Where does exp(-4pi^2) appear in physics?

[franoisbelfor]

This is a somewhat unusual question.
I am looking for any physics problem
in which the number or factor exp(-4pi^2)
appears.

If you know one, I'd like to hear about it!

François

 

[Born2bwire]

Given a potential described as -exp(-x), what is the work done against the potential when moving a unit mass from infinity to x = 4\pi^2 ?

 

[arildno]

Introduce the new mass unit, $\hat{m}\equiv{m}{e}^{4\pi^{2}}$, where m is the standard unit of mass.

Then, Newton's second law of motion reads:
$$F=e^{-4\pi^{2}}\hat{m}a$$

 

[Mapes]

Here's one that's less arbitrary: the solution of

$$\frac{\partial u(x,t)}{\partial t}=\frac{\partial^2 u(t,x)}{\partial x^2}$$

is

$$u(x,t)=\sum_{n=1}^\infty A_n\exp^{-(n\pi/L)^2t}\sin\frac{n\pi x}{L}$$

where $A_n$ is calculated from the initial conditions. But the upshot is, because the above equation governs heat diffusion by conduction, if you had a bar of material with length 0.5 m, thermal diffusivity 1 m² s^-1, end temperatures of 0°C (i.e., $u(0,t)=u(L,t)=0$), and an initial sinusoidal temperature distribution with maximum temperature 1°C (i.e., $u(x,0)=\sin(\pi x/L)$), the temperature at the midpoint after 1 second is predicted to be $\exp(-4\pi^2)$, which is the term you're looking for.

 

[Vanadium 50]

How is that less arbitrary? You still have to pick a material with the right size and properties.

 

[Mapes]

How is that less arbitrary? You still have to pick a material with the right size and properties.

It's less arbitrary because the exponential function and the $\pi^2$ term come out of the physics rather than being input as variables or functions, as in the other two cases (although we do need to specify a sinusoidal initial temperature distribution). And the values aren't fixed; it could be a $10\,\mu m$ long microfabricated silicon beam (thermal diffusivity $8\times 10^{-5}\,\mathrm{m^2}\,\mathrm{s^{-1}}$) after $5\,\mu s$, for example. Don't you think it's interesting that the $\pi^2$ emerges naturally here?

 

[Born2bwire]

It's less arbitrary because the exponential function and the $\pi^2$ term come out of the physics rather than being input as variables or functions, as in the other two cases (although we do need to specify a sinusoidal initial temperature distribution). And the values aren't fixed; it could be a $10\,\mu m$ long microfabricated silicon beam (thermal diffusivity $8\times 10^{-5}\,\mathrm{m^2}\,\mathrm{s^{-1}}$) after $5\,\mu s$, for example. Don't you think it's interesting that the $\pi^2$ emerges naturally here?

Not really. Pretty much any time-harmonic or complex number system you can probably easily massage out an exp(\pi) and exp(\pi^2) dependence of some kind. I could specify a transmission line and given a certain length and loss I could get you an attenuation of exp(-4\pi^2).

The OP is just posing a really bad question. It is completely arbitrary because, like arildno shows, when it comes to picking out a constant you can get it from just about any kind of equation from judicious choice of your units, scale, or choice of parameters.

 

[Mapes]

I could specify a transmission line and given a certain length and loss I could get you an attenuation of exp(-4\pi^2).

Go for it! That's what the poster seems to be looking for: physical circumstances in which the term arises naturally. Maybe he or she saw the term on a blackboard once, or a poster (or a tattoo!), and wants to know what the context might have been. I don't know. But I don't think it's a bad question; in fact, I was looking forward to comparing the responses to get a sense of the... grand interconnectedness of physics. Consider it a challenge: what's the least amount of massaging needed to get $\exp(-4\pi^2)$, without inputting it directly?

 

[arildno]

"physical circumstances in which the term arises naturally"

What is natural, or unnatural about a particular choice of length scale??

Yet, they are also part of physics..

 

[Vanadium 50]

Getting a pi^2 is not horribly difficult; have a 4 or 5 dimensional volume in phase space. You'll have to turn a 1/2 or an 8/15 into a 4, but that shouldn't be hard. Then find a reason to exponentiate it...e.g. as a partition function.

This is contrived, of course, but no more so than a length of one meter, time of one second, capacitance of one farad, etc.

As far as the context, I fear it's not a blackboard, poster or tattoo. A search of other messages here will perhaps provide some enlightenment.

 

[Mapes]

As far as the context, I fear it's not a blackboard, poster or tattoo. A search of other messages here will perhaps provide some enlightenment.

Ah, got it. I was originally thinking it was something innocuous like https://www.physicsforums.com/showthread.php?t=257304".