Thermodynamic stability in relativistic viscous and spin hydrodynamics

In relativistic heavy ion collisions, two nuclei are accelerated to speeds close to that of light, collide with each other, and generate a hot and dense matter known as the quark-gluon plasma (QGP) (BRAHMS:2004adc, ; PHENIX:2004vcz, ; STAR:2005gfr, ; ALICE:2008ngc, ). The evolution of the QGP is well described by relativistic hydrodynamics. Relativistic hydrodynamics serves as a macroscopic effective theory for relativistic many-body systems in the long-wavelength and low-frequency limit. The main equations of relativistic hydrodynamics are the conservation equations for the energy-momentum tensor and other conserved currents in the gradient expansion, e.g. the Israel-Stewart theory (Israel:1979wp, ; Israel:1979, ), the extended Baier-Romatschke-Son-Starinets-Stephanov theory (Baier:2007ix, ), the Denicol-Niemi-Molnar-Rischke theory (Denicol:2012cn, ), and the more recently established Bemfica-Disconzi-Noronha-Kovtun theory (Bemfica:2017wps, ; Kovtun:2019hdm, ; Bemfica:2019knx, ; Hoult:2020eho, ; Bemfica:2020zjp, ). For additional studies and developments, we refer the reader to the recent review papers (Gavassino:2021kpi, ; Rocha:2023ilf, ) and the references therein.

In the early stages of noncentral collisions, the nuclei possess a huge initial orbital angular momentum, on the order of $10^{7}\hbar$. This initial orbital angular momentum is transferred to the spin polarization of quarks and subsequently to the final-state particles through spin-orbital coupling. This mechanism leads to the spin polarization of $\Lambda$ and $\bar{\Lambda}$ hyperons and the spin alignment of vector mesons (Liang:2004ph, ; Liang:2004xn, ; Gao:2007bc, ). The STAR collaboration has observed both the global and local polarization of $\Lambda$ and $\bar{\Lambda}$ hyperons (STAR:2017ckg, ; STAR:2019erd, ), as well as the spin alignment of $\phi$ and $K^{0,*}$ mesons (STAR:2022fan, ).

On the theoretical side, the global polarization can be well described by various phenomenological models (Becattini:2007sr, ; Karpenko:2016jyx, ; Xie:2017upb, ; Li:2017slc, ; Sun:2017xhx, ; Shi:2017wpk, ; Wei:2018zfb, ; Xia:2018tes, ; Vitiuk:2019rfv, ; Fu:2020oxj, ; Ryu:2021lnx, ; Lei:2021mvp, ; Wu:2022mkr, ) through the combination of the modified Cooper-Frye formula (Becattini:2013fla, ; Fang:2016vpj, ) with hydrodynamic simulations under the assumption that the system is close to global equilibrium. To understand local polarization, effects beyond global equilibrium, such as shear-induced polarization (Liu:2021uhn, ; Liu:2021nyg, ; Fu:2021pok, ; Becattini:2021suc, ; Yi:2021ryh, ; Yi:2021unq, ; Yi:2023tgg, ; Wu:2023tku, ), spin Hall effects (Liu:2020dxg, ; Fu:2022myl, ; Wu:2022mkr, ), weak magnetic fields induced polarization (Sun:2024isb, ) and the corrections due to the interactions between quarks and back ground fields (Fang:2023bbw, ), need to be considered. Although hydrodynamic simulations can qualitatively describe local polarization as functions of azimuthal angle, understanding the dependence on centrality and transverse momentum remains challenging (STAR:2019erd, ; ALICE:2021pzu, ; STAR:2023eck, ). Therefore, it is necessary to consider the evolution of spin during collisions. Recently established spin hydrodynamics, which integrates the total angular momentum conservation equation with conventional relativistic hydrodynamic equations, has been developed from various theoretical frameworks, such as from effective action (Montenegro:2017rbu, ; Montenegro:2017lvf, ), entropy principle (Hattori:2019lfp, ; Fukushima:2020ucl, ; Li:2020eon, ; Gallegos:2021bzp, ; She:2021lhe, ; Hongo:2021ona, ; Wang:2021ngp, ; Wang:2021wqq, ; Cao:2022aku, ; Hu:2022azy, ; Biswas:2023qsw, ), kinetic theory (Florkowski:2017ruc, ; Florkowski:2017dyn, ; Florkowski:2018myy, ; Weickgenannt:2019dks, ; Bhadury:2020puc, ; Weickgenannt:2020aaf, ; Shi:2020htn, ; Speranza:2020ilk, ; Bhadury:2020cop, ; Singh:2020rht, ; Peng:2021ago, ; Sheng:2021kfc, ; Hu:2021pwh, ; Weickgenannt:2022zxs, ; Weickgenannt:2022jes, ; Weickgenannt:2022qvh, ; Wagner:2024fhf, ), holography (Gallegos:2020otk, ; Garbiso:2020puw, ), and quantum statistics (Becattini:2023ouz, ). For recent reviews on this topic, see Refs. (Hidaka:2022dmn, ; Shi:2023sxh, ; Becattini:2024uha, ).

As a fundamental requirement, both conventional relativistic hydrodynamics and spin hydrodynamics must exhibit causality and stability. In pioneering works (Hiscock:1985zz, ; Hiscock:1987zz, ), linear mode analysis was implemented to study the causality and stability of hydrodynamic systems. Through linear mode analysis, the causality and stability conditions for various types of hydrodynamics are derived (Hiscock:1985zz, ; Hiscock:1987zz, ; Koide:2006ef, ; Denicol:2008ha, ; Pu:2009fj, ; Brito:2020nou, ; Brito:2021iqr, ; Sarwar:2022yzs, ; Daher:2022wzf, ; Xie:2023gbo, ; Weickgenannt:2023btk, ; Shokri:2023rpp, ; deBrito:2023vzv, ; Fang:2024skm, ; Daher:2024bah, ). These conditions establish inequalities that constrain the range of transport coefficients. Recently, it was found that the conventional causality criterion (Krotscheck1978CausalityC, ) used in linear mode analysis is insufficient to guarantee causality. Consequently, several studies (Heller:2022ejw, ; Gavassino:2023myj, ; Heller:2023jtd, ; Gavassino:2023mad, ; Wang:2023csj, ; Hoult:2023clg, ) have proposed new causality criteria that also explore the deep connection between causality and stability (Bemfica:2020zjp, ; Gavassino:2021owo, ; Wang:2023csj, ).

Very recently, Ref. (Xie:2023gbo, ) has systematically studied the causality and stability for the minimal extended second-order spin hydrodynamics in the linear mode analysis. Later, Ref. (Daher:2024bah, ) also investigates the impact of other second-order terms. It was revealed that the system appears to be unstable at finite wavelengths, even though it satisfies asymptomatic stability conditions derived for both large and small wavelengths (Xie:2023gbo, ). To address this issue, it is essential to explore the stability of spin hydrodynamics through an alternative approach.

In this work, we apply thermodynamic stability analysis (Hiscock:1983zz, ; Olson:1990rzl, ; Gavassino:2021cli, ; Gavassino:2021kjm, ), which is grounded in the second law of thermodynamics and the principle of maximizing total entropy in equilibrium states (Landau:1980mil, ), to spin hydrodynamics. We will derive stability conditions from this thermodynamic stability analysis and compare them with those obtained through linear mode analysis.

The structure of this paper is organized as follows: In Sec. II, we briefly review thermodynamic stability analysis. Next, we apply this analysis to conventional relativistic viscous hydrodynamics as a test case in Sec. III. In Sec. IV, we analyze the thermodynamic stability conditions for spin hydrodynamics and compare the results with those obtained from linear mode analysis. We conclude with a summary in Sec. V.

Throughout this work, we choose the metric $g_{\mu\nu}=\textrm{diag}\{+,-,-,-\}$ and define the projector $\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$ with $u^{\mu}$ being the fluid velocity. For an arbitrary tensor $A^{\mu\nu}$, we introduce the notations $A^{(\mu\nu)}=\frac{1}{2}(A^{\mu\nu}+A^{\nu\mu})$, $A^{[\mu\nu]}=\frac{1}{2}(A^{\mu\nu}-A^{\nu\mu})$, and $A^{<\mu\nu>}\equiv\frac{1}{2}[\Delta^{\mu\alpha}\Delta^{\nu\beta}+\Delta^{\mu\beta}\Delta^{\nu\alpha}]A_{\alpha\beta}-\frac{1}{3}\Delta^{\mu\nu}(\Delta^{\alpha\beta}A_{\alpha\beta})$.

In this section, we briefly review the main idea in Ref. (Gavassino:2021kjm, ). Consider an isolated system near thermodynamic equilibrium, consisting of a fluid connected to a sufficiently large heat-particle bath. According to the second law of thermodynamics, the entropy of the entire system, $S$, must not decrease, i.e., the variation of entropy $\Delta S$ follows:

$$\Delta S=\Delta S_{F}+\Delta S_{B}\geq 0,$$

(1)

where $S_{F,B}$ stand for the entropy for fluid and bath, respectively. Equation (1) is the original condition for the thermodynamic stability.

Now, let us consider conserved quantities $\mathcal{Q}^{a}$ and their thermodynamic conjugates $\alpha^{a}$ in the system, where $a=1,2,...$ label different conserved quantities. For example, if the total number is conserved, then $\mathcal{Q}$ and $\alpha$ correspond to the total number and $\mu/T$, respectively, with $\mu$ and $T$ being the chemical potential and temperature. While, if the total energy is conserved, $\mathcal{Q}$ and $\alpha$ are total energy and $-1/T$, respectively. Then, the variation of entropy can be expressed as

$$dS=-\sum_{a}\alpha^{a}d\mathcal{Q}^{a}.$$

(2)

The $\mathcal{Q}^{a}$ can be divided as the part for fluid $\mathcal{Q}_{F}^{a}$ and the one for the bath $\mathcal{Q}_{B}^{a}$, with the following relationship:

$$d\mathcal{Q}_{B}^{a}=-d\mathcal{Q}_{F}^{a}.$$

(3)

Then the variation of total entropy becomes

$$\Delta S=\Delta S_{F}+\sum_{a}\alpha_{B}^{a}\Delta\mathcal{Q}_{F}^{a}\geq 0.$$

(4)

If defining

$$\Psi\equiv S_{F}+\sum_{a}\alpha_{B}^{a}\mathcal{Q}_{F}^{a},$$

(5)

then Eq. (4) implies that the function $\Psi$ should be maximized in the equilibrium state.

One can also define the information current $E^{\mu}$ as

$$E^{\mu}\equiv-\delta s_{F}^{\mu}-\sum_{a}\alpha_{F}^{a}\delta J_{F}^{a,\mu},$$

(6)

where $s_{F}^{\mu}$ is the entropy current of the fluid and $J_{F}^{a,\mu}$ is the conserved current associated with $\mathcal{Q}_{F}^{a}$, and the symbol $\delta$ denotes the small perturbations from the thermodynamic equilibrium state.

Given that the whole system is near thermodynamic equilibrium and the heat-particle bath is sufficiently large, we can assume that the chemical potential and temperature in the fluid are equal to those in the bath, i.e. $\alpha_{F}^{a}$ for the fluid is approximately equal to $\alpha_{B}^{a}$ in the bath. Under this assumption, $\alpha_{F}^{a}\delta J_{F}^{a,\mu}$ can be simplified to $\delta(\alpha^{a}J_{F}^{a,\mu})$, where we do not distinguish between $\alpha_{F}^{a}$ in the fluid and $\alpha_{B}^{a}$ in the bath. Consequently, Eq. (4) can be further written as

$$E\equiv\int d\Sigma\;E^{\mu}n_{\mu}\geq 0,$$

(7)

holds for an arbitrary spacelike three-dimensional surface $\Sigma$ and its timelike and future-directed normal unit vector $n^{\mu}$. If the thermodynamic equilibrium state is unique, i.e., determined solely by the thermodynamic variables, then from Eq. (7) and the definition of $E^{\mu}$ in Eq. (6), the information current $E^{\mu}$ must satisfy the following conditions:

	$\displaystyle\mathrm{(i)}$		$\displaystyle E^{\mu}n_{\mu}\geq 0\textrm{ for any $n^{\mu}$ with $n_{0}>0,n^{\mu}n_{\mu}=1$},$
	$\displaystyle\mathrm{(ii)}$		$\displaystyle E^{\mu}n_{\mu}=0\textrm{ if and only if all perturbations are zero},$
	$\displaystyle\mathrm{(iii)}$		$\displaystyle\partial_{\mu}E^{\mu}\leq 0.$		(8)

As a remark, the conditions in Eq. (8) can be treated as criteria of thermodynamic stability (Gavassino:2021kjm, ). It has also been found that these criteria in Eq. (8) can guarantee the causality of the system (Gavassino:2021kjm, ). Moreover, when all these conditions are satisfied in one inertial frame of reference, the thermodynamic stability conditions in Eq. (1) are assured across all inertial frames of Refs. (Gavassino:2021owo, ; Wang:2023csj, ). These criteria provide us with a novel tool for analyzing the stability and causality of the system.

In this section, we implement the thermodynamic stability criteria (8) to the relativistic second order viscous hydrodynamics in the gradient expansion. It can be considered as an example to show the connection between the constraints from the thermodynamic stability and conventional linear mode analysis. For convenience, we focus on the quantities for fluid and omit all the lower index $F$ from now on.

The energy momentum conservation equation reads

$$\partial_{\mu}T^{\mu\nu}=0,$$

(9)

and the energy momentum tensor in the Landau or energy frame is given by (landau:1987Fluid, )

$\displaystyle T^{\mu\nu}$

$\displaystyle=$

$\displaystyle(e+P)u^{\mu}u^{\nu}-Pg^{\mu\nu}+\pi^{\mu\nu}-\Pi\Delta^{\mu\nu},$

(10)

where $e,P,\pi^{\mu\nu},\Pi$ are energy density, pressure, shear viscous tensor, and bulk pressure, respectively. Note that the net baryon number density of the QGP produced in relativistic heavy ion collisions is negligible (Yagi:2005yb, ). For simplicity, the following discussions are limited in the cases where (baryon) currents vanish.

In order to compare the constraints from thermodynamic stability and linear mode analysis, we choose the minimal extension of second order viscous hydrodynamics (Koide:2006ef, ). The corresponding entropy current is given by

$$s^{\mu}=su^{\mu}-Q^{\mu}+\mathcal{O}(\partial^{3}),$$

(11)

where $Q^{\mu}$ stands for the possible corrections from the second order. Following Refs. (Israel:1979wp, ; Israel:1979, ), we take

$$Q^{\mu}=\frac{1}{2}\frac{u^{\mu}}{T}(\chi_{\Pi}\Pi^{2}+\chi_{\pi}\pi^{\rho\sigma}\pi_{\rho\sigma}),$$

(12)

as an example. Then the entropy principle $\partial_{\mu}s^{\mu}\geq 0$ gives the constitutive equations for $\pi^{\mu\nu}$ and $\Pi$ as below,

	$\displaystyle\tau_{\Pi}(u\cdot\partial)\Pi+\Pi$	$\displaystyle=$	$\displaystyle-\zeta\left[\partial_{\mu}u^{\mu}+\frac{1}{2}\chi_{\Pi}T\partial_{\rho}(u^{\rho}/T)\Pi\right],$
	$\displaystyle\tau_{\pi}\Delta^{\alpha<\mu}\Delta^{\nu>\beta}(u\cdot\partial)\pi_{\alpha\beta}+\pi^{\mu\nu}$	$\displaystyle=$	$\displaystyle 2\eta\left[\partial^{<\mu}u^{\nu>}-\frac{1}{2}\chi_{\pi}T\partial_{\rho}(u^{\rho}/T)\pi^{\mu\nu}\right],$		(13)

where

$$\zeta,\eta>0,$$

(14)

and

$$\tau_{\Pi}=\zeta\chi_{\Pi},\quad\tau_{\pi}=2\eta\chi_{\pi}.$$

(15)

For the more comprehensive discussions on the second order theories, we refer to Refs. (Israel:1979wp, ; Israel:1979, ). Later, we will compare our results from thermodynamic stability with those from linear mode analysis (Denicol:2008ha, ; Pu:2009fj, ). The terms proportional to $\chi_{\Pi},\chi_{\pi}$ on the right-hand side of Eq. (13) do not appear in the constitutive equations in Refs. (Denicol:2008ha, ; Pu:2009fj, ), but these terms will not contribute to the causality and stability conditions in linear mode analysis.

Next, we choose local rest frame $u^{\mu}=(1,0)$ and assume the fluid reaches the thermodynamic equilibrium state, in which $\Pi$ and $\pi^{\mu\nu}$ are zero. For the macroscopic variables $\varphi=(e,u^{\mu},\Pi,\pi^{\mu\nu})$, we consider the perturbations near the thermodynamic equilibrium, $\delta\varphi$. We can expand the system in the power series of $\delta\varphi$. By using the following relationship,

	$\displaystyle u^{\mu}\delta u_{\mu}$	$\displaystyle=$	$\displaystyle-\frac{1}{2}\delta u^{\mu}\delta u_{\mu},$
	$\displaystyle u_{\mu}\delta\pi^{\mu\nu}$	$\displaystyle=$	$\displaystyle-\delta u_{\mu}\delta\pi^{\mu\nu},$
	$\displaystyle\delta\pi_{\mu}^{\mu}$	$\displaystyle=$	$\displaystyle 0,$		(16)

we find,

	$\displaystyle\delta u^{i},\delta\pi^{ij}$	$\displaystyle\sim$	$\displaystyle\mathcal{O}(\delta),$
	$\displaystyle\delta u^{0},\delta\pi^{0i}$	$\displaystyle\sim$	$\displaystyle\mathcal{O}(\delta^{2}),$
	$\displaystyle\delta\pi^{00}$	$\displaystyle\sim$	$\displaystyle\mathcal{O}(\delta^{3}).$		(17)

With the help of Eq. (11), the information current is then given by (Gavassino:2021cli, ; Gavassino:2021kjm, )

	$\displaystyle E^{\mu}$	$\displaystyle=$	$\displaystyle-\delta s^{\mu}+\frac{u_{\nu}}{T}\delta T^{\mu\nu}$		(18)
		$\displaystyle=$	$\displaystyle u^{\mu}\left(\frac{\delta e}{T}-\delta s\right)+\frac{1}{T}\delta u^{\mu}\delta P+\frac{1}{2}\frac{u^{\mu}}{T}(\chi_{\pi}\delta\pi^{\alpha\beta}\delta\pi_{\alpha\beta}+\chi_{\Pi}\delta\Pi\delta\Pi)$
			$\displaystyle-\frac{1}{2T}(e+P)u^{\mu}\delta u_{\nu}\delta u^{\nu}-\frac{1}{T}\delta u_{\nu}\delta\pi^{\mu\nu}+\frac{1}{T}\delta u^{\mu}\delta\Pi+\mathcal{O}(\delta^{3}).$

By using the thermodynamic relations,

$$ds=\frac{1}{T}de,\;dP=sdT,$$

(19)

we find that

	$\displaystyle\delta s$	$\displaystyle=$	$\displaystyle\frac{1}{T}\delta e+\frac{1}{2}\frac{\partial^{2}s}{\partial e^{2}}(\delta e)^{2}+\mathcal{O}(\delta^{3})$		(20)
		$\displaystyle=$	$\displaystyle\frac{1}{T}\delta e-\frac{1}{2T}\frac{\delta P}{e+P}\delta e+\mathcal{O}(\delta^{3})$
		$\displaystyle=$	$\displaystyle\frac{1}{T}\delta e-\frac{1}{2T}\frac{c_{s}^{2}}{e+P}(\delta e)^{2}+\mathcal{O}(\delta^{3}),$

where $c_{s}^{2}$ is the speed of sound. Then, $E^{\mu}$ can be further simplified,

	$\displaystyle E^{\mu}$	$\displaystyle=$	$\displaystyle\frac{1}{2T}\frac{u^{\mu}c_{s}^{2}}{e+P}(\delta e)^{2}+\frac{c_{s}^{2}}{T}\delta u^{\mu}\delta e-\frac{1}{2T}(e+P)u^{\mu}\delta u_{\nu}\delta u^{\nu}$		(21)
			$\displaystyle-\frac{1}{T}\delta u_{\nu}\delta\pi^{\mu\nu}+\frac{1}{T}\delta u^{\mu}\delta\Pi+\frac{1}{2T}u^{\mu}(\chi_{\pi}\delta\pi^{\alpha\beta}\delta\pi_{\alpha\beta}+\chi_{\Pi}\delta\Pi\delta\Pi).$

Let us now impose the three conditions (8) on $E^{\mu}$. From the definition (6), we have $\partial_{\mu}E^{\mu}=-\partial_{\mu}\delta s^{\mu}$, so that the condition (iii) in Eq. (8) leads to the inequality (14), which is consistent with the requirement from the conventional entropy principle. To analyze the constraints from the conditions (i) and (ii) in Eq. (8), we introduce an arbitrary timelike future-directed vector $n^{\mu}$ with $n_{0}>0,n^{\mu}n_{\mu}=1$. After some tedious and straightforward calculations, we obtain,

	$\displaystyle\frac{2n_{0}TE^{\mu}n_{\mu}}{e+P}$	$\displaystyle=$	$\displaystyle\frac{n_{0}^{2}\tau_{\pi}}{\eta(e+P)}\sum_{i<j}\left[\delta\pi^{ij}-\frac{1}{n_{0}\chi_{\pi}}n_{(i}\delta u_{j)}\right]^{2}$		(22)
			$\displaystyle+\frac{n_{0}^{2}\tau_{\pi}}{\eta(e+P)}\left[\delta\pi^{11}+\frac{1}{2}\delta\pi^{22}+\frac{1}{2n_{0}\chi_{\pi}}(n_{3}\delta u_{3}-n_{1}\delta u_{1})\right]^{2}$
			$\displaystyle+\frac{3n_{0}^{2}\tau_{\pi}}{4\eta(e+P)}\left[\delta\pi^{22}+\frac{1}{3n_{0}\chi_{\pi}}(n_{3}\delta u_{3}+n_{1}\delta u_{1}-2n_{2}\delta u_{2})\right]^{2}$
			$\displaystyle+\sum_{i=1}^{5}a_{i}(\delta A_{i})^{2},$

where the exact expressions for $a_{i}$ and $\delta A_{i}$ can be found in Appendix B. Imposing the conditions (i) and (ii) in Eq. (8) leads to¹¹1In this work, we assume $e+P>0$, while Ref. (Almaalol:2022pjc, ) also explores cases where $e+P<0$. Additionally, we note that the treatment of $\delta\pi^{\mu\nu}$ in Eq. (22) is different with Eq. (C14) of Ref. (Almaalol:2022pjc, ), since the number of independent components of $\delta\pi^{\mu\nu}$ is $5$.

	$\displaystyle c_{s}^{2},\tau_{\pi},\tau_{\Pi}$	$\displaystyle>$	$\displaystyle 0,$
	$\displaystyle 1-c_{s}^{2}-\frac{4\eta}{3\tau_{\pi}(e+P)}-\frac{\zeta}{\tau_{\Pi}(e+P)}$	$\displaystyle>$	$\displaystyle 0,$		(23)

which are exactly the same as those derived from linear mode analysis in the previous literature (Denicol:2008ha, ; Pu:2009fj, ; Pu:2011vr, ).

In general, if the baryon or other conserved current is considered, e.g. $j^{\mu}=nu^{\mu}+\nu^{\mu}$ with $n$ and $\nu^{\mu}$ being number density and diffusive current, the independent fields become $\varphi=(e,u^{\mu},\Pi,\pi^{\mu\nu},n,\nu^{\mu})$ (Gavassino:2023qnw, ). In these cases, the thermodynamic relations (19), constitutive relations (11)-(13), and information current (18) will be modified. More constraints for thermodynamic stability would occur and the final constraints become different with Eq. (23). For the general analysis including baryon currents, one can refer to Refs. (Gavassino:2021cli, ; Gavassino:2021kjm, ; Almaalol:2022pjc, ; Gavassino:2023qnw, ).

In this section, we implement the thermodynamic stability criteria (8) to the spin hydrodynamics. First, let us briefly review the spin hydrodynamics in the canonical form. Besides the energy momentum conservation, we also have the conservation equations for the total angular momentum, i.e.

	$\displaystyle\partial_{\lambda}J^{\lambda\mu\nu}$	$\displaystyle=$	$\displaystyle 0,$
	$\displaystyle\partial_{\mu}\Theta^{\mu\nu}$	$\displaystyle=$	$\displaystyle 0,$		(24)

where $J^{\lambda\mu\nu}$ and $\Theta^{\mu\nu}$ are the total angular momentum tensor and energy momentum tensor in canonical form, respectively. The constitutive equations of $J^{\lambda\mu\nu}$ and $\Theta^{\mu\nu}$ are

	$\displaystyle\Theta^{\mu\nu}$	$\displaystyle=$	$\displaystyle(e+P)u^{\mu}u^{\nu}-Pg^{\mu\nu}+2q^{[\mu}u^{\nu]}+\phi^{\mu\nu}+\pi^{\mu\nu}-\Pi\Delta^{\mu\nu},$
	$\displaystyle J^{\lambda\mu\nu}$	$\displaystyle=$	$\displaystyle x^{\mu}\Theta^{\lambda\nu}-x^{\nu}\Theta^{\lambda\mu}+\Sigma^{\lambda\mu\nu},$		(25)

where $q^{\mu},\phi^{\mu\nu}$ are related to the spin and $\Sigma^{\lambda\mu\nu}$ is the spin tensor. In the following, we will limit our considerations to the cases where (baryon) currents vanish and, therefore, the terms for (baryon) number density do not contribute to constitutive relations and thermodynamic relations.

Inserting Eq. (25) into Eq. (24), yields

$$\partial_{\lambda}\Sigma^{\lambda\mu\nu}=-2\Theta^{[\mu\nu]}.$$

(26)

The spin tensor $\Sigma^{\lambda\mu\nu}$ is usually decomposed as (Hattori:2019lfp, ; Fukushima:2020ucl, )

$$\Sigma^{\lambda\mu\nu}=u^{\lambda}S^{\mu\nu}+\sigma^{\lambda\mu\nu},$$

(27)

where $S^{\mu\nu}$ is named as spin density and $\sigma^{\lambda\mu\nu}$ is perpendicular to the fluid velocity. We follow Ref. (Fukushima:2020ucl, ) to consider the power counting of the spin tensor,

$$S^{\mu\nu}\sim\mathcal{O}(\partial^{0}),\quad\sigma^{\lambda\mu\nu}\sim\mathcal{O}(\partial^{1}).$$

(28)

Analogy to charge chemical potential, one can introduce the spin chemical potential $\omega^{\mu\nu}$, which modifies the thermodynamic relations in the presence of spin density (Hattori:2019lfp, ; Fukushima:2020ucl, ),

	$\displaystyle e+P$	$\displaystyle=$	$\displaystyle Ts+\omega_{\mu\nu}S^{\mu\nu},$
	$\displaystyle de$	$\displaystyle=$	$\displaystyle Tds+\omega_{\mu\nu}dS^{\mu\nu},$
	$\displaystyle dP$	$\displaystyle=$	$\displaystyle sdT+S^{\mu\nu}d\omega_{\mu\nu}.$		(29)

The entropy current in Eq. (11) can also be extended as

$\displaystyle s^{\mu}$

$\displaystyle=$

$\displaystyle su^{\mu}+\frac{1}{T}q^{\mu}-Q^{\mu}.$

(30)

The complete second order terms for the entropy current is complicated, see e.g. Ref. (Biswas:2023qsw, ). For simplicity, we write down the $Q^{\mu}$ analogy to Eq. (12),

$$Q^{\mu}=\frac{1}{2T}u^{\mu}(\chi_{q}q^{\nu}q_{\nu}+\chi_{\phi}\phi^{\alpha\beta}\phi_{\alpha\beta}+\chi_{\Pi}\Pi^{2}+\chi_{\pi}\pi^{\alpha\beta}\pi_{\alpha\beta}).$$

(31)

From the second law of thermodynamics, we can get

	$\displaystyle\tau_{q}\Delta^{\mu\nu}(u\cdot\partial)q_{\nu}+q^{\mu}$	$\displaystyle=$	$\displaystyle\lambda\left[u^{\rho}\partial_{\rho}u^{\mu}-T\Delta^{\mu\nu}\partial_{\nu}\frac{1}{T}-4\omega^{\mu\nu}u_{\nu}+\frac{1}{2}\chi_{q}T\partial_{\rho}\left(\frac{u^{\rho}}{T}\right)q_{\nu}\right],$
	$\displaystyle\tau_{\phi}\Delta^{\mu\alpha}\Delta^{\nu\beta}(u\cdot\partial)\phi_{\alpha\beta}+\phi^{\mu\nu}$	$\displaystyle=$	$\displaystyle 2\gamma_{s}\Delta^{\mu\alpha}\Delta^{\nu\beta}\left[\partial_{[\alpha}u_{\beta]}+2\omega_{\alpha\beta}-\frac{1}{2}\chi_{\phi}T\partial_{\rho}\left(\frac{u^{\rho}}{T}\right)\phi_{\alpha\beta}\right],$		(32)

with the transport coefficients,

$$\tau_{q}=-\lambda\chi_{q},\quad\tau_{\phi}=2\chi_{\phi}\gamma_{s},\quad\lambda,\gamma_{s}>0.$$

(33)

The equation for $\pi^{\mu\nu}$ and $\Pi$ are the same as Eq. (13). We notice that the terms proportional to $\chi_{q,}\chi_{\phi}$ on the right-hand side of Eq. (32) differs with the constitutive equations for $q^{\mu}$ and $\phi^{\mu\nu}$ in the minimal causal extended second order theory in Ref. (Xie:2023gbo, ). However, these new terms proportional to $\chi_{q},\chi_{\phi}$ will not contribute to the causality and stability conditions in linear mode analysis.