diff --git a/README.md b/README.md index fe723b1..3f8ff60 100644 --- a/README.md +++ b/README.md @@ -102,7 +102,7 @@ $$\langle L \rangle (f ) \approx \frac{Nfa^2}{k_BT}$$ may be used. In this final simulation, we seek to find out how the mean lengths $L$ from biased sampling according to the force-dependent $p_+(f)$ align with the theoretical expected value and the small force approximation. ![](./figures/task3_true_vs_sim.png) -**Figure 6:** *$\langle L \rangle(f)$ for various forces $f$. We have $M=10^3$ samples for each $f$, with $k_BT=a=1$ and $N=100$, which gives a $k_{eff}$ of $100$. The analytical formulas are in red and are described above this figure. We can see excellent agreement with the analytical $\langle L \rangle(f)$ across a large range of $f$. However, the small force approximation in the lower figure starts to diverge at $f > 0.3$. A linear fit to $\langle \hat{L} \rangle(f<0.3)$ gives us an estimated $k_{eff} \approx 96.93$. +**Figure 6:** *$\langle L \rangle(f)$ for various forces $f$. We have $M=10^3$ samples for each $f$, with $k_BT=a=1$ and $N=100$, which gives a $k_{eff}$ of $100$. The analytical formulas are in red and are described above this figure. We can see excellent agreement with the analytical $\langle L \rangle(f)$ across a large range of $f$. However, the small force approximation in the lower figure starts to diverge at $f > 0.3$. A linear fit to $\langle \hat{L} \rangle(f<0.3)$ gives us an estimated $k_{eff} \approx 96.93$*. ![](./figures/task3_true_vs_sim_error.png) **Figure 7:** *To clarify Figure 6, the absolute difference between the sampled $\langle \hat{L} \rangle(f)$ and $\langle L \rangle(f)$ as well as $\langle L \rangle(f)$ with the small force approximation. We can very clearly see that the small force approximation breaks down around an