... cycles. ¹

Note that I have unceremoniously changed my definition of ``cycle''. Before, one ``cycle'' was $N$ moves; now it is a single move. This distinction isn't important for now, but I thought you'd like to be made aware.

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... as²

Eq. 270 is a result of the following general relations, given in Appendix C. For some quantity $A$ of a system subject to a small constant perturbation, $\lambda B$, the average of the response of $A$ after the perturbation is removed, $\left<\Delta A\left(t\right)\right>$, is given by

$$\left<\Delta A\left(t\right)\right> = \frac{ \int d{\bf \Gam... ... d{\bf \Gamma}e^{-\beta\left(\mathscr{H}_0-\lambda B\right)} }$$ (266)

$A(t)$ is the trajectory of $A$ in the unperturbed system, and $\Gamma$ is shorthand for phase space point $({\bf q},{\bf p})^N$. $B$ is generally a function of $\Gamma$, but not time. In the limit as $\lambda\rightarrow 0$, and assuming that $\left<A\right>$ = 0, we see that

$$\left<\Delta A\left(t\right)\right> = \beta\lambda\frac{ \int d{\bf \Gamma}e^{-\beta\mathscr{H}_0}BA\left(t\right) }{ \int d{\bf \Gamma}e^{-\beta\mathscr{H}_0} }$$ (267)

$$= \beta\lambda\left<B\left(0\right)A\left(t\right)\right>$$ (268)

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...,³

If that second equality in Eq. 279 looks fishy, but it is due to the fact that the time origin is chosen arbitrarily. For any autocorrelation function

$$\left<A\left(t\right)A\left(t^\prime\right)\right> = \left<A\left(0\right)A\left(t^\prime-t\right)\right> = \left<A\left(t^\prime-t\right)A\left(0\right)\right>$$ (276)

$$\therefore-\left<A\left(0\right)\dot{A}\left(t\right)\right> = \left<\dot{A}\left(0\right)A\left(t\right)\right>$$ (277)

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... 2.0. ⁴

Note that the code tps provided by F&S (written by the eminent Thijs Vlugt) is hard-coded with a path length $T$ = 5 $\tau$. To change $T$ to 2 $\tau$, we edit the file maxarray.inc, and change the parameter Maxtraject from 1000 to 400. This is the number of ``slices'' in a trajectory, and each slice is Nshort = 5 time steps. The reason $T$ = 5 was originally used is likely because the exercise for which this code was originally written was conducted with a total energy of 9 $\epsilon$ vs. the 15 $\epsilon$ we consider here; that is, we run at a higher effective temperature. Dellago's original paper considered $N$ = 9 particles with a total energy of 9 $\epsilon$, so we should be running at the same effective temperature as Dellago with the larger number of particles considered by Vlugt. We are, however, running at a slightly lower density ( $\rho = 0.53 \sigma^{-2}$) than considered by Dellago ( $\rho = 0.6 \sigma^{-2}$).

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...-vectors. ⁵

One modification of this code is a necessary one: the implementation of the real-space energy was left as an exercise. Other modifications made easily include embedding the main program inside a double loop over desired $\alpha$ and $k_{\rm max}$ values.

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...eq:dos). ⁶

Strictly speaking, the density of states has units of $\left\{energy\right\}^{-1}$, because $\overline\Omega(E)dE$ is the number of states with energy between $E$ and $E+dE$. However, in common usage in the literature, the quantity $\Omega (E)$ is referred to as the density of states, although it is correctly termed the microcanonical partition function.

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