# Regular conditional probability

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In plasma physics, a Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity.[1]

## Derivation

Consider a closed, simply-connected, flux-conserving, perfectly conducting surface ${\displaystyle S}$ surrounding a plasma with negligible thermal energy (${\displaystyle \beta \rightarrow 0}$).

As discussed above, the plasma would relax towards a minimum energy state while conserving its magnetic helicity. Since the boundary is perfectly conducting, there cannot be any change in the associated flux. This implies ${\displaystyle \delta {\vec {B}}\cdot {\vec {ds}}=0}$ and ${\displaystyle \delta {\vec {A}}_{||}=0}$ on ${\displaystyle S}$.

We formulate a variational problem of minimizing the plasma energy ${\displaystyle W=\int d^{3}rB^{2}/2\mu _{\circ }}$ while conserving magnetic helicity ${\displaystyle K=\int d^{3}r{\vec {A}}\cdot {\vec {B}}}$.

The variational problem is ${\displaystyle \delta W-\lambda \delta K=0}$.

After some algebra this leads to the following constraint for the minimum energy state ${\displaystyle \nabla \times {\vec {B}}=\lambda {\vec {B}}}$.

## References

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