Relativistic distribution function for particles with spin at local thermodynamical equilibrium

•Single-particle distribution function in local thermodynamical equilibrium with spin.•Polarization of spin 1/2 particles in a fluid at local thermodynamical equilibrium.•Prediction of a new effect: a steady gradient of temperature induces a polarization.•Application to the calculation of polarization in relativistic heavy ion collisions.

We present an extension of relativistic single-particle distribution function for weakly interacting particles at local thermodynamical equilibrium including spin degrees of freedom, for massive spin 1/2 particles. We infer, on the basis of the global equilibrium case, that at local thermodynamical equilibrium particles acquire a net polarization proportional to the vorticity of the inverse temperature four-vector field. The obtained formula for polarization also implies that a steady gradient of temperature entails a polarization orthogonal to particle momentum. The single-particle distribution function in momentum space extends the so-called Cooper–Frye formula to particles with spin 1/2 and allows us to predict their polarization in relativistic heavy ion collisions at the freeze-out.