3478:
31:
2981:
1720:
3477:
1736:
A quantity may decay via two or more different processes simultaneously. In general, these processes (often called "decay modes", "decay channels", "decay routes" etc.) have different probabilities of occurring, and thus occur at different rates with different half-lives, in parallel. The total decay
2575:
of corresponding processes. Terms "partial half-life" and "partial mean life" denote quantities derived from a decay constant as if the given decay mode were the only decay mode for the quantity. The term "partial half-life" is misleading, because it cannot be measured as a time interval for which a
3233:
is exposed to a medium of another temperature, the temperature difference between the object and the medium follows exponential decay (in the limit of slow processes; equivalent to "good" heat conduction inside the object, so that its temperature remains relatively uniform through its volume). See
3385:
a retirement fund will decay exponentially being subject to discrete payout amounts, usually monthly, and an input subject to a continuous interest rate. A differential equation dA/dt = input – output can be written and solved to find the time to reach any amount A, remaining in the
1244:
This is the form of the equation that is most commonly used to describe exponential decay. Any one of decay constant, mean lifetime, or half-life is sufficient to characterise the decay. The notation λ for the decay constant is a remnant of the usual notation for an
2275:
2739:
1491:
2728:
1865:
1567:
3247:
After excitation, the emission intensity – which is proportional to the number of excited atoms or molecules – of a luminescent material decays exponentially. Depending on the number of mechanisms involved, the decay can be mono- or
2068:
817:
Thus, the amount of material left is 2 = 1/2 raised to the (whole or fractional) number of half-lives that have passed. Thus, after 3 half-lives there will be 1/2 = 1/8 of the original material left.
2362:
934:
2999:, the agent of interest might be situated in a decay chain, where the accumulation is governed by exponential decay of a source agent, while the agent of interest itself decays by means of an exponential process.
3469:
assigns each route a weight that gets bigger each time the route changes its state and decays exponentially with time. When the weight reaches a certain limit, no more flapping is done, thus suppressing the
701:
1210:
2477:
2167:
1540:
1061:
352:
143:
1909:
1001:
812:
214:
1374:
444:
1112:
2976:{\displaystyle T_{1/2}={\frac {\ln 2}{\lambda _{c}}}={\frac {\ln 2}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}={\frac {t_{1}t_{2}t_{3}}{(t_{1}t_{2})+(t_{1}t_{3})+(t_{2}t_{3})}}.}
2112:
3186:
3309:
to a different state, the number of atoms in the original state follows exponential decay as long as the remaining number of atoms is large. The decay product is termed a
1936:
516:
3143:
1393:
2139:
2624:
2159:
572:
2616:
2515:
1715:{\displaystyle \tau =\langle t\rangle =\int _{0}^{\infty }t\cdot c\cdot N_{0}e^{-\lambda t}\,dt=\int _{0}^{\infty }\lambda te^{-\lambda t}\,dt={\frac {1}{\lambda }}.}
3009:
In the pharmacology setting, some ingested substances might be absorbed into the body by a process reasonably modeled as exponential decay, or might be deliberately
2569:
2542:
2385:
1751:
1299:
839:
724:
543:
467:
375:
313:
3112:
of exponential decay (when the capacitor is charged from a constant voltage source though a constant resistance). The exponential time-constant for the process is
595:
A more intuitive characteristic of exponential decay for many people is the time required for the decaying quantity to fall to one half of its initial value. (If
3716:
3395:, the (debatable) assumption of a constant decay rate in languages allows one to estimate the age of single languages. (To compute the time of split between
3288:
such as light or X-rays or gamma rays in an absorbent medium, follows an exponential decrease with distance into the absorbing medium. This is known as the
1944:
1313:
of an element of the assembly is the time elapsed between some reference time and the removal of that element from the assembly, the mean lifetime is the
34:
A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant (
2283:
847:
3202:
makes an interesting example of multiple decay processes, with each resistor representing a separate process. In fact, the expression for the
357:
The mean lifetime can be looked at as a "scaling time", because the exponential decay equation can be written in terms of the mean lifetime,
620:
1127:
3358:
3028:
Many decay processes that are often treated as exponential, are really only exponential so long as the sample is large and the
2393:
3645:
3196:
2270:{\displaystyle {\frac {1}{\tau _{c}}}=\lambda _{c}=\lambda _{1}+\lambda _{2}={\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}}
3487:
Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/
3484:
3465:
and, even more, broadcasting useless information to peer routers. To prevent this undesired behavior, an algorithm named
577:
A very similar equation will be seen below, which arises when the base of the exponential is chosen to be 2, rather than
3668:
3275:"alpha half-life" and "beta half-life" of a substance measure how quickly a substance is distributed and eliminated.
1502:
1016:
321:
96:
3434:
1873:
957:
3362:
3220:
decreases approximately exponentially with increasing height above sea level, at a rate of about 12% per 1000m.
736:
280:, it is possible to compute the average length of time that an element remains in the set. This is called the
158:
17:
1554:
1326:
3189:
383:
55:
1384:
1072:
603:) is discrete, then this is the median life-time rather than the mean life-time.) This time is called the
3235:
3496:
2733:
For a decay by three simultaneous exponential processes the total half-life can be computed as above:
2083:
3660:
3285:
3021:
Exponential decay occurs in a wide variety of situations. Most of these fall into the domain of the
476:
3711:
3598:
1486:{\displaystyle 1=\int _{0}^{\infty }c\cdot N_{0}e^{-\lambda t}\,dt=c\cdot {\frac {N_{0}}{\lambda }}}
3148:
3064:
1550:
1914:
486:
3115:
3010:
2723:{\displaystyle T_{1/2}={\frac {\ln 2}{\lambda _{c}}}={\frac {\ln 2}{\lambda _{1}+\lambda _{2}}}.}
1118:
1007:
3203:
3593:
3453:
controlling that path has to repeatedly add and remove the path record from its routing table (
3101:
2117:
2077:
727:
2144:
1860:{\displaystyle -{\frac {dN(t)}{dt}}=N\lambda _{1}+N\lambda _{2}=(\lambda _{1}+\lambda _{2})N.}
548:
3731:
3706:
3206:
of two resistors in parallel mirrors the equation for the half-life with two decay processes.
2586:
2485:
1546:
1254:
59:
2387:
by a constant factor, the same equation holds in terms of the two corresponding half-lives:
3585:
3462:
3272:
3267:
3217:
3029:
2547:
2520:
2370:
1558:
1309:
of the amount of time before an object is removed from the assembly. Specifically, if the
1284:
824:
709:
528:
452:
360:
298:
614:. The half-life can be written in terms of the decay constant, or the mean lifetime, as:
8:
3510:
3409:
3336:
3589:
1553:(i.e. the individual lifetime of each object is exponentially distributed), which has a
3611:
3515:
3450:
3089:
3046:
1731:
3344:
2063:{\displaystyle N(t)=N_{0}e^{-(\lambda _{1}+\lambda _{2})t}=N_{0}e^{-(\lambda _{c})t}.}
3689:
3677:
3664:
3641:
3615:
3607:
3520:
3317:
3306:
3195:
Furthermore, the particular case of a capacitor or inductor changing through several
3056:
1277:
Given an assembly of elements, the number of which decreases ultimately to zero, the
277:
58:
to its current value. Symbolically, this process can be expressed by the following
3603:
3391:
3063:. Reactions whose rate depends only on the concentration of one reactant (known as
3003:
2996:
1250:
3032:
holds. For small samples, a more general analysis is necessary, accounting for a
3576:
Leike, A. (2002). "Demonstration of the exponential decay law using beer froth".
3279:
3085:
3033:
3022:
2992:
1870:
The solution to this equation is given in the previous section, where the sum of
1314:
3523:
for the mathematics of chains of exponential processes with differing constants
3335:
Some vibrations may decay exponentially; this characteristic is often found in
3100:) discharges with exponential decay (when the capacitor experiences a constant
3079:
1306:
3437:
can be deviated to. When one of these paths repeatedly changes its state from
3725:
3430:
3399:
languages requires additional assumptions, independent of exponential decay).
3340:
3296:
3224:
3052:
1266:
291:
2357:{\displaystyle \tau _{c}={\frac {\tau _{1}\tau _{2}}{\tau _{1}+\tau _{2}}}.}
3302:
3289:
3252:
3242:
3109:
940:
929:{\displaystyle \tau ={\frac {t_{1/2}}{\ln(2)}}\approx 1.4427\cdot t_{1/2}.}
3685:
3637:
3230:
30:
3348:
3330:
3323:
3310:
3262:
3256:
3212:
1246:
3093:
3072:
590:
315:, relates to the decay rate constant, λ, in the following way:
3422:
3199:
3060:
2577:
47:
3351:
system will simply return to equilibrium via an exponential decay.
3261:
It is found that many administered substances are distributed and
469:
is the time at which the population of the assembly is reduced to
3417:
3380:
3693:
3322:
The decline in resistance of a
Negative Temperature Coefficient
3068:
841:
is equal to the half-life divided by the natural log of 2, or:
696:{\displaystyle t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2).}
71:
3655:
Serway, Raymond A.; Moses, Clement J.; Moyer, Curt A. (1989),
1205:{\displaystyle N(t)=e^{C}e^{-\lambda t}=N_{0}e^{-\lambda t}\,}
943:
has a half-life of 138 days, and a mean lifetime of 200 days.
3682:
Differential
Equations with Applications and Historical Notes
483:≈ 0.367879441 times its initial value. This is equivalent to
27:
Decrease in value at a rate proportional to the current value
3067:) consequently follow exponential decay. For instance, many
3366:
2583:
In terms of separate decay constants, the total half-life
2076:
associated with individual processes is by definition the
149:
3458:
3426:
2472:{\displaystyle T_{1/2}={\frac {t_{1}t_{2}}{t_{1}+t_{2}}}}
1745:
of the decay routes; thus, in the case of two processes:
3499:, hover over a graph to highlight it and its complement.
946:
521:
For example, if the initial population of the assembly,
248:
is the initial quantity, that is, the quantity at time
3151:
3118:
2742:
2627:
2589:
2550:
2523:
2488:
2396:
2373:
2286:
2170:
2147:
2120:
2086:
1947:
1917:
1876:
1754:
1570:
1505:
1396:
1329:
1287:
1130:
1075:
1019:
960:
850:
827:
739:
712:
623:
551:
531:
489:
455:
386:
363:
324:
301:
161:
99:
3634:
2517:
is the combined or total half-life for the process,
581:. In that case the scaling time is the "half-life".
276:), is the number of discrete elements in a certain
3180:
3137:
2975:
2722:
2610:
2563:
2536:
2509:
2471:
2379:
2356:
2269:
2153:
2133:
2106:
2062:
1930:
1903:
1859:
1714:
1534:
1485:
1368:
1293:
1272:
1204:
1106:
1055:
1006:or, by rearranging (applying the technique called
995:
928:
833:
806:
730:is absorbed into the base, this equation becomes:
718:
695:
566:
537:
510:
461:
438:
369:
346:
307:
208:
137:
3457:the path), thus spending local resources such as
1725:
951:The equation that describes exponential decay is
3723:
3654:
3540:
3271:) according to exponential decay patterns. The
2986:
3059:depend on the concentration of one or another
3712:A stochastic simulation of exponential decay
3016:
1583:
1577:
1535:{\displaystyle c={\frac {\lambda }{N_{0}}}.}
1249:. In this case, λ is the eigenvalue of the
1226:, is obtained by evaluating the equation at
1056:{\displaystyle {\frac {dN}{N}}=-\lambda dt.}
347:{\displaystyle \tau ={\frac {1}{\lambda }}.}
138:{\displaystyle {\frac {dN}{dt}}=-\lambda N.}
3632:
3564:
1904:{\displaystyle \lambda _{1}+\lambda _{2}\,}
996:{\displaystyle {\frac {dN}{dt}}=-\lambda N}
258:
1383:be the normalizing factor to convert to a
525:(0), is 1000, then the population at time
3597:
3369:froth obeys the law of exponential decay.
3159:
3155:
3128:
2080:of corresponding partial decay constant:
1911:is treated as a new total decay constant
1900:
1689:
1642:
1450:
1365:
1269:. The units of the decay constant are s.
1201:
1103:
807:{\displaystyle N(t)=N_{0}2^{-t/t_{1/2}}.}
377:, instead of the decay constant, λ:
209:{\displaystyle N(t)=N_{0}e^{-\lambda t},}
1369:{\displaystyle N=N_{0}e^{-\lambda t},\,}
29:
3676:
3552:
2367:Since half-lives differ from mean life
726:in the exponential equation above, and
439:{\displaystyle N(t)=N_{0}e^{-t/\tau },}
14:
3724:
3359:Ludwig Maximilian University of Munich
3575:
3188:The same equations can be applied to
1320:Starting from the population formula
947:Solution of the differential equation
706:When this expression is inserted for
1237:is defined as being the quantity at
1107:{\displaystyle \ln N=-\lambda t+C\,}
3403:
3039:
3002:These systems are solved using the
148:The solution to this equation (see
24:
3373:
1662:
1599:
1413:
607:, and often denoted by the symbol
25:
3743:
3700:
3433:in order to remember the paths a
3108:) and similarly charges with the
3013:to have such a release profile.
38:) of 25, 5, 1, 1/5, and 1/25 for
3476:
2107:{\displaystyle \tau =1/\lambda }
1557:. We can compute it here using
263:
74:) is a positive rate called the
3541:Serway, Moses & Moyer (1989
1273:Derivation of the mean lifetime
3569:
3558:
3546:
3534:
3172:
3166:
2964:
2941:
2935:
2912:
2906:
2883:
2049:
2036:
2007:
1981:
1957:
1951:
1848:
1822:
1773:
1767:
1726:Decay by two or more processes
1215:where the final substitution,
1140:
1134:
890:
884:
749:
743:
687:
681:
660:
654:
561:
555:
396:
390:
171:
165:
13:
1:
3626:
3337:damped mechanical oscillators
3181:{\displaystyle R\,C\,\ln(2).}
1317:of the individual lifetimes.
821:Therefore, the mean lifetime
3707:Exponential decay calculator
3326:as temperature is increased.
2987:Decay series / coupled decay
1931:{\displaystyle \lambda _{c}}
1385:probability density function
584:
511:{\displaystyle \log _{2}{e}}
7:
3636:(10th ed.). New York:
3578:European Journal of Physics
3504:
3192:of current in an inductor.
3138:{\displaystyle \tau =R\,C,}
10:
3748:
3717:Tutorial on time constants
3608:10.1088/0143-0807/23/1/304
3407:
3075:reactions behave this way.
1737:rate of the quantity
1729:
1301:, (also called simply the
588:
268:If the decaying quantity,
76:exponential decay constant
54:if it decreases at a rate
3661:Harcourt Brace Jovanovich
3286:electromagnetic radiation
3017:Applications and examples
2141:can be given in terms of
2134:{\displaystyle \tau _{c}}
1555:well-known expected value
3527:
2154:{\displaystyle \lambda }
1551:exponential distribution
567:{\displaystyle N(\tau )}
259:Measuring rates of decay
230:is the quantity at time
3495:approximations. In the
3365:for demonstrating that
3339:, and used in creating
3236:Newton's law of cooling
3088:(or, equivalently, the
2611:{\displaystyle T_{1/2}}
2510:{\displaystyle T_{1/2}}
1545:Exponential decay is a
1265:) as the corresponding
1119:constant of integration
1008:separation of variables
518:≈ 1.442695 half-lives.
88:transformation constant
80:disintegration constant
3467:route flapping damping
3182:
3139:
2977:
2724:
2612:
2565:
2538:
2511:
2473:
2381:
2358:
2271:
2155:
2135:
2108:
2078:multiplicative inverse
2064:
1932:
1905:
1861:
1716:
1536:
1487:
1370:
1295:
1206:
1108:
1057:
997:
930:
835:
808:
720:
697:
568:
539:
512:
463:
440:
371:
348:
309:
210:
139:
43:
3273:biological half-lives
3204:equivalent resistance
3183:
3140:
3065:first-order reactions
2978:
2725:
2613:
2566:
2564:{\displaystyle t_{2}}
2539:
2537:{\displaystyle t_{1}}
2512:
2474:
2382:
2380:{\displaystyle \tau }
2359:
2272:
2156:
2136:
2109:
2065:
1933:
1906:
1862:
1717:
1537:
1488:
1371:
1296:
1294:{\displaystyle \tau }
1255:differential operator
1207:
1109:
1066:Integrating, we have
1058:
998:
931:
836:
834:{\displaystyle \tau }
809:
721:
719:{\displaystyle \tau }
698:
569:
540:
538:{\displaystyle \tau }
513:
464:
462:{\displaystyle \tau }
441:
372:
370:{\displaystyle \tau }
349:
310:
308:{\displaystyle \tau }
211:
140:
60:differential equation
33:
3429:, has to maintain a
3229:If an object at one
3218:Atmospheric pressure
3149:
3145:so the half-life is
3116:
3055:of certain types of
3030:law of large numbers
2740:
2625:
2587:
2576:certain quantity is
2548:
2521:
2486:
2394:
2371:
2284:
2168:
2145:
2118:
2084:
1945:
1915:
1874:
1752:
1568:
1559:integration by parts
1503:
1496:or, on rearranging,
1394:
1327:
1285:
1128:
1073:
1017:
958:
848:
825:
737:
710:
621:
549:
529:
487:
453:
384:
361:
322:
299:
159:
97:
66:is the quantity and
3590:2002EJPh...23...21L
3511:Exponential formula
3410:Exponential backoff
3357:Arnd Leike, of the
2618:can be shown to be
1666:
1603:
1417:
1311:individual lifetime
3678:Simmons, George F.
3565:McGraw-Hill (2007)
3516:Exponential growth
3248:multi-exponential.
3178:
3135:
3057:chemical reactions
3047:Chemical reactions
2973:
2720:
2608:
2573:partial half-lives
2561:
2534:
2507:
2469:
2377:
2354:
2267:
2151:
2131:
2104:
2060:
1928:
1901:
1857:
1732:Branching fraction
1712:
1652:
1589:
1532:
1483:
1403:
1366:
1291:
1202:
1104:
1053:
993:
926:
831:
804:
716:
693:
564:
535:
508:
459:
436:
367:
344:
305:
206:
135:
44:
3647:978-0-07-144143-8
3521:Radioactive decay
3318:Thermoelectricity
3307:radioactive decay
3301:In a sample of a
3284:The intensity of
3092:) contained in a
2968:
2843:
2787:
2715:
2672:
2467:
2349:
2265:
2245:
2186:
2074:Partial mean life
1785:
1707:
1527:
1481:
1033:
979:
894:
667:
339:
118:
52:exponential decay
16:(Redirected from
3739:
3696:
3673:
3651:
3620:
3619:
3601:
3573:
3567:
3562:
3556:
3550:
3544:
3538:
3480:
3418:routing protocol
3404:Computer science
3392:glottochronology
3187:
3185:
3184:
3179:
3144:
3142:
3141:
3136:
3040:Natural sciences
3023:natural sciences
3004:Bateman equation
2997:pharmacokinetics
2982:
2980:
2979:
2974:
2969:
2967:
2963:
2962:
2953:
2952:
2934:
2933:
2924:
2923:
2905:
2904:
2895:
2894:
2881:
2880:
2879:
2870:
2869:
2860:
2859:
2849:
2844:
2842:
2841:
2840:
2828:
2827:
2815:
2814:
2804:
2793:
2788:
2786:
2785:
2776:
2765:
2760:
2759:
2755:
2729:
2727:
2726:
2721:
2716:
2714:
2713:
2712:
2700:
2699:
2689:
2678:
2673:
2671:
2670:
2661:
2650:
2645:
2644:
2640:
2617:
2615:
2614:
2609:
2607:
2606:
2602:
2570:
2568:
2567:
2562:
2560:
2559:
2543:
2541:
2540:
2535:
2533:
2532:
2516:
2514:
2513:
2508:
2506:
2505:
2501:
2478:
2476:
2475:
2470:
2468:
2466:
2465:
2464:
2452:
2451:
2441:
2440:
2439:
2430:
2429:
2419:
2414:
2413:
2409:
2386:
2384:
2383:
2378:
2363:
2361:
2360:
2355:
2350:
2348:
2347:
2346:
2334:
2333:
2323:
2322:
2321:
2312:
2311:
2301:
2296:
2295:
2276:
2274:
2273:
2268:
2266:
2264:
2263:
2251:
2246:
2244:
2243:
2231:
2226:
2225:
2213:
2212:
2200:
2199:
2187:
2185:
2184:
2172:
2160:
2158:
2157:
2152:
2140:
2138:
2137:
2132:
2130:
2129:
2113:
2111:
2110:
2105:
2100:
2069:
2067:
2066:
2061:
2056:
2055:
2048:
2047:
2027:
2026:
2014:
2013:
2006:
2005:
1993:
1992:
1972:
1971:
1937:
1935:
1934:
1929:
1927:
1926:
1910:
1908:
1907:
1902:
1899:
1898:
1886:
1885:
1866:
1864:
1863:
1858:
1847:
1846:
1834:
1833:
1818:
1817:
1802:
1801:
1786:
1784:
1776:
1759:
1741:is given by the
1721:
1719:
1718:
1713:
1708:
1700:
1688:
1687:
1665:
1660:
1641:
1640:
1625:
1624:
1602:
1597:
1541:
1539:
1538:
1533:
1528:
1526:
1525:
1513:
1492:
1490:
1489:
1484:
1482:
1477:
1476:
1467:
1449:
1448:
1433:
1432:
1416:
1411:
1375:
1373:
1372:
1367:
1361:
1360:
1345:
1344:
1300:
1298:
1297:
1292:
1211:
1209:
1208:
1203:
1200:
1199:
1184:
1183:
1171:
1170:
1155:
1154:
1113:
1111:
1110:
1105:
1062:
1060:
1059:
1054:
1034:
1029:
1021:
1002:
1000:
999:
994:
980:
978:
970:
962:
935:
933:
932:
927:
922:
921:
917:
895:
893:
876:
875:
871:
858:
840:
838:
837:
832:
813:
811:
810:
805:
800:
799:
798:
797:
793:
780:
764:
763:
725:
723:
722:
717:
702:
700:
699:
694:
668:
663:
646:
641:
640:
636:
573:
571:
570:
565:
544:
542:
541:
536:
517:
515:
514:
509:
507:
499:
498:
482:
481:
473:
468:
466:
465:
460:
445:
443:
442:
437:
432:
431:
427:
411:
410:
376:
374:
373:
368:
353:
351:
350:
345:
340:
332:
314:
312:
311:
306:
254:
247:
233:
229:
215:
213:
212:
207:
202:
201:
186:
185:
144:
142:
141:
136:
119:
117:
109:
101:
69:
65:
41:
37:
21:
3747:
3746:
3742:
3741:
3740:
3738:
3737:
3736:
3722:
3721:
3703:
3671:
3648:
3629:
3624:
3623:
3599:10.1.1.693.5948
3574:
3570:
3563:
3559:
3551:
3547:
3539:
3535:
3530:
3507:
3502:
3501:
3500:
3486:
3481:
3412:
3406:
3376:
3374:Social sciences
3305:that undergoes
3280:Physical optics
3150:
3147:
3146:
3117:
3114:
3113:
3086:electric charge
3042:
3034:Poisson process
3019:
2993:nuclear science
2989:
2958:
2954:
2948:
2944:
2929:
2925:
2919:
2915:
2900:
2896:
2890:
2886:
2882:
2875:
2871:
2865:
2861:
2855:
2851:
2850:
2848:
2836:
2832:
2823:
2819:
2810:
2806:
2805:
2794:
2792:
2781:
2777:
2766:
2764:
2751:
2747:
2743:
2741:
2738:
2737:
2708:
2704:
2695:
2691:
2690:
2679:
2677:
2666:
2662:
2651:
2649:
2636:
2632:
2628:
2626:
2623:
2622:
2598:
2594:
2590:
2588:
2585:
2584:
2555:
2551:
2549:
2546:
2545:
2528:
2524:
2522:
2519:
2518:
2497:
2493:
2489:
2487:
2484:
2483:
2460:
2456:
2447:
2443:
2442:
2435:
2431:
2425:
2421:
2420:
2418:
2405:
2401:
2397:
2395:
2392:
2391:
2372:
2369:
2368:
2342:
2338:
2329:
2325:
2324:
2317:
2313:
2307:
2303:
2302:
2300:
2291:
2287:
2285:
2282:
2281:
2259:
2255:
2250:
2239:
2235:
2230:
2221:
2217:
2208:
2204:
2195:
2191:
2180:
2176:
2171:
2169:
2166:
2165:
2146:
2143:
2142:
2125:
2121:
2119:
2116:
2115:
2096:
2085:
2082:
2081:
2043:
2039:
2032:
2028:
2022:
2018:
2001:
1997:
1988:
1984:
1977:
1973:
1967:
1963:
1946:
1943:
1942:
1922:
1918:
1916:
1913:
1912:
1894:
1890:
1881:
1877:
1875:
1872:
1871:
1842:
1838:
1829:
1825:
1813:
1809:
1797:
1793:
1777:
1760:
1758:
1753:
1750:
1749:
1734:
1728:
1699:
1677:
1673:
1661:
1656:
1630:
1626:
1620:
1616:
1598:
1593:
1569:
1566:
1565:
1547:scalar multiple
1521:
1517:
1512:
1504:
1501:
1500:
1472:
1468:
1466:
1438:
1434:
1428:
1424:
1412:
1407:
1395:
1392:
1391:
1350:
1346:
1340:
1336:
1328:
1325:
1324:
1315:arithmetic mean
1286:
1283:
1282:
1275:
1236:
1221:
1189:
1185:
1179:
1175:
1160:
1156:
1150:
1146:
1129:
1126:
1125:
1117:where C is the
1074:
1071:
1070:
1022:
1020:
1018:
1015:
1014:
971:
963:
961:
959:
956:
955:
949:
913:
909:
905:
877:
867:
863:
859:
857:
849:
846:
845:
826:
823:
822:
789:
785:
781:
776:
769:
765:
759:
755:
738:
735:
734:
711:
708:
707:
647:
645:
632:
628:
624:
622:
619:
618:
613:
593:
587:
550:
547:
546:
530:
527:
526:
503:
494:
490:
488:
485:
484:
475:
471:
470:
454:
451:
450:
423:
416:
412:
406:
402:
385:
382:
381:
362:
359:
358:
331:
323:
320:
319:
300:
297:
296:
284:(or simply the
266:
261:
249:
241:
235:
231:
220:
191:
187:
181:
177:
160:
157:
156:
110:
102:
100:
98:
95:
94:
67:
63:
39:
35:
28:
23:
22:
15:
12:
11:
5:
3745:
3735:
3734:
3720:
3719:
3714:
3709:
3702:
3701:External links
3699:
3698:
3697:
3674:
3669:
3659:, Fort Worth:
3657:Modern Physics
3652:
3646:
3628:
3625:
3622:
3621:
3568:
3557:
3545:
3543:, p. 384)
3532:
3531:
3529:
3526:
3525:
3524:
3518:
3513:
3506:
3503:
3483:
3482:
3475:
3474:
3473:
3472:
3471:
3405:
3402:
3401:
3400:
3387:
3375:
3372:
3371:
3370:
3363:Ig Nobel Prize
3352:
3341:ADSR envelopes
3327:
3314:
3293:
3276:
3249:
3239:
3221:
3209:
3208:
3207:
3177:
3174:
3171:
3168:
3165:
3162:
3158:
3154:
3134:
3131:
3127:
3124:
3121:
3104:of resistance
3080:Electrostatics
3076:
3041:
3038:
3018:
3015:
2988:
2985:
2984:
2983:
2972:
2966:
2961:
2957:
2951:
2947:
2943:
2940:
2937:
2932:
2928:
2922:
2918:
2914:
2911:
2908:
2903:
2899:
2893:
2889:
2885:
2878:
2874:
2868:
2864:
2858:
2854:
2847:
2839:
2835:
2831:
2826:
2822:
2818:
2813:
2809:
2803:
2800:
2797:
2791:
2784:
2780:
2775:
2772:
2769:
2763:
2758:
2754:
2750:
2746:
2731:
2730:
2719:
2711:
2707:
2703:
2698:
2694:
2688:
2685:
2682:
2676:
2669:
2665:
2660:
2657:
2654:
2648:
2643:
2639:
2635:
2631:
2605:
2601:
2597:
2593:
2558:
2554:
2531:
2527:
2504:
2500:
2496:
2492:
2480:
2479:
2463:
2459:
2455:
2450:
2446:
2438:
2434:
2428:
2424:
2417:
2412:
2408:
2404:
2400:
2376:
2365:
2364:
2353:
2345:
2341:
2337:
2332:
2328:
2320:
2316:
2310:
2306:
2299:
2294:
2290:
2278:
2277:
2262:
2258:
2254:
2249:
2242:
2238:
2234:
2229:
2224:
2220:
2216:
2211:
2207:
2203:
2198:
2194:
2190:
2183:
2179:
2175:
2150:
2128:
2124:
2103:
2099:
2095:
2092:
2089:
2071:
2070:
2059:
2054:
2051:
2046:
2042:
2038:
2035:
2031:
2025:
2021:
2017:
2012:
2009:
2004:
2000:
1996:
1991:
1987:
1983:
1980:
1976:
1970:
1966:
1962:
1959:
1956:
1953:
1950:
1925:
1921:
1897:
1893:
1889:
1884:
1880:
1868:
1867:
1856:
1853:
1850:
1845:
1841:
1837:
1832:
1828:
1824:
1821:
1816:
1812:
1808:
1805:
1800:
1796:
1792:
1789:
1783:
1780:
1775:
1772:
1769:
1766:
1763:
1757:
1727:
1724:
1723:
1722:
1711:
1706:
1703:
1698:
1695:
1692:
1686:
1683:
1680:
1676:
1672:
1669:
1664:
1659:
1655:
1651:
1648:
1645:
1639:
1636:
1633:
1629:
1623:
1619:
1615:
1612:
1609:
1606:
1601:
1596:
1592:
1588:
1585:
1582:
1579:
1576:
1573:
1543:
1542:
1531:
1524:
1520:
1516:
1511:
1508:
1494:
1493:
1480:
1475:
1471:
1465:
1462:
1459:
1456:
1453:
1447:
1444:
1441:
1437:
1431:
1427:
1423:
1420:
1415:
1410:
1406:
1402:
1399:
1377:
1376:
1364:
1359:
1356:
1353:
1349:
1343:
1339:
1335:
1332:
1307:expected value
1290:
1274:
1271:
1234:
1219:
1213:
1212:
1198:
1195:
1192:
1188:
1182:
1178:
1174:
1169:
1166:
1163:
1159:
1153:
1149:
1145:
1142:
1139:
1136:
1133:
1115:
1114:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1064:
1063:
1052:
1049:
1046:
1043:
1040:
1037:
1032:
1028:
1025:
1004:
1003:
992:
989:
986:
983:
977:
974:
969:
966:
948:
945:
937:
936:
925:
920:
916:
912:
908:
904:
901:
898:
892:
889:
886:
883:
880:
874:
870:
866:
862:
856:
853:
830:
815:
814:
803:
796:
792:
788:
784:
779:
775:
772:
768:
762:
758:
754:
751:
748:
745:
742:
715:
704:
703:
692:
689:
686:
683:
680:
677:
674:
671:
666:
662:
659:
656:
653:
650:
644:
639:
635:
631:
627:
611:
589:Main article:
586:
583:
563:
560:
557:
554:
534:
506:
502:
497:
493:
458:
447:
446:
435:
430:
426:
422:
419:
415:
409:
405:
401:
398:
395:
392:
389:
366:
355:
354:
343:
338:
335:
330:
327:
304:
265:
262:
260:
257:
239:
217:
216:
205:
200:
197:
194:
190:
184:
180:
176:
173:
170:
167:
164:
146:
145:
134:
131:
128:
125:
122:
116:
113:
108:
105:
50:is subject to
26:
18:Decay constant
9:
6:
4:
3:
2:
3744:
3733:
3730:
3729:
3727:
3718:
3715:
3713:
3710:
3708:
3705:
3704:
3695:
3691:
3687:
3683:
3679:
3675:
3672:
3670:0-03-004844-3
3666:
3662:
3658:
3653:
3649:
3643:
3639:
3635:
3631:
3630:
3617:
3613:
3609:
3605:
3600:
3595:
3591:
3587:
3583:
3579:
3572:
3566:
3561:
3555:, p. 15)
3554:
3553:Simmons (1972
3549:
3542:
3537:
3533:
3522:
3519:
3517:
3514:
3512:
3509:
3508:
3498:
3494:
3490:
3485:
3479:
3468:
3464:
3460:
3456:
3452:
3448:
3444:
3443:not available
3440:
3436:
3432:
3431:routing table
3428:
3424:
3420:
3419:
3414:
3413:
3411:
3398:
3394:
3393:
3388:
3384:
3382:
3378:
3377:
3368:
3364:
3360:
3356:
3353:
3350:
3346:
3342:
3338:
3334:
3332:
3328:
3325:
3321:
3319:
3315:
3312:
3308:
3304:
3300:
3298:
3297:Radioactivity
3294:
3291:
3287:
3283:
3281:
3277:
3274:
3270:
3269:
3264:
3260:
3258:
3254:
3250:
3246:
3244:
3240:
3237:
3232:
3228:
3226:
3225:Heat transfer
3222:
3219:
3216:
3214:
3210:
3205:
3201:
3198:
3194:
3193:
3191:
3175:
3169:
3163:
3160:
3156:
3152:
3132:
3129:
3125:
3122:
3119:
3111:
3107:
3103:
3102:external load
3099:
3096:(capacitance
3095:
3091:
3087:
3083:
3081:
3077:
3074:
3070:
3066:
3062:
3058:
3054:
3050:
3048:
3044:
3043:
3037:
3035:
3031:
3026:
3024:
3014:
3012:
3007:
3005:
3000:
2998:
2994:
2970:
2959:
2955:
2949:
2945:
2938:
2930:
2926:
2920:
2916:
2909:
2901:
2897:
2891:
2887:
2876:
2872:
2866:
2862:
2856:
2852:
2845:
2837:
2833:
2829:
2824:
2820:
2816:
2811:
2807:
2801:
2798:
2795:
2789:
2782:
2778:
2773:
2770:
2767:
2761:
2756:
2752:
2748:
2744:
2736:
2735:
2734:
2717:
2709:
2705:
2701:
2696:
2692:
2686:
2683:
2680:
2674:
2667:
2663:
2658:
2655:
2652:
2646:
2641:
2637:
2633:
2629:
2621:
2620:
2619:
2603:
2599:
2595:
2591:
2581:
2579:
2574:
2571:are so-named
2556:
2552:
2529:
2525:
2502:
2498:
2494:
2490:
2461:
2457:
2453:
2448:
2444:
2436:
2432:
2426:
2422:
2415:
2410:
2406:
2402:
2398:
2390:
2389:
2388:
2374:
2351:
2343:
2339:
2335:
2330:
2326:
2318:
2314:
2308:
2304:
2297:
2292:
2288:
2280:
2279:
2260:
2256:
2252:
2247:
2240:
2236:
2232:
2227:
2222:
2218:
2214:
2209:
2205:
2201:
2196:
2192:
2188:
2181:
2177:
2173:
2164:
2163:
2162:
2148:
2126:
2122:
2114:. A combined
2101:
2097:
2093:
2090:
2087:
2079:
2075:
2057:
2052:
2044:
2040:
2033:
2029:
2023:
2019:
2015:
2010:
2002:
1998:
1994:
1989:
1985:
1978:
1974:
1968:
1964:
1960:
1954:
1948:
1941:
1940:
1939:
1923:
1919:
1895:
1891:
1887:
1882:
1878:
1854:
1851:
1843:
1839:
1835:
1830:
1826:
1819:
1814:
1810:
1806:
1803:
1798:
1794:
1790:
1787:
1781:
1778:
1770:
1764:
1761:
1755:
1748:
1747:
1746:
1744:
1740:
1733:
1709:
1704:
1701:
1696:
1693:
1690:
1684:
1681:
1678:
1674:
1670:
1667:
1657:
1653:
1649:
1646:
1643:
1637:
1634:
1631:
1627:
1621:
1617:
1613:
1610:
1607:
1604:
1594:
1590:
1586:
1580:
1574:
1571:
1564:
1563:
1562:
1560:
1556:
1552:
1548:
1529:
1522:
1518:
1514:
1509:
1506:
1499:
1498:
1497:
1478:
1473:
1469:
1463:
1460:
1457:
1454:
1451:
1445:
1442:
1439:
1435:
1429:
1425:
1421:
1418:
1408:
1404:
1400:
1397:
1390:
1389:
1388:
1386:
1382:
1362:
1357:
1354:
1351:
1347:
1341:
1337:
1333:
1330:
1323:
1322:
1321:
1318:
1316:
1312:
1308:
1304:
1288:
1280:
1279:mean lifetime
1270:
1268:
1267:eigenfunction
1264:
1260:
1256:
1252:
1248:
1242:
1240:
1233:
1229:
1225:
1218:
1196:
1193:
1190:
1186:
1180:
1176:
1172:
1167:
1164:
1161:
1157:
1151:
1147:
1143:
1137:
1131:
1124:
1123:
1122:
1120:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1069:
1068:
1067:
1050:
1047:
1044:
1041:
1038:
1035:
1030:
1026:
1023:
1013:
1012:
1011:
1009:
990:
987:
984:
981:
975:
972:
967:
964:
954:
953:
952:
944:
942:
939:For example,
923:
918:
914:
910:
906:
902:
899:
896:
887:
881:
878:
872:
868:
864:
860:
854:
851:
844:
843:
842:
828:
819:
801:
794:
790:
786:
782:
777:
773:
770:
766:
760:
756:
752:
746:
740:
733:
732:
731:
729:
713:
690:
684:
678:
675:
672:
669:
664:
657:
651:
648:
642:
637:
633:
629:
625:
617:
616:
615:
610:
606:
602:
598:
592:
582:
580:
575:
558:
552:
532:
524:
519:
504:
500:
495:
491:
480:
479:
456:
433:
428:
424:
420:
417:
413:
407:
403:
399:
393:
387:
380:
379:
378:
364:
341:
336:
333:
328:
325:
318:
317:
316:
302:
294:
293:
292:time constant
288:), where the
287:
283:
282:mean lifetime
279:
275:
271:
264:Mean lifetime
256:
252:
245:
238:
227:
223:
203:
198:
195:
192:
188:
182:
178:
174:
168:
162:
155:
154:
153:
151:
132:
129:
126:
123:
120:
114:
111:
106:
103:
93:
92:
91:
89:
85:
84:rate constant
81:
77:
73:
61:
57:
53:
49:
32:
19:
3732:Exponentials
3684:, New York:
3681:
3656:
3633:
3584:(1): 21–26.
3581:
3577:
3571:
3560:
3548:
3536:
3492:
3488:
3466:
3454:
3446:
3442:
3438:
3416:
3396:
3390:
3379:
3354:
3345:synthesizers
3329:
3316:
3303:radionuclide
3295:
3290:Beer-Lambert
3278:
3266:
3253:Pharmacology
3251:
3243:Luminescence
3241:
3223:
3211:
3110:mirror image
3105:
3097:
3078:
3045:
3027:
3020:
3008:
3001:
2990:
2732:
2582:
2572:
2481:
2366:
2073:
2072:
1869:
1742:
1738:
1735:
1544:
1495:
1380:
1378:
1319:
1310:
1302:
1278:
1276:
1262:
1258:
1243:
1238:
1231:
1227:
1223:
1216:
1214:
1121:, and hence
1116:
1065:
1005:
950:
941:polonium-210
938:
820:
816:
705:
608:
604:
600:
596:
594:
578:
576:
522:
520:
477:
448:
356:
290:exponential
289:
285:
281:
273:
269:
267:
250:
243:
236:
225:
221:
218:
147:
87:
83:
79:
75:
56:proportional
51:
45:
42:from 0 to 5.
3686:McGraw-Hill
3638:McGraw-Hill
3497:SVG version
3449:), the BGP
3355:Beer froth:
3263:metabolized
3231:temperature
152:below) is:
3627:References
3447:vice versa
3408:See also:
3389:In simple
3349:overdamped
3331:Vibrations
3324:Thermistor
3311:radiogenic
3257:toxicology
3213:Geophysics
3011:formulated
1730:See also:
1379:first let
1247:eigenvalue
574:, is 368.
150:derivation
3616:250873501
3594:CiteSeerX
3439:available
3415:The core
3361:, won an
3268:clearance
3200:resistors
3164:
3120:τ
3094:capacitor
3090:potential
3073:catalyzed
2834:λ
2821:λ
2808:λ
2799:
2779:λ
2771:
2706:λ
2693:λ
2684:
2664:λ
2656:
2375:τ
2340:τ
2327:τ
2315:τ
2305:τ
2289:τ
2257:τ
2237:τ
2219:λ
2206:λ
2193:λ
2178:τ
2149:λ
2123:τ
2102:λ
2088:τ
2041:λ
2034:−
1999:λ
1986:λ
1979:−
1920:λ
1892:λ
1879:λ
1840:λ
1827:λ
1811:λ
1795:λ
1756:−
1705:λ
1682:λ
1679:−
1668:λ
1663:∞
1654:∫
1635:λ
1632:−
1614:⋅
1608:⋅
1600:∞
1591:∫
1584:⟩
1578:⟨
1572:τ
1515:λ
1479:λ
1464:⋅
1443:λ
1440:−
1422:⋅
1414:∞
1405:∫
1355:λ
1352:−
1305:) is the
1289:τ
1194:λ
1191:−
1165:λ
1162:−
1092:λ
1089:−
1080:
1042:λ
1039:−
988:λ
985:−
903:⋅
897:≈
882:
852:τ
829:τ
771:−
728:ln 2
714:τ
679:
673:τ
665:λ
652:
605:half-life
591:Half-life
585:Half-life
559:τ
533:τ
501:
457:τ
449:and that
429:τ
418:−
365:τ
337:λ
326:τ
303:τ
196:λ
193:−
127:λ
124:−
3726:Category
3694:75173716
3680:(1972),
3640:. 2007.
3505:See also
3423:Internet
3313:nuclide.
3197:parallel
3190:the dual
3061:reactant
1303:lifetime
1251:negative
1230:= 0, as
286:lifetime
62:, where
48:quantity
3586:Bibcode
3491:and 72/
3421:on the
3381:Finance
1549:of the
1253:of the
474:⁄
3692:
3667:
3644:
3614:
3596:
3470:route.
3451:router
3435:packet
3347:. An
3069:enzyme
2578:halved
2482:where
900:1.4427
219:where
72:lambda
3612:S2CID
3528:Notes
3455:flaps
3445:(and
3386:fund.
3265:(see
3234:also
3053:rates
1257:with
1241:= 0.
86:, or
3690:LCCN
3665:ISBN
3642:ISBN
3461:and
3367:beer
3292:law.
3255:and
3084:The
3051:The
2995:and
2544:and
3604:doi
3463:RAM
3459:CPU
3441:to
3427:BGP
3397:two
3343:in
3006:.
2991:In
2161:s:
1743:sum
1010:),
612:1/2
492:log
278:set
253:= 0
246:(0)
3728::
3688:,
3663:,
3610:.
3602:.
3592:.
3582:23
3580:.
3425:,
3161:ln
3036:.
3025:.
2796:ln
2768:ln
2681:ln
2653:ln
2580:.
1938:.
1561:.
1387::
1281:,
1222:=
1077:ln
879:ln
676:ln
649:ln
545:,
295:,
255:.
242:=
234:,
90::
82:,
78:,
46:A
3650:.
3618:.
3606::
3588::
3493:t
3489:t
3383::
3333::
3320::
3299::
3282::
3259::
3245::
3238:.
3227::
3215::
3176:.
3173:)
3170:2
3167:(
3157:C
3153:R
3133:,
3130:C
3126:R
3123:=
3106:R
3098:C
3082::
3071:-
3049::
2971:.
2965:)
2960:3
2956:t
2950:2
2946:t
2942:(
2939:+
2936:)
2931:3
2927:t
2921:1
2917:t
2913:(
2910:+
2907:)
2902:2
2898:t
2892:1
2888:t
2884:(
2877:3
2873:t
2867:2
2863:t
2857:1
2853:t
2846:=
2838:3
2830:+
2825:2
2817:+
2812:1
2802:2
2790:=
2783:c
2774:2
2762:=
2757:2
2753:/
2749:1
2745:T
2718:.
2710:2
2702:+
2697:1
2687:2
2675:=
2668:c
2659:2
2647:=
2642:2
2638:/
2634:1
2630:T
2604:2
2600:/
2596:1
2592:T
2557:2
2553:t
2530:1
2526:t
2503:2
2499:/
2495:1
2491:T
2462:2
2458:t
2454:+
2449:1
2445:t
2437:2
2433:t
2427:1
2423:t
2416:=
2411:2
2407:/
2403:1
2399:T
2352:.
2344:2
2336:+
2331:1
2319:2
2309:1
2298:=
2293:c
2261:2
2253:1
2248:+
2241:1
2233:1
2228:=
2223:2
2215:+
2210:1
2202:=
2197:c
2189:=
2182:c
2174:1
2127:c
2098:/
2094:1
2091:=
2058:.
2053:t
2050:)
2045:c
2037:(
2030:e
2024:0
2020:N
2016:=
2011:t
2008:)
2003:2
1995:+
1990:1
1982:(
1975:e
1969:0
1965:N
1961:=
1958:)
1955:t
1952:(
1949:N
1924:c
1896:2
1888:+
1883:1
1855:.
1852:N
1849:)
1844:2
1836:+
1831:1
1823:(
1820:=
1815:2
1807:N
1804:+
1799:1
1791:N
1788:=
1782:t
1779:d
1774:)
1771:t
1768:(
1765:N
1762:d
1739:N
1710:.
1702:1
1697:=
1694:t
1691:d
1685:t
1675:e
1671:t
1658:0
1650:=
1647:t
1644:d
1638:t
1628:e
1622:0
1618:N
1611:c
1605:t
1595:0
1587:=
1581:t
1575:=
1530:.
1523:0
1519:N
1510:=
1507:c
1474:0
1470:N
1461:c
1458:=
1455:t
1452:d
1446:t
1436:e
1430:0
1426:N
1419:c
1409:0
1401:=
1398:1
1381:c
1363:,
1358:t
1348:e
1342:0
1338:N
1334:=
1331:N
1263:t
1261:(
1259:N
1239:t
1235:0
1232:N
1228:t
1224:e
1220:0
1217:N
1197:t
1187:e
1181:0
1177:N
1173:=
1168:t
1158:e
1152:C
1148:e
1144:=
1141:)
1138:t
1135:(
1132:N
1101:C
1098:+
1095:t
1086:=
1083:N
1051:.
1048:t
1045:d
1036:=
1031:N
1027:N
1024:d
991:N
982:=
976:t
973:d
968:N
965:d
924:.
919:2
915:/
911:1
907:t
891:)
888:2
885:(
873:2
869:/
865:1
861:t
855:=
802:.
795:2
791:/
787:1
783:t
778:/
774:t
767:2
761:0
757:N
753:=
750:)
747:t
744:(
741:N
691:.
688:)
685:2
682:(
670:=
661:)
658:2
655:(
643:=
638:2
634:/
630:1
626:t
609:t
601:t
599:(
597:N
579:e
562:)
556:(
553:N
523:N
505:e
496:2
478:e
472:1
434:,
425:/
421:t
414:e
408:0
404:N
400:=
397:)
394:t
391:(
388:N
342:.
334:1
329:=
274:t
272:(
270:N
251:t
244:N
240:0
237:N
232:t
228:)
226:t
224:(
222:N
204:,
199:t
189:e
183:0
179:N
175:=
172:)
169:t
166:(
163:N
133:.
130:N
121:=
115:t
112:d
107:N
104:d
70:(
68:λ
64:N
40:x
36:λ
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
