mean lifetime

• 平均寿命

関

average lifespan、average lifetime、mean lifespan

WordNet

1. have a specified degree of importance; "My ex-husband means nothing to me"; "Happiness means everything"
2. an average of n numbers computed by adding some function of the numbers and dividing by some function of n (同)mean_value
3. mean or intend to express or convey; "You never understand what I mean!"; "what do his words intend?" (同)intend
4. denote or connote; "`maison means `house in French"; "An example sentence would show what this word means" (同)intend, signify, stand_for
5. (used of persons or behavior) characterized by or indicative of lack of generosity; "a mean person"; "he left a miserly tip" (同)mingy, miserly, tight
6. destine or designate for a certain purpose; "These flowers were meant for you"
7. excellent; "famous for a mean backhand"
8. rich in significance or implication; "a meaning look" (同)pregnant, significant
9. the message that is intended or expressed or signified; "what is the meaning of this sentence"; "the significance of a red traffic light"; "the signification of Chinese characters"; "the import of his announcement was ambiguous" (同)significance, signification, import
10. the idea that is intended; "What is the meaning of this proverb?" (同)substance
11. considerable capital (wealth or income); "he is a man of means" (同)substance
12. how a result is obtained or an end is achieved; "a means of control"; "an example is the best agency of instruction"; "the true way to success" (同)agency, way
13. an instrumentality for accomplishing some end

PrepTutorEJDIC

1. 〈人が〉…‘を'『意図する』,‘の'つもりである;…‘を'言おうとある;示そうとする / 〈単語・句・文章などが〉…‘を'『意味を持つ』,‘を'『意味する』 / (人に対して)…‘を'当てる,向ける《+『名』+『for』+『名』》 / 《受動態で》〈人・物〉‘を'(…)向きにする / 〈物事が〉…‘を'示す,表す / (結果として)〈物事が〉…‘を'もたらす,生じさせる / 《『mean』+『for』+『名』+『to』 do》《米話》(…が…するような)つもりでいる
2. 《名詞の前にのみ用いて》(品質・価値・才能などが)『劣った』,つまらない / (人・行為などが)『卑劣な』,さもしい / 《古》(身分・地位などが)卑しい / 《まれ》(身なり・建物などが)みずぼらしい,見るもあわれな / 《米話》(動物が)御しにくい;(一般に)扱いにくい
3. 中庸,中道 / (数学で) / 平均[値];(特に)算術平均,相加平均 / (比例式の)中項,内項 / (両極端の)中間の,中央に位置する / (大きさ・品質・程度などが)平均的な,平均の
4. (・・・の)『目的』,意義《+『of』+『名』》 / (吾・文の)『意味』(sense)《+『of』+『名』》 / 意味ありげな様子,意味深長 / 《名詞の前にのみ用いて》意味ありげな,意味常長な / 《複合吾を作って》・・・するつもりの,の意図をもった
5. 〈C〉《単数・複数扱い》『方法』,『手段』 / 《複数扱い》『資力』,『財産』,富
6. 『生涯』,一生 / 一生の,終生の
7. 『私を』(『に』) / 《話》『私』(=『I』)

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/11/20 09:17:21」(JST)

wiki en

[Wiki en表示]

A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constants (λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5.

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the decay constant:

The solution to this equation (see derivation below) is: Exponential rate of change

Here N(t) is the quantity at time t, and N₀ = N(0) is the initial quantity, i.e. the quantity at time t = 0.

Measuring rates of decay

Mean lifetime

If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime), τ, and it can be shown that it relates to the decay rate, λ, in the following way:

The mean lifetime (also called the exponential time constant) can be looked at as a "scaling time", because we can write the exponential decay equation in terms of the mean lifetime, τ, instead of the decay constant, λ:

We can see that τ is the time at which the population of the assembly is reduced to 1/e = 0.367879441 times its initial value.

E.g., if the initial population of the assembly, N(0), is 1000, then at time τ, the population, N(τ), is 368.

A very similar equation will be seen below, which arises when the base of the exponential is chosen to be 2, rather than e. In that case the scaling time is the "half-life".

Half-life

Main article: Half-life

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. This time is called the half-life, and often denoted by the symbol t_1/2. The half-life can be written in terms of the decay constant, or the mean lifetime, as:

When this expression is inserted for in the exponential equation above, and ln 2 is absorbed into the base, this equation becomes:

Thus, the amount of material left is 2⁻¹ = 1/2 raised to the (whole or fractional) number of half-lives that have passed. Thus, after 3 half-lives there will be 1/2³ = 1/8 of the original material left.

Therefore, the mean lifetime is equal to the half-life divided by the natural log of 2, or:

E.g. Polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days.

Solution of the differential equation

The equation that describes exponential decay is

or, by rearranging,

Integrating, we have

where C is the constant of integration, and hence

where the final substitution, N₀ = e^C, is obtained by evaluating the equation at t = 0, as N₀ is defined as being the quantity at t = 0.

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 eigenvalue. In this case, λ is the eigenvalue of the negative of the differentiation operator with N(t) as the corresponding eigenfunction. The units of the decay constant are s⁻¹.

Derivation of the mean lifetime

Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime, , (also called simply the lifetime) is the expected value of the amount of time before an object is removed from the assembly. Specifically, if the individual lifetime 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 arithmetic mean of the individual lifetimes.

Starting from the population formula

we firstly let c be the normalizing factor to convert to a probability density function:

or, on rearranging,

We see that exponential decay is a scalar multiple of the exponential distribution (i.e. the individual lifetime of each object is exponentially distributed), which has a well-known expected value. We can compute it here using integration by parts.

Decay by two or more processes

See also: Branching fraction

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 rate of the quantity N is given by the sum of the decay routes; thus, in the case of two processes:

The solution to this equation is given in the previous section, where the sum of is treated as a new total decay constant .

Partial mean live associated with an individual processes is by definition the multiplicative inverse of corresponding partial decay constant: . A combined can be given in terms of s:

Since half-lives differ from mean life by a constant factor, the same equation holds in terms of the two corresponding half-lives:

where is the combined or total half-life for the process, and are so-named partial half-lives 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 certain quantity is halved.

In terms of separate decay constants, the total half-life can be shown to be

For a decay by three simultaneous exponential processes the total half-life can be computed as above:

Applications and examples

Exponential decay occurs in a wide variety of situations. Most of these fall into the domain of the natural sciences.

Many decay processes that are often treated as exponential, are really only exponential so long as the sample is large and the law of large numbers holds. For small samples, a more general analysis is necessary, accounting for a Poisson process.

Natural sciences

• Chemical reactions: The rates of certain types of chemical reactions depend on the concentration of one or another reactant. Reactions whose rate depends only on the concentration of one reactant (known as first-order reactions) consequently follow exponential decay. For instance, many enzyme-catalyzed reactions behave this way.

• Electrostatics: The electric charge (or, equivalently, the potential) stored on a capacitor (capacitance C) decays exponentially, if the capacitor experiences a constant external load (resistance R). The exponential time-constant τ for the process is R C, and the half-life is therefore R C ln2. (Furthermore, the particular case of a capacitor discharging through several parallel resistors makes an interesting example of multiple decay processes, with each resistor representing a separate process. In fact, the expression for the equivalent resistance of two resistors in parallel mirrors the equation for the half-life with two decay processes.)

• Fluid Dynamics: A fluid emptying from a tube with an opening at the bottom will empty at a rate which depends on the pressure at the opening (which in turn depends on the height of the fluid remaining). Thus the height of the column of fluid remaining will follow an exponential decay.

• Geophysics: Atmospheric pressure decreases approximately exponentially with increasing height above sea level, at a rate of about 12% per 1000m.

• Heat transfer: If an object at one temperature 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 also Newton's law of cooling.

• Luminescence: 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 multi-exponential.

• Pharmacology and toxicology: It is found that many administered substances are distributed and metabolized (see clearance) according to exponential decay patterns. The biological half-lives "alpha half-life" and "beta half-life" of a substance measure how quickly a substance is distributed and eliminated.

• Physical optics: The intensity of electromagnetic radiation 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 Beer-Lambert law.

• Radioactivity: In a sample of a radionuclide that undergoes radioactive decay 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 radiogenic nuclide.

• Thermoelectricity: The decline in resistance of a Negative Temperature Coefficient Thermistor as temperature is increased.

• Vibrations: Some vibrations may decay exponentially; this characteristic is often found in damped mechanical oscillators, and used in creating ADSR envelopes in synthesizers. An overdamped system will simply return to equilibrium via an exponential decay.

Social sciences

• Finance: 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 fund.
• In simple glottochronology, the (debatable) assumption of a constant decay rate in languages allows to estimate the age of single languages. (To compute the time of split between TWO languages requires additional assumptions, independent of exponential decay).

Computer science

• The core routing protocol on the Internet, BGP, has to maintain a routing table in order to remember the paths a packet can be deviated to. When one of these paths repeatedly changes its state from available to not available (and vice-versa), the BGP router controlling that path has to repeatedly add and remove the path record from its routing table (flaps the path), thus spending local resources such as CPU and RAM and, even more, broadcasting useless information to peer routers. To prevent this undesired behavior, an algorithm named route flapping damping 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 route.

See also

• Exponential formula
• Exponential growth
• Radioactive decay for the mathematics of chains of exponential processes with differing constants

External links

• A stochastic simulation of exponential decay
• Tutorial on time constants

UpToDate Contents

全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.

• 1. 自動血液分析装置 automated hematology instrumentation
• 2. 平均赤血球容積 mean corpuscular volume
• 3. 貧血を有する成人患者に対するアプローチ approach to the adult patient with anemia
• 4. 成人における双極性障害：疫学および病因 bipolar disorder in adults epidemiology and pathogenesis
• 5. Eating disorders: Overview of epidemiology, clinical features, and diagnosis

English Journal

• Psychosocial and metabolic function by smoking status in individuals with binge eating disorder and obesity.

• Udo T1, White MA2, Barnes RD2, Ivezaj V2, Morgan P2, Masheb RM2, Grilo CM3.
• Addictive behaviors.Addict Behav.2016 Feb;53:46-52. doi: 10.1016/j.addbeh.2015.09.018. Epub 2015 Sep 30.
• Individuals with binge eating disorder (BED) report smoking to control appetite and weight. Smoking in BED is associated with increased risk for comorbid psychiatric disorders, but its impact on psychosocial functioning and metabolic function has not been evaluated. Participants were 429 treatment-s
• PMID 26451703

• How many TCR clonotypes does a body maintain?

• Lythe G1, Callard RE2, Hoare RL2, Molina-París C3.
• Journal of theoretical biology.J Theor Biol.2016 Jan 21;389:214-24. doi: 10.1016/j.jtbi.2015.10.016. Epub 2015 Nov 7.
• We consider the lifetime of a T cell clonotype, the set of T cells with the same T cell receptor, from its thymic origin to its extinction in a multiclonal repertoire. Using published estimates of total cell numbers and thymic production rates, we calculate the mean number of cells per TCR clonotype
• PMID 26546971

• Concentrations and potential health risks of methyl tertiary-butyl ether (MTBE) in air and drinking water from Nanning, South China.

• Zhang L1, Qin J2, Zhang Z2, Li Q1, Huang J3, Peng X4, Qing L5, Liang G1, Liang L2, Huang Y6, Yang X2, Zou Y7.
• The Science of the total environment.Sci Total Environ.2016 Jan 15;541:1348-54. doi: 10.1016/j.scitotenv.2015.10.038. Epub 2015 Nov 11.
• Levels of methyl tertiary-butyl ether (MTBE) in occupational air, ambient air, and drinking water in Nanning, South China, were investigated, and then their potential health risks to occupational workers and the general public were evaluated. Results show that the MTBE concentration in occupational
• PMID 26479908

Japanese Journal

• Landau-like theory for universality of critical exponents in quasistationary states of isolated mean-field systems

• Physical Review E 91(6), 2015-06-08
• NAID 120005649320

• 炭素鋼環状切欠き材のねじり疲労に及ぼす組合せ平均応力の影響

• 法政大学大学院紀要（理工学・工学研究科編） (56), 2015-03-24
• NAID 120005608564

• 鋼材及びアルミニウム合金材のねじり疲労における静的荷重の影響

• 法政大学大学院紀要（理工学・工学研究科編） (56), 2015-03-24
• NAID 120005608538

Related Links

• Mean lifetime - definition of Mean lifetime by Medical dictionary

Mean lifetime - definition of Mean lifetime by Medical dictionary http://medical-dictionary.thefreedictionary.com/Mean+lifetime Printer Friendly Dictionary, Encyclopedia and Thesaurus - The Free Dictionary 7,492,760,914 visitors served ...

• Mean lifetime - definition of Mean lifetime by The Free Dictionary

mean life n 1. (General Physics) physics the average time of existence of an unstable or reactive entity, such as a nucleus, elementary particle, charge carrier ... Females were found to mate up to 3 times in a mean lifetime of 6 d ...

Related Pictures

---

★リンクテーブル★

リンク元	「平均寿命」「mean lifespan」「average lifespan」「average lifetime」
関連記事	「mean」「means」「meaning」「lifetime」

「平均寿命」

　　[★]

英

mean lifetime、average lifetime、mean lifespan、average lifespan, natural span, life span

関

平均余命、生命表

「mean lifespan」

　　[★]

• 平均寿命

関

average lifespan、average lifetime、mean lifetime

「average lifespan」

　　[★]

• 平均寿命

関

average lifetime、mean lifespan、mean lifetime

「average lifetime」

　　[★]

• 平均寿命

関

average lifespan、mean lifespan、mean lifetime

「mean」

　　[★]

• n.

• 平均値、 手段

• adj.

• 平均の

• v.

(過去-過去分詞meant)

• 意味する

関

avenue、average、even、gateway、implication、imply、instrumental、mean value、meaning、means、policy、purport、represent、semantic、sense、tool

　 　 　 　 　 　 　 　 　 　

「means」

　　[★]

• np.

• 手段、方法

関

avenue、fashion、gateway、instrumental、manner、mean、method、mode、policy、procedure、process、tool、way

　 　 　 　

「meaning」

　　[★]

• n.

• 意味、意義

関

consequence、implication、imply、mean、purport、represent、semantic、sense、significance

　 　 　 　

「lifetime」

　　[★]

• 寿命、生涯、生存期間

関

length of life、life、life span、lifelong、lifespan