18
votes
Accepted
How to turn a sum into an integral?
So the trapezoid rule states that, if we partition an interval $(a, b)$ into $N + 1$ equally spaced points,
$$\Delta x = \frac{b - a}{N},~~~x_k = a + k~\Delta x,$$
then $$ \int_a^b \mathrm dx ~f(x) = ...
- 38.3k
14
votes
How to turn a sum into an integral?
$$Z_{tot}=\sum_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}$$
in the limit that $kT\gg\epsilon$, thus he writes
$$Z_{tot}\approx\int_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}\,dj$$
But how is this correct? There ...
- 21.9k
8
votes
How to turn a sum into an integral?
Here is an answer as a mathematician.
The situation you have is that you want to compare
$$
\sum_{j=0}^\infty f_s(j)\qquad\text{ and} \qquad \int_0^\infty f_s(t)\,dt,
$$
where $f_s(t)$ is a function ...
- 1,823
3
votes
How to turn a sum into an integral?
Really, why $j$ , not any other variable like $\epsilon$ and even $T$ .
See , as we have summation variables in integrals , there is always some variable in pure sum ( $\Sigma$ ) also. They are ...
- 378
2
votes
How to turn a sum into an integral?
Since $j$ increases by $1$ from term to term in the series, trivially, its increment is also $\Delta j= \left(j+1\right) - j = 1$, and it is correct to write
$$
\begin{aligned}
Z_\mathrm{tot} & = \...
- 1,082
2
votes
Accepted
How does the extra term and sign change come from in Harold Grad’s derivation of Boltzmann’s equation?
The minus sign in the third term of the first framed expression is due to the direction of the normal, which was taken outside the sphere $|q_2-q_1| = \sigma$. The second term in the first framed ...
- 5,697
2
votes
Accepted
Which higher-order terms require 4th-order integration of quadratically-constrained dynamics?
You need to be careful in applying leapfrog to your EoM's since you have a velocity dependence on the force. You can have an appropriate second-order solver if you derive it using the same principles.
...
- 13.1k
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