# From Phase Space to Non-Equilibrium Dynamics: Exploring [PERSON]'s Theorem and Its Implications
## 4 Adaptability of the Formulation
### The Mechanics of Magnetic Helicity in Plasma  
The helicity associated with ions and electrons in plasma has opposite signs. This is because helicity is a measure of the handedness of the magnetic field, and the ions and electrons have opposite charges and therefore move in opposite directions in a magnetic field. As a result, the magnetic fields generated by the two populations will have opposite handedness. In the expression for the free energy that includes both ions and electrons, we need to take into account the opposite signs of the helicity densities. A more general expression for the free energy that accounts for this is:  
\[F=\frac{B^{2}}{2\mu_{0}}+(\alpha_{i}-\alpha_{e})K, \tag{75}\]  
where \(\alpha_{i}\) and \(\alpha_{e}\) are the helicity densities associated with the ion and electron populations, respectively, and \(K\) is the kinetic energy density. The term \((\alpha_{i}-\alpha_{e})\) accounts for the opposite signs of ion and electron helicities. Taking the gradient of \(F\) with respect to position, we obtain:  
\[\mathbf{\
abla}F=\frac{1}{2\mu_{0}}\mathbf{\
abla}\Big{(}B^{2}\Big{)}+(\alpha_{i}- \alpha_{e})\mathbf{\
abla}(K). \tag{76}\]  
Using the same identity as before, \(\mathbf{\
abla}\big{(}B^{2}\big{)}=4\alpha\mathbf{B}\), where \(\alpha\) is the total helicity density, including contributions from both ions and electrons, we can write:  
\[\mathbf{\
abla}F=\left(\frac{4\alpha}{\mu_{0}}\right)B^{2}+(\alpha_{i}-\alpha_{e} )\mathbf{\
abla}(K). \tag{77}\]  
The total helicity density in plasma physics is a measure of the twistedness or knotting of magnetic field lines, and it is a conserved quantity in ideal magnetohydrodynamics (MHD). The gradient of free energy is related to the total helicity density, but with an additional term that accounts for the difference between ion and electron helicities. This term reflects how the dynamics of ion and electron populations can impact the overall plasma behavior. Mathematically, the total helicity density is defined as the volume integral of the dot product between the magnetic field \(\mathbf{B}\) and its vector potential \(\mathbf{A}\), i.e., \(\alpha=\int_{V}(\mathbf{A}\cdot\mathbf{B})dV\), where \(V\) is the volume of the plasma. In general, the total helicity density can be both positive and negative depending on the orientation and topology of the magnetic field lines. In plasma, the total helicity density is related to the free energy of the system and plays a crucial role in determining the stability and dynamics of the plasma. The gradient of the total helicity density is related to the Lorentz force that acts on the plasma, and it can drive various instabilities and reconnection events in the magnetic field. Therefore, the total helicity density is an important quantity in plasma physics and is often used in theoretical and experimental studies of plasmas. In a plasma, both ions and electrons can contribute to the helicity of the magnetic field. The ion and electron helicities are defined as the volume integrals of the dot products between the magnetic field and the velocity of the respective species, i.e., \(\alpha_{i}=\int_{V}(\mathbf{v}_{i}\cdot\mathbf{B})dV\) and \(\alpha_{e}=\int_{V}(\mathbf{v}_{e}\cdot\mathbf{B})dV\), where \(\mathbf{v}_{i}\) and \(\mathbf{v}_{e}\) are the velocities of the ions and electrons, respectively. The total helicity density is the sum of the ion and electron helicities, that is, \(\alpha=\alpha_{i}+\alpha_{e}\). As shown in Figure 6, the complexity of magnetic field lines increases with their writhe (a measure of the total amount of coiling or twisting in a knot) and twist (a measure of the local twisting or rotation of a knot). Using \"arbitrary units\" (a.i.) for both axes, Figure 6 provides a conceptual representation of the link between helicity and free energy in plasma physics.