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| -rw-r--r-- | data/glow_series_caselle.dat | 3 | ||||
| -rw-r--r-- | ising_scaling.tex | 39 |
2 files changed, 40 insertions, 2 deletions
diff --git a/data/glow_series_caselle.dat b/data/glow_series_caselle.dat index 0e059ed..449d771 100644 --- a/data/glow_series_caselle.dat +++ b/data/glow_series_caselle.dat @@ -15,4 +15,5 @@ 14 -6808.202478556653 15 28115.088082925722 16 -118184.25755733704 -17 504553.6128947737
\ No newline at end of file +17 504553.6128947737 +18 -2.183603153706728e6
\ No newline at end of file diff --git a/ising_scaling.tex b/ising_scaling.tex index 161a4e6..54bada0 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -264,7 +264,7 @@ where $t$ and $h$ are polynomial functions selected so as to associate different \begin{align} \label{eq:schofield.funcs} t(\theta)=1-\theta^2 && - h(\theta)=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^\infty h_i\theta^{2i+1} + h(\theta)=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^\infty h_iP_{2i+1}(\theta/\theta_c) \end{align} This means that $\theta=0$ corresponds to the high-temperature zero-field line, $\theta=1$ to the critical isotherm at nonzero field, and $\theta=\theta_c$ to @@ -778,4 +778,41 @@ The successful smooth description of the Ising free energy produced in part by a \bibliography{ising_scaling} +\appendix + +\section{Legendre series to power series} + +In the work, we examine a series of the form +\[ + h(\theta)=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^\infty h_iP_{2i+1}(\theta/\theta_c), +\] +where $P_n$ is the $n$th Legendre polynomial. This form is useful for iteratively fitting the $h_i$, since these polynomials are orthogonal on the domain $[-\theta_c,\theta_c]$, but is not useful for evaluating the convergence of the function. For this a simple power expansion in $\theta$ is better. To make the conversion, first recall that +\[ + P_n(x)=2^n\sum_{k=0}^n\binom nk\binom{\frac{n+k-1}2}nx^k +\] +\[ + \begin{aligned} + \sum_{i=0}^\infty h_iP_{2i+1}(\theta/\theta_c) + &=\sum_{i=0}^\infty h_i2^{2i+1}\sum_{k=0}^{2i+1}\frac{\theta^k}{\theta_c^k}\binom{2i+1}k\binom{\frac{2i+k}2}{2i+1} \\ + &=\sum_{k=0}^\infty\theta^k\frac1{\theta_c^k}\sum_{i=(k-1)/2}^\infty h_i2^{2i+1}\binom{2i+1}k\binom{\frac{2i+k}2}{2i+1} \\ + &=\sum_{j=0}^\infty\theta^{2j+1}\frac1{\theta_c^{2j+1}}\sum_{i=j}^\infty h_i2^{2i+1}\binom{2i+1}{2j+1}\binom{\frac{2i+2j+1}2}{2i+1} + \end{aligned} +\] +We saw in our fitting that at higher order the coefficients $h_i$ approached roughly the pattern $h_i/h_{i-1}=-b+m/i$ for positive constants $b$ and $m$. This recurrence relation can be solved to give +\[ + h_i\simeq\frac{(-1)^i}{i!}b^{i-1}(b-m)h_0\frac{\Gamma(2-\frac mb+i-1)}{\Gamma(2-\frac mb)} +\] +\[ + \sum_{i=0}^\infty h_iP_{2i+1}(\theta/\theta_c) + =\sum_{j=0}^\infty\theta^{2j+1}\frac1{\theta_c^{2j+1}} + \frac{\Gamma(1+j-m/b)}{\Gamma(1-m/b)\Gamma(1+j)}2^{1+2j}(-b)^jh_0\binom{2j+\frac12}{2j+1}{}_2F_1(2j+\tfrac32,1+j-m/b;1+j;b) +\] +If we call this coefficient $H_j$, then +\[ + h(\theta)=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^\infty H_i\theta^{2i+1} + =\sum_{i=0}^\infty H_i\theta^{2i+1}-\frac1{\theta_c^2}\sum_{i=0}^\infty H_i\theta^{2i+3} + =\sum_{i=0}^\infty(H_i-H_{i-1}/\theta_c^2)\theta^{2i+1} +\] +for $H_{-1}=0$. + \end{document} |