# 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
\[\frac{d\rho}{dt}=\int(\mathbf{\
abla}\times\mathbf{A})\cdot\bigg{(}\mathbf{\
abla} \times\frac{d\mathbf{A}}{dt}\bigg{)}dV \tag{78}\]  
Using the equation of motion for the plasma, which is given by:  
\[\rho\mathbf{v}=[\mathbf{j}\times\mathbf{B}]-\epsilon_{0}\frac{d\mathbf{E}}{ dt}, \tag{79}\]  
where \(v\) is the plasma velocity, \(j\) is the current density, \(E\) is the electric field, and \(\epsilon_{0}\) is the electric permittivity of free space, we can rewrite the time derivative of the helicity density as:  
\[\frac{d\rho}{dt}=-2\int(\mathbf{j}\cdot\mathbf{B})dV. \tag{80}\]  
This equation shows that the time rate of change of the helicity density is proportional to the current density and the magnetic field. Thus, if we increase the twist in the magnetic field, we will also increase the current density, which will in turn increase the rate of change of the helicity density and hence the free energy of the plasma. Similarly, if we perturb the magnetic field by adding a small amount of writhe to it, the helicity density will again increase, and this will lead to an increase in the free energy of the plasma. This can be seen by considering the writhe of the magnetic field lines, which is given by:  
\[Wr=\int\bigg{[}\mathbf{B}\cdot\mathbf{\
abla}\times\bigg{(}\frac{\mathbf{B}}{B^{2 }}\bigg{)}\bigg{]}dV, \tag{81}\]  
where the integral is taken over the volume of the plasma. This equation quantifies the degree of linking of the magnetic field lines, and it is related to the helicity density through the equation \(\rho=2 Wr\); the greater the degree of linking between magnetic field lines, the higher the helicity density in the system. Thus, we can see that changes in the free energy of the plasma can lead to changes in the twist and writhe of the magnetic-field lines and that the helicity density provides a fundamental link between these quantities. This detailed investigation of the mechanics of magnetic helicity in plasma not only emphasizes the intricacy of plasma behavior but also emphasizes the importance of magnetic helicity in more general physical contexts, including plasma fusion and astrophysical events. According to studies like those of [PERSON] (1987) [PERSON] (1987) and [PERSON] et al. (2014) [PERSON] et al. (2014), optimising confinement and stability strategies in tokamak and stellarator configurations in plasma fusion research requires an understanding of the specifics of magnetic helicity. Furthermore, the role of magnetic helicity in astrophysical settings--ranging from solar flares to galactic dynamics--has been extensively documented, showcasing its fundamental contribution to magnetic reconnection processes and the evolution of cosmic magnetic fields ([PERSON], 1984 [12]; Blackman & Field, 2000 [13]). These diverse applications not only demonstrate the universal relevance of magnetic helicity considerations but also emphasize the necessity of a robust theoretical framework for analyzing out-of-equilibrium systems, offering profound insights into the intricate dance of energy and entropy in the cosmos.