Israel-Stewart framework [@Israel:1976tn; @Israel:1979wp; @Monnai:2010qp], reads $$\begin{aligned}
V_L^\mu &=& \kappa_V \nabla_L^\mu \frac{\mu}{T} - \tau_V (\Delta_{L})^{\mu}_{\ \nu} D_L V_L^\nu \nonumber \\
&+& \chi_V^a V_L^\mu D_L \frac{\mu}{T} + \chi_V^b V_L^\mu D_L \frac{1}{T} + \chi_V^c V_L^\mu \nabla^L_\nu u_L^\nu \nonumber \\
&+& \chi_V^d V_L^\nu \nabla^L_\nu u_L^\mu + \chi_V^e V_L^\nu \nabla_L^\mu u^L_\mu, 
\label{eq:diffusion}\end{aligned}$$ where $\kappa_V \geq 0$ is the baryon conductivity, $\tau_V \geq 0$ is the relaxation time for the baryon diffusion, and $\chi_V^{a,b,c,d,e}$ are second-order transport coefficients. $D = u^\mu \partial_\mu$ and $\nabla^\mu = \Delta^{\mu \nu} \partial_\nu$ are the time- and the space-like derivatives, respectively. Similarly, in the Eckart frame, the energy dissipation reads $$\begin{aligned}
W_E^\mu &=& - \kappa_W \bigg( \nabla_E^\mu \frac{1}{T} + \frac{1}{T} D_E u_E^\mu \bigg) - \tau_W (\Delta_{E})^{\mu}_{\ \nu} D_E W_E^\nu \nonumber \\
&+& \chi_W^a W_E^\mu D_E \frac{\mu}{T} + \chi_W^b W_E^\mu D_E \frac{1}{T} + \chi_W^c W_E^\mu \nabla^E_\nu u_E^\nu  \nonumber \\
&+& \chi_W^d W_E^\nu \nabla^E_\nu u_E^\mu + \chi_W^e W_E^\nu \nabla_E^\mu u^E_\nu, 
\label{eq:dissipation}\end{aligned}$$ where $\kappa_W \geq 0$ is the energy conductivity and $\tau_W \geq 0$ is the relaxation time for the energy dissipation, and $\chi_W^{a,b,c,d,e}$ are second-order transport coefficients. For the full expression of the second-order hydrodynamic equations including the scalar and the tensor dissipative currents, see for example Ref. [@Monnai:2010qp].

The second law 