@Kaumudi so you're integrating $t dN$ to get the total lifetimes of all the atoms. Just integrate $t dN$ and divide by $N_0$.

The trick is to convert $dN$ into an expression in $dt$ so you can do the integration.

Anyhow my alarm has just nagged me! Back later this afternoon.

gday

@Kaumudi

:)

let's derive the mean life

ok

imagine every particle in the system

each one has some probabilyt of decaying into nothing

we want to determine the average time it takes for one of these guys to decay

to do this let $f(t)$ be the probability the particle dies between time $t$ and $t+dt$

Then $f(t)$ is a probability density function

make sense so far?

I haven't yet said what the equation is for $f(t)$ but I'm happy to make it ke^-lambdat if you want

all I've said so far is let us define some quantity $f(t)$ to represent the probabilty a particle decays between $t$ and $t+dt$

so clearly from my sentence above $f(t)$ is a probability density function

because

it represents

the probabilty of decay between $t$ and $t+dt$

as per DavidZ's answer

a probability density function is one in which the integral of it gives 1

now if we know decay is definite at some time, and we know $f(t)$ is the probability of decay between $t$ and $t+dt$ then by summing up all $f(t)$ we will get the total probability of decay which must be 1

no

lamda is the probability of decay per unit time

not the probability of decay

but wait

correct

now calculate the derivative of that

@MAFIA36790 Well AFAIK it is kind of the point of Bourbaki's publications: they wanted to reformulate mathematics in abstract yet self-contained form

it's both

see I started explaining the derivation using a particular terminology

In my terminology let $f(t)$ be the probability of decay between $t$ and $t + dt$

now we can show that $f(t) = Ae^{-lambda t}$

now if we integrate we can show the probability will be

your answer

do you understand where your confusion arose

because the form of $e^{-\lambda t}$ arises in both the density function and the cummulative distribution function

So let us start from the beginning again, and let's not make reference to external sources as they may use different notation and it will be confusing

ok then

let us start from DavidZ's answer

to derive the maen life time

DavidZ states that the probabilty of survival to time t is given by

$P = e^{-\lambda t}$

Now we want the probabliity of surviving up to some time $t'$ and then drying

because this represents the probability of having a lifetime of $t'$

so this quantity is simply $P(t') \lambda dt$

that is, $P(t')$ is the probability of surviving up to time $t'$ and $\lambda dt$ is the probability of dying between $t'$ and $t'+dt$

so that quantity represents the probabilty of surviving to t'

now in order to work out the mean we need to do the following:

probability of surviving to time t' * time t' (summed over all possible t')

That is

$/int tP(t)\lambda dt$

$\int tP(t) \lambda dt$

yep i'l wait

ok

ready to proceed or still understanding?

btw when I said prob of surviving to time ' * time t', may be better to replace "surviving" with dying at time t

the words are off but the maths is right

anyway

let is plug in the value of $P(t)$ to give:

$\int t e^{-\lambda t} \lambda dt$

$= \lambda \int t e^{-\lambda t}$

dt

note the integral is from 0 to infinity (all of time)

do you know how to do integration by parts?

because that is required to solve the integrla

k it's a bit hard to type the integral here in chat

but the answer is $\frac{1}{\lambda}$

where do the values come from?

@Kaumudi your integral looks right you have an error in substituting in ur limits

expanding your integral gives $e^{-\lambda t} t + e^{-\lambda t}/\lambda$

yep

now @Kaumudi expand the brackets

and you will see that one term goes to 0 and the other term becomes $\frac{1}{\lambda}$

when the limits are calculated

yah

expand the brackets

to get

@MAFIA36790 do you think I could benefit from studying Bourbaki's textbooks? I am interested in Lie algebras, Kac-Moody algebras, Hopf algebras & quantum groups, noncommutative geometry

not yet

you know how to expand brackets don't u

I personally don't think so, but I'd like to hear it our if you disagree

oh

so u got it?

good job

yeah the mean life time is $1/\lambda$

np

the key part of the derivation is realizing $P(t')\lambda dt$ is the probability of life time being between $t'$ and $t'+ dt$

why doesn't sweet = candy

= Prob of surving to t' multiply by prob of dying just after t' ($\lambda dt$)

easy stuff

wats dat

dat looks nice

is that sweet common in ur country

ur pretty smart tho

oh that's weird

np :)

we say lolly

for candy

candy = american

@MAFIA36790 I don't really want to waste my time on conceptual issues of infinite-dimensional set theory or something like that. Well, at least it is how I see Bourbaki's textbooks. Though I don't really know since I haven't read them.

ty i say that one alot

np

anyone interested in psychology here?

hello everyone

SUP HEATHER

hello

heather are you interested in psychology?

well, not particularly, but I suppose I'd be interested in discussing it =)

oh

sorry

i don't wanna send u crazy

cos if u think too much about how ur mind works it can get weird

But i'm interested in the psychology of motivation

but when i went to research it i couldn't find much

very annoying

hmm, interesting

the only thing that exists in the literature I can find is something called "Maslow's heirachy of needs"

but as a physicist/mathematician it's very deficient to me

@Mew I don't know much about phychology. Isn't it supposed to predict who molested you when you were a child by analyzing aspects of which flavor of ice cream you like?

oh, i think i've heard of that. isn't it like, 1) food/water/shelter 2) companionship 3) soomething

@SolenodonParadoxus I dunno

@heather yeah that's right

like it starts at necessities and gets to more complex things as you have them

but that is just some list some guy invented

right

it doesn't seem very scientific to me

and it was invented like 50+ years ago

and relaly nothing has been done on a full theory of motivation since that I can find

what motivates you @heather?

is learning a motivator in itself?

or is learning only a means to an end?

hmm, to some extent both

wow ty @Kaumudi

looks interesting

let me think about this.

@Kaumudi yeah that's true, it will be good to have that knowledge to set me up well for my future groundbreaking theories

@Kaumudi lots of low hanging fruit there, they've barely developed in 50 years

but I guess physics hasn't either :p

there's mathematical psychology? cool!

oh yeah I didn't know that

hey, found this

might be helpful, that and other courses

yeah ty

may be interesting

but I think the field is deficient with respect to motivation

ocw.mit.edu/courses/find-by-topic/…

from my research I think they gave up at the Maslow Pyramid (despite the many criticisms)

^full list of psychology courses

like if Maslow's theory was true, people wouldn't go on hunger strikes for their beliefs

because they would want to satisfy their need for food above the needs morality etc.

hmm, but remember self-actualization.

but yes, good point.

yeah cos MAslow says you won't go for self-actualisation until you've been fed

that might be worth a paper in and of itself: the flaws of maslow

yeah but I think they're already well publicised

the flaws are well known

and that's what's annoying

cos the next more satisfying theory hasn't come to the rescue yet

you know, what if you reformulated the pyramid and put "belief system" on the bottom?

but then it would topple over

why?

cos a pyramid can't stand on a point

jks

@heather I dont' think the mind works like a pyramid at all

@Mew is it possible to use psychology to read somebody's mind? Nothing complicated, just simple things - like whether he's holding a monster hand or a complete garbage in a game of poker? No kidding, I really want to know.

but rather there are different distinct modes that all need to be satisfied

hmm, you know, just thought of this, you might want to read a bit of economics: they analyze all the time why people want to buy something. So maybe that might help you.

@heather I think we have a need for curiosity, need for affiliation, need for psiological desires etc.

@heather I work in finance and finance and economics are my specialities :) that's kinda what got me into psychology of motivation

we have many needs and we have many wants. so how do the wants progress from the needs, and how do the actions progress from the wants?

@Mew, oh =)

@heather in finance it is assumed that money is proportional to utility and happiness

not necessary a linear relationship, but still it is really the only factor that's considered

economics*

But doesn't factor in the need for affiliation, the need for learning etc which I think the mind has all these independent needs

I think a failure of modern society is to think that you can seek out money and you can neglect your other needs

but if one need is not fulfilled u wont' be happy

@SolenodonParadoxus pscyhology can't read minds

@SolenodonParadoxus not every thought would display physical cues

@Mew that's true, but maybe you can use tricks to induce information leaks? Like ask an opponent something and observe his reactions.

@SolenodonParadoxus a hyper advanced understanding of psychology may allow you to best guess at what another person would do under various situation, and poker players use this to find odds of various holdings given their oponent's actions

a need to understand who we were made by/where we came from, perhaps

@heather yes that's a good one

@heather perhaps more generally a need to understand?

anything/everything

but i guess not, cos there are many things i don't care about

perhaps, but for some this is dulled

but I think who made us/where we came from isn't necessarily the goal for all either

as a three year old, though, they always ask why

@heather that's an interesting thought. I personally tend to think of "who made us" being a wrongly posed question.

@heather yes the need is to find MEANING

@SolenodonParadoxus, how so?

@Mew, is it? how is that different from what we've been saying?

@heather I think what we are trying to say is we all have a need to finding MEANING in ourselves, in our world

@heather well who made us <> meaning

but I think meaning is more encompassing

right

if we found out who made us, we will still ask what's the meaning of it all

and you know, for some evolution cannot fulfill this aspect:

we would then be an accident

no, because evolution doesn't provide us with meaning

well, i personally disagree with evolution myself

but that's another thing altogether

@heather if evolution was true, why would we evolve with a desire for meaning?

and why would we evolve with a set of morals?

I mean let's assume evolution is true for now (because it is the scientific consensuse), then is it possible to explain why humans seek out meaning?

ooh, morals is a need, isn't it?

@heather morals can potentially be explained by arguing the groups that looked out for one another survived and the groups that didn't work as a team died out

@heather but I can't work out why evolution would result in creatures looking for their own meaning

@heather I don't know if morality is a need or not but it's an instinct. I think belonging to a group is a need tho.

@Mew, eh...that's incredibly weak, the morals bit, i mean. survival of the fittest, right, or who produces the most strong descendents. so why didn't we evolve with the idea that, say, rape was okay, or multiple wives (I know there are religions that believe that is okay, but I'm focusing on generalities here)?

@heather We (humans) have a long history of asking the wrong questions. In this case we implicitly assume that 1) our existence is due to somebody's decision 2) our existence is due to something (has a cause).

I don't see why either of these should hold true

@heather is multiple wives instinctually wrong though?

or just wrong in the community

@heather rape is wrong because I think we understand violating another in anyway is wrong

the more general moral feature isn't that "rape" per se is wrong, but rather violating another's will is wrong

@Mew isn't the term "wrong" determined by the community?

@Mew, wrong in the community progressed from multiple people thinking it was wrong, right? And as for rape and all that, but doesn't that mean we evolved wrong?

because that means less descendants?

@SolenodonParadoxus we have instincts too that determine right and wrong.

@SolenodonParadoxus this is why ancient philosphers ponded for years on wher morality comes from. Because they felt it internally

@Mew these are determined by those who did the upbringing

By our parents, for example

@SolenodonParadoxus no, I believe we have an instinct of morality

I'm not saying it wasn't evolution tho

If I were raised in a vacuum I would know nothing about the wrongness of killing

@SolenodonParadoxus I disagree

Children don't want to harm there parents even before they learn right and wrong

evolutionary speaking, if they did, they would have been killed

so people evolved to have the instinct to not harm their parents

similarly other instincts of not harming others can have evolved

Because they see no reason for doing it, not because they see it as wrong or immoral.

Your argument doesn't support your claim

You claim that morality is internal

@SolenodonParadoxus no

we need to be clear on our terms here

I'm saying that humans have an instinct to not kill or harm others unnecessarily

now we also develop laws and moral codes as a community

this stems from (1) mutual contract and (2) innate instinct

(1) represents that we don't want to be killed, so we create laws that take away our freedom to kill in exchange to not be killed

but at the same time, many would not want to kill out of instinct alone

I'm saying that humans have an instinct not to do anything unnecessarily

@SolenodonParadoxus I don't agree with that, children are very experimental

they will do lots of random stuff just to see what happens

as a child and friend of children, I can testify to do this: we do weird stuff.

@heather I don't think wrong is determined by the community, but rather by individuals first, and then the community is shaped by individuals

the "community" then has the power to supress those individuals who fall far outiside the norms (e.g. those who have a desire to harm others)

i think i agree with that.

the other thing i might point out here, is we also kind of have a want for meta-thought: the analyzation of our thought.

@heather evolutionary speaking, if you have a human who loves to kill, then that human would die off quickly because he would always be picking fights with others