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I'm learning about something in Physics and after a few days of searching around I've realized my issue is a mathematical one. The quantity in question is called the "microcanonical partition function", and is written as follows, where $\{p,q\}$ are all independent variables and called the "phase space", and $E\equiv E(\{p,q\})$,

$$\Omega(E)=\int...\int_{\{p,q\}}\delta(E-H(p_1,q_1,...,p_N,q_N))dp_1dq_1...dp_Ndq_N \tag{1}$$

The paper I was reading in particular manipulated this by taking the Laplace transform of it (this part I understand): $$Z(\beta)=\int^{\infty}_{0} e^{-\beta E}\Omega(E)dE \tag{2}$$

And due to the properties of the dirac-delta function, we evaluate our exponential function at $E=H$, $$Z(\beta)=\int...\int_{\{p,q\}}e^{-\beta H}dp_1dq_1...dp_Ndq_N \tag{3}$$

This itself is a physically important quantity (canonical ensemble partition function) and therefore (1) is related to (3) via the L.T., even though physically they can be derived entirely separately. Quite an interesting revelation.

Upon reflection, I realize that I have no idea how to evaluate (1), given that the dirac-delta function is a function E, which is shifted by a function in and of itself (H). I would know how to evaluate this for example,

$$\int_{a}^{b}f(E)\delta(E-H)dE=f(H)\tag{4}$$

Where $a < H<b$. But in (1) the differential components are written as the components that H depends on.

Is there a term for this type of dirac-delta integral? Is there any theory that can be leveraged to expand it?

michael b
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  • Working in phase space is complicated, in particular if your system has many particles. In order to obtain the average value of a quantity, you have to integrate over all positions and momenta. Equation (1) focuses on that part of phase space where H = E. The Dirac function acts as a filter. Of course eq (1) is a formal expression, but it can be manipulated further. You don't want to actually evaluate the integration over all variables. – M. Wind May 27 '21 at 02:53
  • I guess I am struggling with the mathematical interpretation of (1). That function, $\Omega$, is considered the total number of possible microstates of a system (it seems like you are aware of this). How does performing 6N integrals over the phase space on the dirac function give us the number of microstates? As a function of E, it would only produce one "spike". – michael b May 27 '21 at 17:03
  • You can think of the Dirac delta function as a density 1/c for H in the interval (E - c/2, E + c/2) and zero outside of the interval. Where c is arbitrarily small. This way formula (1) selects all microstates in the given interval around H = E. Please check the Wikipedia page on the microcanonical partition function, where this is explained in the first sentence. – M. Wind May 27 '21 at 17:24
  • Oh believe me, I have read that Wikipedia page many times. I don't understand technically how (1) selects all microstates in the interval around H = E. In my example above in (4) if f = 1, then the result would just be 1. I don't know why performing the integral 6N times producers a larger number, since the dirac function only has one instance and even if it were there in 6N instances it would just be 6N*1. – michael b May 27 '21 at 17:40
  • The point is that phase space has (in general) states with different values of energy E. By introducing the delta function you can select those states (still very many) that meet the criterion of H = E. The resulting quantity is useful, because now you can vary E and for example average over a Boltzman factor, as you did in eq. (2). You then get a density in terms of the temparature T. – M. Wind May 27 '21 at 17:53
  • Okay, thank you for your explanations. – michael b May 27 '21 at 18:21

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