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-rw-r--r-- | ising_scaling.bib | 16 | ||||
-rw-r--r-- | ising_scaling.tex | 10 |
2 files changed, 22 insertions, 4 deletions
diff --git a/ising_scaling.bib b/ising_scaling.bib index a971fbd..4185f2f 100644 --- a/ising_scaling.bib +++ b/ising_scaling.bib @@ -341,3 +341,19 @@ } +info: 'Griffiths_1967' has been autocompleted into 'Griffiths_1967_Thermodynamic'. +@article{Griffiths_1967_Thermodynamic, + author = {Griffiths, Robert B.}, + title = {Thermodynamic Functions for Fluids and Ferromagnets near the Critical Point}, + journal = {Physical Review}, + publisher = {American Physical Society (APS)}, + year = {1967}, + month = {6}, + number = {1}, + volume = {158}, + pages = {176--187}, + url = {https://doi.org/10.1103%2Fphysrev.158.176}, + doi = {10.1103/physrev.158.176} +} + + diff --git a/ising_scaling.tex b/ising_scaling.tex index 8488f9d..2b1fb62 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -171,7 +171,7 @@ use $\xi=u_h|u_t|^{-\Delta}$ and $\eta=u_t|u_h|^{-1/\Delta}$. The analyticity of the free energy at places away from the critical point implies that the functions $\mathcal F_\pm$ and $\mathcal F_0$ have power-law expansions of their arguments about zero, the result of so-called Griffiths -analyticity. For instance, when $u_t$ goes to zero for nonzero $u_h$ there is +analyticity \cite{Griffiths_1967_Thermodynamic}. For instance, when $u_t$ goes to zero for nonzero $u_h$ there is no phase transition, and the free energy must be an analytic function of its arguments. It follows that $\mathcal F_0$ is analytic about zero. This is not the case at infinity: since @@ -896,7 +896,7 @@ values of both are plotted. Free parameters in the fit of the parametric coordinate transformation and scaling form to known values of the scaling function series coefficients for $\mathcal F_\pm$. The fit at stage $n$ matches those coefficients up to - and including order $n$. Error estimates are difficult to quantify directly. + and including order $n$. Uncertainty estimates are difficult to quantify directly. } \label{tab:fits} \end{table} @@ -1013,7 +1013,7 @@ Fig.~\ref{fig:phi.series}. The series coefficients for the scaling function $\mathcal F_-$ as a function of polynomial order $m$. The numeric values are from Table \ref{tab:data}, and those of Caselle \textit{et al.} are from the most - accurate scaling function listed in \cite{Caselle_2001_The}. The deviation at high polynomial order illustrates the lack of the essential singularity in Caselle's form. + accurate scaling function listed in \cite{Caselle_2001_The}. The deviation at high polynomial order illustrates the lack of the essential singularity in the form of Caselle \textit{et al.}. } \label{fig:glow.series} \end{figure} @@ -1186,7 +1186,9 @@ the ratio. Sequential ratios of the series coefficients of the scaling function $\mathcal F_-$ as a function of inverse polynomial order $m$. The extrapolated $y$-intercept of this plot gives the radius of convergence of - the series, which should be zero due to the essential singularity (as seen in the known numeric values and in this work). Cassel {\em et al} do not incorporate the essential singularity. + the series, which should be zero due to the essential singularity (as seen + in the known numeric values and in this work). Caselle \textit{et al.} do + not incorporate the essential singularity. } \label{fig:glow.radius} \end{figure} |