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| \documentclass[onecolumn,preprintnumbers,amsmath,amssymb]{revtex4} | ||
| \usepackage{graphicx} | ||
| \usepackage{float} | ||
| \usepackage[usenames,dvipsnames]{color} | ||
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| \begin{document} | ||
| %\preprint{Draft.13} | ||
| \title{Fluctuation of Magnetization} | ||
| \author{Cheng Tao Yang$^a$, Andrew Steinmetz$^a$, Johann Rafelski$^a$} | ||
| \affiliation{$^a$Department of Physics, The University of Arizona, Tucson, Arizona 85721, USA} | ||
| \date{\today} | ||
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| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | ||
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| \begin{abstract} | ||
| \end{abstract} | ||
| \maketitle | ||
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| \section{Statistical fluctuation of magnetization} | ||
| In general, given the magnetic moment $\mu$ the magnetization density of the quantum system is defined as | ||
| \begin{align} | ||
| \langle M\rangle\equiv\frac{1}{V}\langle \mu\rangle=\frac{1}{V}\left(T\frac{\partial \ln\mathcal Z}{\partial B}\right),\qquad \langle\mu\rangle=\left(T\frac{\partial \ln\mathcal Z}{\partial B}\right) | ||
| \end{align} | ||
| In statistical mechanical, the mean-square fluctuation of any quantity magnetic moment $\mu$ can be written as | ||
| \begin{align} | ||
| \langle\Delta \mu^2\rangle=\langle \mu^2\rangle-\langle \mu\rangle^2=T^2\frac{\partial^2 \ln\mathcal Z }{\partial B^2} | ||
| \end{align} | ||
| In this scenario, the fluctuation of magnetization can be written as | ||
| \begin{align} | ||
| {\langle\Delta M^2\rangle}=\frac{\langle\Delta \mu^2\rangle}{V}=\frac{T^2}{V}\frac{\partial^2 \ln\mathcal Z }{\partial B^2}=T\frac{\partial}{\partial B}\left(\frac{T}{V}\frac{\partial \ln\mathcal Z}{\partial B}\right)=T\frac{\partial\langle M\rangle}{\partial B}. | ||
| \end{align} | ||
| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | ||
| \section{The fluctuation of magnetized $e^\pm$ plasma } | ||
| In our study of magnetization of electron-positron plasma, the partition function of electron-positron plasma is given by | ||
| \begin{align} | ||
| \begin{split} | ||
| \label{boltzmann} | ||
| \ln{\cal Z}&\simeq\frac{T^{3}V}{\pi^{2}}\sum_{s'}^{\pm1}\left[\xi_{s'}\cosh{\frac{\mu}{T}}\right]\left(x_{s'}^{2}K_{2}(x_{s'})+\frac{b_{0}}{2}x_{s'}K_{1}(x_{s'})+\frac{b_{0}^{2}}{12}K_{0}(x_{s'})\right)\,, | ||
| \end{split}\\ | ||
| \label{xfunc} | ||
| x_{s'}&=\frac{{\tilde m}_{s'}}{T}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+b_{0}\left(1-\frac{g}{2}s'\right)},\,\qquad b_0=\frac{eB}{T^2}. | ||
| \end{align} | ||
| We show that the total dimensionless magnetization ${\mathfrak M}$ for the case $g=2$ can be broken into the sum of magnetic moment parallel ${\mathfrak M}_{+}$ and magnetic moment anti-parallel ${\mathfrak M}_{-}$ contributions as follows | ||
| \begin{align} | ||
| \label{g2mag} | ||
| {\mathfrak M}={\mathfrak M}_{+}+{\mathfrak M}_{-},\,\qquad {\mathfrak M}=\frac{e\langle M\rangle}{m_e^2},\qquad \langle M\rangle=\frac{T}{V}\frac{\partial \ln{\mathcal Z}}{\partial B},\end{align} | ||
| where ${\mathfrak M}_{+}$ and ${\mathfrak M}_{-}$ are given by | ||
| \begin{align} | ||
| {\mathfrak M}_{+}&=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi\cosh{\frac{\mu}{T}}\left[\frac{1}{2}x_{+}K_{1}(x_{+})+\frac{b_{0}}{6}K_{0}(x_{+})\right],\qquad x_{+}=\frac{m_{e}}{T}, | ||
| \end{align} | ||
| and | ||
| \begin{align} | ||
| {\mathfrak M}_{-}&=-\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi^{-1}\cosh{\frac{\mu}{T}}\left[\left(\frac{1}{2}+\frac{b_{0}^{2}}{12x_{-}^{2}}\right)x_{-}K_{1}(x_{-})+\frac{b_{0}}{3}K_{0}(x_{-})\right], \qquad x_{-}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+2b_{0}} | ||
| \end{align} | ||
| In this scenario, the fluctuation of magnetized electron-positron plasma can be obtain | ||
| \begin{align} | ||
| \langle\Delta {M}^2\rangle=T\frac{\partial {\langle M\rangle} }{\partial B}=T\frac{\partial b_0}{\partial B}\frac{\partial {\langle M\rangle} }{\partial b_0}=T\left(\frac{e}{T^2}\right)\frac{\partial}{\partial b_0}\left(\frac{m^2_e}{e}{\mathfrak M}\right)=\frac{m_e^2}{T}\left(\frac{\partial {\mathfrak M}_{+} }{\partial b_0}+\frac{\partial {\mathfrak M} _{-}}{\partial b_0}\right) | ||
| \end{align} | ||
| where | ||
| \begin{align} | ||
| \frac{\partial {\mathfrak M}_{+} }{\partial b_0}=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi\cosh{\frac{\mu}{T}}\,\left[\frac{1}{6}K_{0}(x_{+})\right],\qquad x_{+}=\frac{m_{e}}{T} | ||
| \end{align} | ||
| and | ||
| \begin{align} | ||
| \frac{\partial {\mathfrak M}_{-} }{\partial b_0}=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi^{-1}\cosh{\frac{\mu}{T}}\left[\left(\frac{1}{6}+\frac{b^2_0}{12x^2_{-}}\right)K_0(x_{-})+\left(\frac{b_0}{6x_{-}}+\frac{b^2_0}{6x^3_{-}}\right)K_1(x_{-})\right],\qquad x_{-}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+2b_{0}} | ||
| \end{align} | ||
| In Fig.~\ref{Susc_fig} we plot the $\partial{\mathfrak M}_\pm/\partial b_0$ of the primordial $e^{+}e^{-}$-plasma as a function of temperature, with $g=2$, $\xi=1$, $b_0=10^{-11}$ and $b_0=10^{-3}$. It shows that in the magnetic field we consider $10^{-11}<b_0<10^{-3}$, the dominate term for $\partial{\mathfrak M}_{-}/\partial b_0$ is given by | ||
| \begin{align} | ||
| \frac{\partial {\mathfrak M}_{-} }{\partial b_0}&\approx\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi^{-1}\cosh{\frac{\mu}{T}}\left[\left(\frac{1}{6}\right)K_0(x_{-})\right]\approx \frac{\partial {\mathfrak M}_{+} }{\partial b_0}, \qquad \mathrm{for} \,\,10^{-11}<b_0<10^{-3}, | ||
| \end{align} | ||
| then the fluctuation of magnetized electron-positron plasma can be approximated by | ||
| \begin{align} | ||
| \langle\Delta {M}^2\rangle=\frac{m_e^2}{T}\left(\frac{\partial {\mathfrak M}_{+} }{\partial b_0}+\frac{\partial {\mathfrak M} _{-}}{\partial b_0}\right)\approx \frac{T}{6}\left(\frac{e^{2}}{\pi^{2}}\right)\cosh{\frac{\mu}{T}}\bigg[\xi K_0(x_+)+\xi^{-1}K_0(x_{-})\bigg] | ||
| \end{align} | ||
| In Fig.~\ref{Flu_fig} we plot the fluctuation $ \langle\Delta {\mathcal M}^2\rangle/m_e$ nd $\langle\Delta {\mathcal M}^2\rangle/\sqrt{\langle M\rangle}$ as a function of temperature with $g=2$, $\xi=1$, $b_0=10^{-11}$ and $b_0=10^{-3}$. | ||
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| %~~~Figure~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
| \begin{figure}[ht] | ||
| \centering | ||
| \includegraphics[width=0.5\textwidth]{Susceptibility001}\includegraphics[width=0.5\textwidth]{Susceptibility002} | ||
| \caption{The $\partial{\mathfrak M}_\pm/\partial b_0$ of the primordial $e^{+}e^{-}$-plasma as a function of temperature, with $g=2$, $\xi=1$, $b_0=10^{-11}$ and $b_0=10^{-3}$.} | ||
| \label{Susc_fig} | ||
| \end{figure} | ||
| %~~~Figure~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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| %~~~Figure~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
| \begin{figure}[ht] | ||
| \centering | ||
| \includegraphics[width=0.5\textwidth]{Fluctuation_Magnetization}\includegraphics[width=0.5\textwidth]{Fluctuation_M002} | ||
| \caption{The dimensionless fluctuation $ \langle\Delta {M}^2\rangle/m_e$ and $\langle\Delta {\mathcal M}^2\rangle/\sqrt{\langle M\rangle}$ of the primordial $e^{+}e^{-}$-plasma as a function of temperature, with $g=2$, $\xi=1$, $b_0=10^{-11}$ and $b_0=10^{-3}$.} | ||
| \label{Flu_fig} | ||
| \end{figure} | ||
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| %~~~Figure~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | ||
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| \begin{thebibliography}{99} | ||
| \end{thebibliography} | ||
| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | ||
| \end{document} |
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