Magnetic Free-Energy Contributions

• Magnetic free-energy contributions are defined as the excess energy above a potential configuration, quantifying the work available for phase transitions, solar flares, and mechanical stresses.
• They are computed using statistical and nonequilibrium methods—such as high/low temperature expansions, the Jarzynski equality, and thermodynamic integration—which enable precise field-dependent free energy evaluations.
• These contributions play a crucial role in forecasting solar eruptions and analyzing magnetocrystalline anisotropy, linking theoretical frameworks to practical energy release diagnostics in complex magnetic systems.

Magnetic free-energy contributions describe the excess thermodynamic or field-theoretic energy above a specified reference (often potential or non-interacting) configuration, arising from the presence of magnetic structures—spin alignments, field shears, or current-carrying topologies—in systems ranging from spin models to astrophysical plasmas. Magnetic free energy quantifies the capacity for work extraction (e.g., phase transitions, solar flares, or mechanical stresses in materials), links to fundamental symmetries and constraints (helicity, topology, gauge invariance), and underpins analytic and computational frameworks in statistical mechanics, condensed matter, and plasma astrophysics. The topic is informed by fluctuation relations, partition function-based formalisms, nonlinear force-free field reconstructions, and nonequilibrium thermodynamic methodologies.

1. Foundational Definitions and General Formalism

In both statistical mechanics and classical field theory, the magnetic free energy $F$ is defined as the Helmholtz or thermodynamic free energy in the presence of magnetic degrees of freedom, specifically including contributions from field–matter, internal interactions, and imposed external fields. For a magnetic system (e.g., spin lattice, coronal field), one distinguishes:

• Total magnetic energy:

$$E_{\rm tot} = \frac{1}{8\pi}\int_V |\mathbf{B}|^2\,dV$$

where $\mathbf{B}$ is the total field (physical or modeled).

• Potential (current-free) field energy:

$$E_{\rm pot} = \frac{1}{8\pi}\int_V |\mathbf{B}_{\rm pot}|^2\,dV$$

with $\nabla \times \mathbf{B}_{\rm pot} = 0$.

• Magnetic free energy:

$$E_{\rm free} = E_{\rm tot} - E_{\rm pot} = \frac{1}{8\pi}\int_V \left(|\mathbf{B}|^2 - |\mathbf{B}_{\rm pot}|^2\right) dV$$

This captures the excess energy available for processes such as phase transitions (in statistical models) or magnetic reconnection and eruptive events (in plasmas) (Liokati et al., 2023, Thalmann et al., 2013, Regnier, 2011).

The free-energy functional is central in quantifying phase behavior, quantifying stability against instabilities, and predicting energy release in non-equilibrium phenomena.

2. Statistical Mechanics: Spin Models and Field Dependence

Spin systems such as the Ising and Heisenberg models provide canonical frameworks for analyzing magnetic free-energy contributions. The equilibrium free energy, $F(T, H) = -k_B T \log Z(T,H)$, encodes all thermodynamic information, with the magnetic field $H$ coupling to magnetization $M$. Key results include:

• High- and low-temperature expansions: The free energy is systematically expanded as a bivariate series in coupling strength and field (Butera et al., 2010). For the 3D Ising model:

$F(K, h) = \sum_{n=0}^{24} f_n(h) K^n + O(K^{25})$

where $f_n(h)$ are polynomials in the field.

• Universal scaling: Near criticality, magnetic free energy obeys scaling relations linking field, temperature, and response functions (e.g., susceptibilities $\chi_n$ as field derivatives of $F$) (Butera et al., 2010).
• Nonequilibrium methods: Advanced methods such as the Jarzynski equality (JE) and Crooks fluctuation theorem (CFT) enable computation of magnetic-field-dependent free energy differences in systems driven out of equilibrium:

$$e^{-\beta \Delta F} = \bigl\langle e^{-\beta W} \bigr\rangle$$

where $W$ is the work performed by switching the magnetic field according to a protocol, and $\Delta F$ is the free-energy change (Cajahuaringa et al., 2018).

• Thermodynamic integration: The field-dependent free energy can be integrated via

$\Delta F(H) = -\int_0^H \langle M \rangle_{H'} dH'$

either quasi-statically (equilibrium) or via JE estimates (nonequilibrium) (Cajahuaringa et al., 2018).

These techniques yield precise, field-resolved $F(T,H)$ surfaces, facilitating phase-boundary identification, susceptibility calculations, and studies of transition phenomena.

3. Magnetic Free Energy in Solar and Astrophysical Plasmas

In astrophysical contexts, particularly solar active regions and the corona, magnetic free-energy contributions control eruption dynamics, reconnection, and energy-release processes. Quantitative frameworks involve:

• Coronal Free Energy (CFME): Defined as the energy above a potential field for the observed boundary conditions, typically modeled as a nonlinear force-free field (NLFFF):

$$E_{\rm free} = \frac{1}{8\pi} \int_V \left(|\mathbf{B}_{\rm NLFFF}|^2 - |\mathbf{B}_{\rm pot}|^2\right) dV$$

(Liokati et al., 2023, Georgoulis et al., 2012, Thalmann et al., 2013).

• Self and mutual terms: For connectivity-based models, the free energy decomposes into "self" (twist, writhe of individual flux tubes) and "mutual" (inter-tube interaction) contributions (Liokati et al., 2023, Moraitis et al., 2014):

$$E_{\rm f} = A d^2 \sum_{l=1}^N \alpha_l^2 \Phi_l^{2\lambda} + \frac{1}{8\pi} \sum_{l\neq m} \alpha_l \mathcal{L}_{lm}^{\rm arch} \Phi_l \Phi_m$$

with $\alpha_l$ the force-free parameter, $\Phi_l$ the tube flux, and $\mathcal{L}_{lm}^{\rm arch}$ the mutual-helicity coefficient.

• Scaling relations: Emerged flux $\Phi$, free energy $E_{\rm free}$, and relative helicity $H_R$ obey robust power-law relationships (e.g., $E_{\rm free} \propto \Phi^2$), enabling estimation of coronal free energy from photospheric flux measurements alone (Magara, 2014).
• Eruption forecasting: Abrupt losses of $10$–$60\%$ in free energy, co-temporal with magnetic helicity changes, accompany major flare/CME events. Fast, connectivity-based free-energy budgeting enables real-time diagnosis and forecasting of plasma eruptions (Liokati et al., 2023).
• Temporal and spatial partition: Most free energy is stored in mutual terms between major polarity partitions, with localized enhancements (e.g., $\delta$-spots) preconditioning flares (Liokati et al., 2023, Zhang, 2016).

4. Nonequilibrium and Microscopic Approaches in Materials

In condensed matter, free energy contributions arise from microscopic spin interactions, field-induced deformations, and topological constraints:

• Anisotropic and higher-order terms: In systems such as noncollinear antiferromagnets (e.g., Mn$_3$Sn), the magnetic free energy includes nontrivial terms:

$$F(H, \phi) = F_0 - \mu_{\rm eff} H \cos \phi + \frac{1}{2}\alpha H^2 + \frac{1}{3}\beta H^3 \cos 6\phi + \frac{1}{5}\gamma H^5 \cos 12\phi + \cdots$$

The $H^3$ and $H^5$ contributions drive the observed magnetocrystalline anisotropy, in contrast to the pure quadratic dependence in conventional magnets (Li et al., 2021).

• Quantum dissipation: For magnetized, dissipative quantum oscillators, exact field-dependent free energy expressions reflect coupling mechanisms (e.g., momentum vs coordinate coupling to heat baths). The free-energy corrections can be extracted by:

$$\Delta F_{\rm mag}(T, B) = -\frac{1}{\pi}\int_0^\infty d\omega\, f(\omega, T) \operatorname{Im} \frac{d}{d\omega} \ln\left[1 - \frac{\omega_c^2 G^2(\omega)} {(\alpha^{(0)}(\omega))^2}\right]$$

Characteristic scaling and magnetization/susceptibility corrections follow (Gupta et al., 2012).

• Structure-integration approaches: Efficient computation of magnetic free energies in large spin systems may be achieved via structure integration, approximating the density of states as lattice-dependent Gaussian distributions in cluster correlation variables (Takeuchi et al., 2015).

5. Observational, Computational, and Diagnostic Techniques

Accurate evaluation of magnetic free energy—particularly in solar/astrophysical applications or large-scale simulations—requires careful methodology:

• Data-driven Extrapolation: NLFFF methods, VCA4-NLFFF, and partitioning-simulated annealing allow 3D field reconstruction from surface vector magnetograms, enabling direct integration of $$|\mathbf{B}_{\rm NLFFF}|^2 - |\mathbf{B}_{\rm pot}|^2$$ over observed volumes (Thalmann et al., 2013, Aschwanden et al., 2020).
• Energy density decompositions: The local free-energy density can be expressed as

$$\rho_{\rm free} = \frac{B_n^2}{8\pi} + \frac{\mathbf{B}_n \cdot \mathbf{B}_p}{4\pi}$$

where $\mathbf{B}_n$ is the non-potential component and $\mathbf{B}_p$ the potential (reference) field. This highlights the roles of shear and current-carrying contributions; regions with negative $\rho_{\rm free}$ (dominant shear anti-alignment) are often sites for reconnection and eruption onset (Zhang, 2016).

• Solenoidality constraints: Accurate free energy computation requires fields that are (numerically) divergence-free. Deviation from $\nabla\cdot\mathbf{B}=0$ introduces spurious nonsolenoidal contributions, manifest in negative or anomalous free energies, and demands proper Helmholtz decomposition and divergence-cleaning algorithms (Valori et al., 2013).
• Gauge invariance: Surface-integral expressions for free-energy flux, particularly the MHD Poynting flux across boundaries,

$$\frac{dE_{\rm free}}{dt} = \frac{c}{4\pi} \oint_{S} [\mathbf{E} \times \mathbf{B}_c ]\cdot d\mathbf{S}$$

(where $\mathbf{B}_c$ is the current-carrying field) are gauge-invariant and enable model-independent assessment of energy inflow and boundary constraints (Schuck et al., 2019).

6. Scaling Laws and Universalities

Power-law relationships emerge universally across theoretical and computational studies:

Quantity	Scaling Law	Context
$E_{\rm free}$	$C_E \Phi^{2}$	Solar emerging flux regions (Magara, 2014)
$H_R$	$C_H \Phi^{2}$	(Ditto)
$E_{\rm free}$	$\propto \alpha^2 E_{\rm pot}$	Linear FF fields, low $\alpha$ (Regnier, 2011)

Such scaling enables field- or flux-based proxies for the "invisible" reservoir of energy available for eruptions or phase changes.

7. Physical Interpretation and Broader Implications

Magnetic free energy is not simply a measure of available "work," but encodes deep connections to stability, symmetry, and transport:

• In spin models, magnetic free energy controls phase boundaries, response, and universality (e.g., 3D Ising scaling laws) (Butera et al., 2010).
• In solar/astrophysical plasmas, the buildup and release of free energy underpins flare/CME phenomenology and provides the fundamental limit for space-weather event forecasting (Liokati et al., 2023, Moraitis et al., 2014, Schuck et al., 2019).
• Magnetic free energy within nonlinear topologies (e.g. braid, twist, shear) explains the localization of energy release, the onset of reconnection, and the sustainment of dissipationless wave modes (e.g., torsional Alfvén oscillations) observed in flaring loops (Aschwanden et al., 2020, Thalmann et al., 2013).
• The quantification and partitioning of magnetic free energy, combined with diagnostics of helicity and topological invariants, are essential for the self-consistent modeling of complex plasma or solid-state systems.

• Nonequilibrium free energy methods applied to magnetic systems: the degenerate Ising model (Cajahuaringa et al., 2018)
• Magnetic Helicity and Free Magnetic Energy as Tools to Probe Eruptions in two Differently Evolving Solar Active Regions (Liokati et al., 2023)
• Torsional Alfvénic Oscillations Discovered in the Magnetic Free Energy During Solar Flares (Aschwanden et al., 2020)
• Scaling laws of free magnetic energy stored in a solar emerging flux region (Magara, 2014)
• Photospheric Magnetic Free Energy Density of Solar Active Regions (Zhang, 2016)
• Force-free field modeling of twist and braiding-induced magnetic energy in an active-region corona (Thalmann et al., 2013)
• Magnetic Energy Storage and Current Density Distributions for Different Force-Free Models (Regnier, 2011)
• Validation and Benchmarking of a Practical Free Magnetic Energy and Relative Magnetic Helicity Budget Calculation in Solar Magnetic Structures (Moraitis et al., 2014)
• The free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model (Butera et al., 2010)
• The free energy of twisting spins in Mn$_3$Sn (Li et al., 2021)
• Free energy of a charged oscillator in a magnetic field and coupled to a heat bath through the momentum variables (Gupta et al., 2012)
• Extension of structure integration to magnetic system (Takeuchi et al., 2015)
• Magnon Energy Renormalization and Low-Temperature Thermodynamics of O(3) Heisenberg Ferromagnets (Radosevic et al., 2013)
• Accuracy of magnetic energy computations (Valori et al., 2013)
• Determining the Transport of Magnetic Helicity and Free Energy in the Sun's Atmosphere (Schuck et al., 2019)