How Do Lambda and Beta Relate in the Exponential Decay of Foam?

[LucasGB]

Hello,

Suppose you observe some foam. The foam is formed by a set of bubbles, and each bubble blows up after a random time. The density function of the time each bubble will take to blow up is probably exponential, with rate lambda. The total amount of foam (Q) must also decay exponentially, at a rate given by the decay constant beta.

So, we have to exponencial functions:

1) the exponencial density function that determines the lifetime of each individual buble:

f(t) = lambda*exp(-lambda*t)

2) the exponencial decay that determines the amount of foam we have at time t:

Q(t)=Q[0]*exp(-beta*t)

The relation between these two functions is not clear to me. What is the relation between lambda and beta?

Thanks in advance,

Estêvão

 

[bpet]

Hello,

Suppose you observe some foam. The foam is formed by a set of bubbles, and each bubble blows up after a random time. The density function of the time each bubble will take to blow up is probably exponential, with rate lambda. The total amount of foam (Q) must also decay exponentially, at a rate given by the decay constant beta.

So, we have to exponencial functions:

1) the exponencial density function that determines the lifetime of each individual buble:

f(t) = lambda*exp(-lambda*t)

2) the exponencial decay that determines the amount of foam we have at time t:

Q(t)=Q[0]*exp(-beta*t)

The relation between these two functions is not clear to me. What is the relation between lambda and beta?

Thanks in advance,

Estêvão

The probability that the bubble survives to time t is exp(-lambda*t), so if the number of bubbles is large then by the law of large numbers you'd expect that fraction of the bubbles to have survived.

 

[LucasGB]

The probability that the bubble survives to time t is exp(-lambda*t), so if the number of bubbles is large then by the law of large numbers you'd expect that fraction of the bubbles to have survived.

So lambda = beta?