Density of states summation?

[pivoxa15]

If an infinite discrete sum is calculated via integrating over a density of states factor, is this integral an approximation to the discrete sum? i.e the discrete sums could be partition functions or Debye solids.

 

[Mr.Brown]

you mean like when you have a huge amount of particles and instead of summing over all these particles to get the partition function you integrate?
Yes then it´s an approximation :)

 

[Manchot]

Yes, it's an approximation, but it's a very, very good one. Suppose you were numerically integrating a slowly-varying function from 0 to 1e24. You'd probably do so using the definition of a Riemann integral, choosing a finite dx. That is, you'd partition the function into intervals, add their values together, and multiply by dx. So, if you chose dx=1e23, you'd add up the values of the function at 0, 1e23, 2e23, ..., 9e23 and multiply by 1e23. Of course, if you chose a smaller dx, (say 1e22) you'd get a better approximation to the integral. If you chose a dx=1, the error would be extremely small (differing by about 1e-24), and you'd get your answer by summing up the value at every integer and multiplying by 1. Of course, this means that the summation approximates the integral, and vice versa.

 

[Slaviks]

Within the context your question referes to, it is an approximation which becomes exact in the thermodynamic limit (infinite system with fixed finite density). This is precisely the limit in which standard results of statistical mechanics make sense. Mathematically, by taking the thermodynamic limit the sum becomes an integral by the very definition of the latter.It is also worth keeping in mind that caution is necessary in taking a sum to an integral when there are special modes like the condensate state in a bosonic system.

 

[quetzalcoatl9]

remember that when the partition function was derived, the assumption of large N was made in order to invoke the sterling approximation:

$$\ln(N!) \approx N \ln N - N$$

numerical analysis shows that the approximation becomes valid very quickly. you can use the gamma function to investigate more thoroughly.

it is for this reason that we can get away with computer simulations of small particles and periodic boundaries (and still calculate meaningful averages). in fact, you can get reasonable results for a monoatomic gas of only 16 particles with periodic boundaries (at most state points, depending upon the potential).