In thermodynamics and thermal physics, the theoretical formulation of magnetic systems entails expressing the behavior of the systems using the Laws of Thermodynamics. Common magnetic systems examined through the lens of Thermodynamics are ferromagnets and paramagnets as well as the ferromagnet to paramagnet phase transition. It is also possible to derive thermodynamic quantities in a generalized form for an arbitrary magnetic system using the formulation of magnetic work. Simplified thermodynamic models of magnetic systems include the Ising model, the mean field approximation, and the ferromagnet to paramagnet phase transition expressed using the Landau Theory of Phase Transitions.
Arbitrary magnetic systems In order to incorporate magnetic systems into the first law of thermodynamics, it is necessary to formulate the concept of magnetic work. The magnetic contribution to the quasi-static work done by an arbitrary magnetic system is
W = − 1 4 π ∫ V H ⋅ Δ B d V {\displaystyle W=-{\frac {1}{4\pi }}{\int _{V}{H\cdot \Delta BdV}}}
where H {\displaystyle H} is the magnetic field and B {\displaystyle B} is the magnetic flux density. So the first law of thermodynamics in a reversible process can be expressed as
Δ U = ∫ S T d S − ∫ V P d V + 1 4 π ∫ V H ⋅ Δ B d V {\displaystyle \Delta U={\int _{S}{TdS}}-{\int _{V}{PdV}}+{\frac {1}{4\pi }}{\int _{V}{H\cdot \Delta BdV}}}
Accordingly the change during a quasi-static process in the Helmholtz free energy, F {\displaystyle F} , and the Gibbs free energy, G {\displaystyle G} , will be
Δ F = − ∫ T S d T − ∫ V P d V + 1 4 π ∫ V H ⋅ Δ B d V {\displaystyle \Delta F=-{\int _{T}SdT}-{\int _{V}PdV}+{\frac {1}{4\pi }}{\int _{V}{H\cdot \Delta BdV}}}
Δ G = − ∫ T S d T + ∫ P V d P − 1 4 π ∫ V B ⋅ Δ H d V {\displaystyle \Delta G=-{\int _{T}SdT}+{\int _{P}VdP}-{\frac {1}{4\pi }}{\int _{V}{B\cdot \Delta HdV}}}
Paramagnetic systems
In a paramagnetic system, that is, a system in which the magnetization vanishes without the influence of an external magnetic field, assuming some simplifying assumptions (such as the sample system being ellipsoidal), one can derive a few compact thermodynamic relations. Assuming the external magnetic field is uniform and shares a common axis with the paramagnet, the extensive parameter characterizing the magnetic state is I {\displaystyle I} , the magnetic dipole moment of the system. The fundamental thermodynamic relation describing the system will then be of the form U = U ( S , V , I , N ) {\displaystyle U=U(S,V,I,N)} . In the more general case where the paramagnet does not share an axis with the magnetic field, the extensive parameters characterizing the magnetic state will be I x , I y , I z {\displaystyle I_{x},I_{y},I_{z}} . In this case, the fundamental relation describing the system will be U = U ( S , V , I x , I y , I z , N ) {\displaystyle U=U(S,V,I_{x},I_{y},I_{z},N)} . The intensive parameter corresponding to the magnetic moment I {\displaystyle I} is the external magnetic field acting on the paramagnet, B e {\displaystyle B_{e}} . The relation between them is:
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