Math Is Fun Forum

This is a partition function from thermodynamics.
Partition function (wikipedia)
It's the formula under Classical discrete systems, however, it's written a bit differently, since in my case it's a continuous function.

I'm currently working on my PhD in physics and I've always evaluated this integral numerically. I was just curious whether there is an actual analytic solution to this function.

The formula is:

I think it's usually found in this form:

So when you do the following transformation

and use the relativistic energy-momentum-mass equality in natural units, i.e.

you get

It's a well-behaved integral, without any singularities or oscillations or something like that so a numerical evaluation is pretty straightforward, however, it's a bit annoying since I have to evaluate it every time I change the temperature.

You are right

I edited my post and removed the mistake. So what is the distance between these two parallel lines?

You have two equations with two variables each.
You can plot these in a two-dimensional coordinate system with the axes x and y.

In this case it would be sensible to rewrite the equations in such a way that you could plot them by plugging in numbers, e.g.

You can see that you have a linear equation. And then you do the same with the other equation and take the difference.

Well, let's say we have two parallel lines

We would figure out whether they are parallel or not by looking for a point where they meet, i.e.

Obviously this is not correct so there is no point where they meet, i.e. they are parallel.

We would calculate the distance by taking the difference between these two lines