diff --git a/plasma-partition.tex b/plasma-partition.tex index 18338f0..c49744e 100644 --- a/plasma-partition.tex +++ b/plasma-partition.tex @@ -414,7 +414,10 @@ \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 of matter evolution are often non-trivial and generally be 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}. +Magnetization arising from other components in the cosmic gas (protons, neutrinos, etc.) could in principle also be included. Localized inhomogeneities of matter evolution are often non-trivial and generally be 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}. {\color{blue} 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. +} + We introduce dimensionless units for magnetization ${\mathfrak M}$ by defining the critical field strength \begin{align} @@ -611,16 +614,12 @@ \subsection{Self-magnetization} \subsection{Macroscopic magnetization length scale and statistical fluctuations} \label{sec:lengthscale} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent It is of interest to consider in the $e^{+}e^{-}$ medium the spatial length scale $\lambda_\mathcal{M}$ over which the dipole induced magnetization is constant, referred in literature on Amp{\`e}rian magnetization as coherence length. Similarly, we are interested to understand the thermal fluctuations $\langle(\Delta\mathcal{M})^{2}\rangle$ present. As mentioned prior in \rsec{sec:introduction}, we expect that -The two different mechanisms for magnetogenesis (through Amp{\`e}rian matter currents, or through Gilbertian magnetic moment alignment produce different spectra of magnetic fields across differing length scales. Moreover prior Amp{\`e}rian work was considering ionic plasma and not $e^{+}e^{-}$ +\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. -In principle there are two field scales: The first is the correlation length~\cite{Kahniashvili:2012uj} associated with the `external' PMF $\lambda_{B}$, and the other related to the possibly spontaneously occurring magnetization in the plasma, $\lambda_\mathcal{M}$. Under the paramagnetic response described in \rsec{sec:magnetization}, the coherent length of the magnetization naturally mirrors that of the external PMF whatever mechanism is responsible for its generation. However, if magnetization is spontaneous as discussed in \rsec{sec:ferro}, then the two scales, $\lambda_{B}$ and $\lambda_\mathcal{M}$ may differ. Literature in general refers to $\lambda_{B}$ and in following discussion we address this quantity. +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. -The observational restriction on the coherent length scale of IGMFs are not well constrained~\cite{Giovannini:2022rrl,Durrer:2013pga,AlvesBatista:2021sln} and are bounded by the range $\lambda_{B}\sim10^{-2}-10^{3}$ Mpc for the external field strengths we considered. Inhomogeneous PMFs of $\lambda_{B}\lesssim400$ pc are subject to dissipating effects~\cite{Jedamzik:1999bm}. Inhomogeneous field dissipation would affect not only the external PMFs, but also the induced magnetization as well restricting the spectrum of fluctuations. However, length scales above $\lambda_{B}\gtrsim10^{3}$ 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. - -As our model quantifies the relativistic paramagnetism of the $e^{+}e^{-}$ medium, the induced magnetization should then be subject to the variations and spectra of the external PMF present. If an observational signature of the $e^{+}e^{-}$ magnetization could be ascertained, then this would provide a way to characterize the length scale and coherence of the original PMF. If the polarization fugacity was nonzero as per \req{ferro} and \req{hiTferro}, then the spectra of the magnetization would match the variation in spatial spin polarization $\eta(x)$. This would likely be associated with the long-range interaction length scale which generates a nonzero $\eta$. We have yet to consider the effect of the induced magnetization on the conservation of magnetic helicity~\cite{Boyarsky:2011uy}. We plan to return to the question of thermally induced spontaneous magnetization more fully in a future work. +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. } - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Summary and discussion} \label{sec:conclusions} diff --git a/steinmetz_magnetization_prd_response.pdf b/steinmetz_magnetization_prd_response.pdf new file mode 100644 index 0000000..1af16dc Binary files /dev/null and b/steinmetz_magnetization_prd_response.pdf differ