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plasma-partition.tex

@@ -414,7 +414,7 @@ \section{Gilbertian magnetization of electron-positron plasma}
 \label{defmagetization}
  {\cal M}\equiv\frac{T}{V}\frac{\partial}{\partial{\cal B}}\ln{{\cal Z}_{e^{+}e^{-}}} = \frac{T}{V}\left(\frac{\partial b_{0}}{\partial{\cal B}}\right)\frac{\partial}{\partial b_{0}}\ln{{\cal Z}_{e^{+}e^{-}}}\,,
 \end{align}
-Magnetization arising from other components in the cosmic gas (protons, neutrinos, etc.) could in principle also be included. Localized inhomogeneities and matter evolution are often non-trivial and generally are solved numerically using magneto-hydrodynamics (MHD)~\cite{melrose2008quantum,Vazza:2017qge,Vachaspati:2020blt,Stoneking:2020egj}. In the context of MHD, primordial magnetogenesis from fluid flows in the electron-positron epoch was considered in~\cite{Gopal:2004ut,Perrone:2021srr}.  {\xblue We note in passing that the possible conservation of magnetic helicity~\cite{Boyarsky:2011uy}
+Magnetization arising from other components in the cosmic gas (protons, neutrinos, etc.) could in principle also be included. Localized inhomogeneities and matter evolution are often non-trivial and generally are solved numerically using magneto-hydrodynamics (MHD)~\cite{melrose2008quantum,Vazza:2017qge,Vachaspati:2020blt,Stoneking:2020egj}. In the context of MHD, primordial magnetogenesis from fluid flows in the electron-positron epoch was considered in~\cite{Gopal:2004ut,Perrone:2021srr}. {\xblue We note in passing that the possible conservation of magnetic helicity~\cite{Boyarsky:2011uy}
 relates to current induced magnetic fields. We do not expect this conservation law to hold for our Gilbertian spin based magnetization. 
 }
 
@@ -554,11 +554,9 @@ \subsection{Magnetic moment chemical potential}
  +{\cal O}\left(b_{0}^{2}\right)
 \end{multline}
 
-Given~\req{ferro}, we can understand the polarization potential as a kind of `ferromagnetic' influence on the primordial gas which allows for magnetization even in the absence of external magnetic fields. This interpretation is reinforced by the fact that the leading coefficient is $g/2$.
+Given~\req{ferro}, we can understand the polarization potential as a kind of `ferromagnetic' influence on the primordial gas which allows for magnetization in the absence of external magnetic fields. This interpretation is reinforced by the fact that the leading coefficient is $g/2$. We suggest that a variety of physics could produce a small nonzero $\eta$ within a domain of the gas. Such asymmetries could also originate statistically as while the expectation value of free gas polarization is zero, the variance is likely not.
 
-We suggest that a variety of physics could produce a small nonzero $\eta$ within a domain of the gas. Such asymmetries could also originate statistically as while the expectation value of free gas polarization is zero, the variance is likely not.
-
-As $\sinh{\eta/T}$ is an odd function, the sign of $\eta$ also controls the alignment of the magnetization. In the high temperature limit~\req{ferro}, with strictly $b_{0}=0$, is to lowest order for brevity
+As $\sinh{\eta/T}$ is an odd function, the sign of $\eta$ also controls the alignment of the magnetization. In the high temperature limit~\req{ferro}, with strictly $b_{0}=0$, magnetization at lowest order is
 \begin{align}
  \label{hiTferro}
  \lim_{m_{e}/T\rightarrow0}{\mathfrak M}\vert_{b_{0}=0}=\frac{g}{2}\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\frac{\eta}{T}\,,
@@ -613,11 +611,11 @@ \subsection{Self-magnetization}
 \subsection{Macroscopic magnetization length scale and statistical fluctuations}
 \label{sec:lengthscale}
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-\noindent It is of interest to consider in the $e^{+}e^{-}$ medium the spatial length scale $\lambda $ over which the dipole induced magnetization is constant, referred in Amp{\`e}rian magnetization literature as coherence length.  As noted in \rsec{sec:introduction}, the two different mechanisms for magnetogenesis (through Amp{\`e}rian matter currents, or through Gilbertian magnetic moment alignment) are two different physical sources of magnetic field and  produce different spectra of magnetic fields across differing length scales.
+\noindent It is of interest to consider the spatial length scale $\lambda$ over which the dipole induced magnetization is constant. When generated by Amp{\`e}rian currents, this is referred to as coherence length in literature. As noted in \rsec{sec:introduction}, magnetogenesis through Amp{\`e}rian matter currents or Gilbertian dipole alignment are distinct physical sources of magnetic field and produce different spectra across varying length scales.
 
-For these reasons we do not know what the typical length scale of induced magnetization could be. Similarly, the observational situation is in flux.  The length scale of IGMFs are not well constrained~\cite{Giovannini:2022rrl,Durrer:2013pga,AlvesBatista:2021sln} bounded by the range  $\lambda  \sim 10^{-2}-10^{3}$ Mpc for the external field strengths we considered. It has been argued that inhomogeneous PMFs of $\lambda  \lesssim 400$ pc are subject to dissipating effects~\cite{Jedamzik:1999bm}. Inhomogeneous field dissipation would affect not only the external PMFs, but also the induced magnetization as well restricting the spectrum of fluctuations. However, length scales above $\lambda \gtrsim10^{3}$ Mpc are not disallowed, if generated during an sufficiently early epoch such as inflation, but would require that the coherence of the PMF is beyond the size of the present day visible universe.
+We have yet to determine what the typical length scale of induced magnetization could be. Similarly, the observational situation is in flux. The length scale of IGMFs are not well constrained~\cite{Giovannini:2022rrl,Durrer:2013pga,AlvesBatista:2021sln} and are bounded by the range $\lambda\sim 10^{-2}-10^{3}$ Mpc for the external field strengths we considered. It has been argued that inhomogeneous PMFs of $\lambda  \lesssim400$ pc are subject to dissipating effects~\cite{Jedamzik:1999bm}. Inhomogeneous field dissipation would affect not only the external PMFs, but also the induced magnetization which restricts the spectrum of fluctuations. However, length scales above $\lambda\gtrsim10^{3}$ Mpc are not disallowed if generated during a sufficiently early epoch such as inflation. Such large scales would require that the coherence of the PMF is beyond the size of the present day visible universe.
 
-Our theoretical model needs to evolve by consideration of interactions including with the background nuclear dust in order to allow for the introduction of a collective magnetization length scale sourced by magnetic dipoles. Moreover, we note possibility of coupling to the cosmological expansion dynamics~\cite{Kahniashvili:2012uj}. Similarly, we are working to understand the thermal fluctuations $\langle(\Delta\mathcal{M})^{2}\rangle$  which  could be required in models we are exploring to characterize the Gilbertian $e^{+}e^{-}$ plasma magnetization coherence length.
+Our theoretical model should be improved by consideration of interactions including the background nuclear dust from BBN in order to allow for the introduction of a collective magnetization length scale sourced by dipoles. Moreover, we note possibility of coupling to the cosmological expansion dynamics~\cite{Kahniashvili:2012uj}. Similarly, we are working to understand the thermal fluctuations $\langle(\Delta\mathcal{M})^{2}\rangle$ which could be important to exploring the $e^{+}e^{-}$ plasma magnetization coherence length.
 }
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Summary and discussion}