Perfect spin hydrodynamics at all orders in spin polarization
Zbigniew Drogosz
TL;DR
This work addresses whether classical spin hydrodynamics and quantum spin hydrodynamics yield equivalent conserved currents for spin-$1/2$ particles. By comparing the classical generating function $n_{\rm cl}$ with its quantum counterpart $n_{\rm qt}$ and expanding in the spin polarization tensor $\omega_{\mu\nu}$, the authors show that the current tensors $N^\mu$, $T^{\mu\nu}$, and $S^{\lambda,\mu\nu}$ share identical structures at each order, differing only by a multiplicative factor. The key result is that the $2n$-th order terms differ by a factor $3^n/(2n+1)$ when the classical spin-length satisfies $\mathfrak{s}^2=3/4$, with exact agreement at the second order and a progression toward a different infinite-order limit; the analysis extends to Fermi–Dirac statistics without altering this ratio. Practically, this clarifies the domain of validity for classical spin hydrodynamics and provides a quantitative bridge to the quantum description, informing future 3D numerical simulations of spin-polarized fluids.
Abstract
We compare two recently developed frameworks of perfect spin hydrodynamics for spin-$1/2$ particles, based respectively on classical kinetic theory and the Wigner function. We show that the conserved currents in both approaches have the same form at each order of the expansion in the components of the spin polarization tensor $ω$. The only difference is a relative multiplicative factor, which is equal to 1 at the lowest nontrivial order and increases monotonically with the expansion order.