. A drug's plasma concentration falls exponentially as the body clears it, and the elimination half-life — the time for the concentration to drop by half — is the parameter that governs dosing. It is why the schedule follows the half-life rather than the dose size: on repeated dosing, steady state is reached after roughly four to five half-lives, and after the last dose the drug is about 97% gone over the same span. A long half-life means once-a-day dosing and a side effect that lingers for days; a short one means frequent dosing and fast clearance. **Inference**: To reason about *how often* to dose or *how long* an effect lingers, reach for the half-life, not the dose. The dose sets the level; the half-life sets the timescale. Ebbinghaus, memorizing and re-testing lists of nonsense syllables on himself, found that retention drops fast at first and then ever more slowly — the forgetting curve, commonly modeled as R = e^(−t/S) with S a characteristic memory-strength time constant. It is the psychological instance of proportional decay: a roughly constant fraction of the still-retrievable material is lost per unit time, giving memory an approximate half-life. Spaced repetition works by resetting the curve before it bottoms out — each review lengthens the effective time constant, so the same material decays more slowly on the next pass.