- n_L$ and $\Delta e_{E-L} = e_E - e_L$ are, at a given space-time point, $$\begin{aligned}
\Delta n_{E-L} &=& \frac{1}{2} \frac{V_L^\mu V^L_\mu}{n_L} + \mathcal{O}(\delta^3), \label{eq:dn}\\
\Delta e_{E-L} &=& - \frac{W_E^\mu W^E_\mu}{e_E + P_E} + \mathcal{O}(\delta^3), \label{eq:de}\end{aligned}$$ where the correction is of the second order in dissipative quantities. They indicate that the corrections to other thermodynamic variables, *i.e.*, the pressure $P$, the entropy density $s$, the temperature $T$, and the chemical potential $\mu$ are of the second order. The corrections to the transport coefficients should also be of the second order because they are functions of the energy and the conserved charge densities. Hereafter the subscripts $L$ and $E$ are dropped for those variables for simplicity unless otherwise required. The flow difference $\Delta u^\mu_{E-L} = u_E^\mu - u_L^\mu$ is $$\begin{aligned}
\Delta u^\mu_{E-L} = \frac{V_L^\mu }{n} + \mathcal{O}(\delta^2) = - \frac{W_E^\mu}{e+P} + \mathcal{O}(\delta^2), \end{aligned}$$ where the leading order correction is of the first order.

The macroscopic variables are estimated using the conservation laws $\partial_\mu T^{\mu \nu} = 0$ and $\partial_\mu N^\mu = 0$, the equation of state $P=P(e,n_B)$, and the constitutive relations for the dissipative currents. In the Landau frame, the second-order causal expression of the baryon diffusion, based on an extended 