# Rheological behavior of a far-from-equilibrium expanding plasma

###### Abstract

For the Bjorken flow we investigate the hydrodynamization of different modes of the one-particle distribution function by analyzing its relativistic kinetic equations. We calculate the constitutive relations of each mode written as a multi-parameter trans-series encoding the nonperturbative dissipative contributions quantified by the Knudsen and inverse Reynolds numbers. At any given order in the asymptotic expansion of each mode, the transport coefficients get effectively renormalized by summing over all non-perturbative sectors appearing in the trans-series. This gives an effective description of the transport coefficients that provides a new renormalization scheme with an associated renormalization group equation, going beyond the realms of linear response theory. As a result, the renormalized transport coefficients feature a transition to their equilibrium fixed point, which is a neat diagnostics of transient non-Newtonian behavior. As a proof of principle, we verify the predictions of the effective theory with the numerical solutions of their corresponding evolution equations. Our studies strongly suggest that the phenomenological success of fluid dynamics far from local thermal equilibrium is due to the transient rheological behavior of the fluid.

###### pacs:

25.75-q,51.10.+y,52.27.Ny,98.80.EsIntroduction. The regime of validity and applicability of relativistic hydrodynamics is linked with the proximity of the system to a local thermal equilibrium. It has, however, been observed in proton-proton collisions Khachatryan et al. (2016); Aad et al. (2016) that phenomenological models based on fluid dynamics give rise to quite satisfactory results, despite the system size being so small Weller and Romatschke (2017); Werner et al. (2011); Bozek (2011). Furthermore, various theoretical models Kurkela and Zhu (2015); Kurkela and Lu (2014); Kurkela and Zhu (2015); Critelli et al. (2017); Denicol et al. (2014a); Florkowski et al. (2013a, b); Denicol et al. (2014b); Chesler and Yaffe (2010); Heller et al. (2012); Wu and Romatschke (2011); van der Schee (2013); Casalderrey-Solana et al. (2013); Behtash et al. (2018) have shown that hydrodynamics is valid even when large pressure anisotropies are present.

These findings hint at the existence of a new theory for far-from-equilibrium fluid dynamics. Although little is known of this theory, it has been identified that the onset of hydrodynamics depends on the decay of new degrees of freedom, the so-called non-hydrodynamic modes Heller and Spalinski (2015); Heller and Svensson (2018); Romatschke (2017a); Heller et al. (2016); Spaliński (2016), whose nature is purely non-perturbative Heller and Spalinski (2015); Heller and Svensson (2018); Heller et al. (2016); Casalderrey-Solana et al. (2017); Behtash et al. (2018); Denicol and Noronha (2018a); Romatschke (2017b); Strickland et al. (2018); Basar and Dunne (2015); Aniceto and Spaliński (2016); Spaliński (2018); Romatschke (2018); Lublinsky and Shuryak (2007); Denicol and Noronha (2018b). To this end, it is necessary to study how the nonlinear microscopic dynamics between different modes influences its macroscopic response. But a full-fledged study of non-equilibrium physics is not practically possible unless an effective truncation scheme is instead proposed. Thus, the challenge is to identify a tractable truncation scheme which captures the relevant non-equilibrium dynamics while being simple enough to perform concrete calculations. This task requires creating new computational tools and techniques, which is not otherwise achievable via the usual linear response theory.

In this Letter, we develop a new approach to studying hydrodynamization and nonlinear transport processes, which goes beyond linear response regime. Our treatment is highly nonlinear in the Knudsen and inverse Reynolds numbers, and thus, appropriate for the study of far-from-equilibrium dynamics and understanding the non-perturbative information of short-lived non-hydrodynamic modes. In our framework, due to the history deformation of the fluid, the transport coefficients are treated as dynamical quantities evolving with time toward their respective asymptotic equilibrium values obtained by linear response theory. Therefore, knowing the late-time behavior of the fluid, we back-trace the flow of transport coefficients all the way into the UV by including as many non-perturbative contributions as possible, to the transient non-hydrodynamic modes, which naturally build up the formal exponential solutions to the fluid evolution equations. This consequently provides renormalization group (RG) equations governing the RG flows of the dynamical transport coefficients, which by construction, converge to the correct values as the fluid reaches the hydrodynamic equilibrium. This simple situation is analogous to the case of non-Newtonian fluids, where transport coefficients such as shear viscosity, become nonlinear functions of the gradient velocity and thus, their evolution is dictated by the macroscopic deformation history of the fluid Larson and Larson (1999). Thereby, the aforementioned strategy explains the rheological behavior of the expanding fluid.

We consider an expanding conformal plasma undergoing Bjorken flow. The microscopic description of this system is governed by the relativistic Boltzmann equation within the relaxation time approximation (RTA-BE). The problem of solving the RTA-BE is transformed into determining the evolution of the modes through their nonlinear kinetic equations. The solutions of the moments can be written in terms of a multi-parameter trans-series Costin and Costin (2001), which correctly yields the information of the attracting section around the thermal equilibrium fixed point for the gas at late time, otherwise known as the “attractor”. Being able to go beyond the attracting region distinguishes the predictions of linear response theory and the physics of non-equilibrated fluid dynamics.

Kinetic model. The symmetry group of the Bjorken flow becomes explicitly manifest in the Milne coordinates , with longitudinal proper time and space-time rapidity , where the metric is . The four-momentum of particles is decomposed like where the fluid velocity is , and Molnar et al. (2016). In the Milne coordinates the Bjorken flow symmetries reduce the RTA-BE to a simple relaxation type equation Baym (1984)

(1) |

Here , where the transverse momentum is given by and the longitudinal momentum along the -direction is . We consider where is the energy of the particle, and represents the local temperature. For massless particles, the relaxation time scale is , with Denicol et al. (2011), and being the equilibrium value of the shear viscosity over entropy ratio.

The nonlinear relaxation process described by the Boltzmann equation can be understood in terms of the evolution of the moments of the distribution function Grad (1949). Thus, we use the following ansatz for ^{1}^{1}1A more general ansatz of this kind has been used in Lattice Boltzmann methods for relativistic systems Romatschke et al. (2011). Blaizot and Li Blaizot and Yan (2017, 2018) recently studied the thermalization of similar moments within the relaxation time and a more general nonlinear collisional kernel within the small angle approximation Blaizot and Yan (2017).

(2) |

where and denote the generalized Laguerre and Legendre polynomials, respectively. From this expression the moments are directly read as

(3) |

where and . The hydrodynamic equilibrium fixed point is given by . In this Letter, we only study the low energy modes, i.e. , thus we denote by ^{2}^{2}2An extended version of this work including a thorough analysis of the dynamics of the low and high energy modes will be presented in an upcoming paper Behtash et al. (). Furthermore, we shall consider that the initial values of can be obtained from the RS distribution function ansatz Romatschke and Strickland (2003) where measures the momentum anisotropy and is given by Eq. (7a) in Ref. Martinez and Strickland (2010).

For the ansatz (2) the energy-momentum tensor is where the energy density , the transverse () and longitudinal () pressures are given as follows:

(4) |

The Landau matching condition implies . The only independent component of the normalized effective shear tensor is .

The formal trans-series solution. In a scale invariant system governed by (1), an asymptotic series solution can be expanded in terms of the dimensionless parameter Heller and Spalinski (2015). Then, the underlying non-autonomous dynamical system consisting of evolution equations of the moments and temperature has its dimensions (with ) reduced by one. Furthermore, can be interpreted as an ‘energy’ scale for the renormalization scheme arising naturally from summing all the non-perturbative sectors around the (perturbative) asymptotic expansion of the modes . Hence, preparing a dynamical system for the modes automatically amounts to having a renormalization group equation on the phase space of the Bjorken flow built out of these modes. These flows emanate from the vicinity of a UV point discussed below and converge to , alternatively referred to as the IR fixed point due to being reached when (or, equivalently, ).

The evolution equations of the moments obtained from the conservation laws and the RTA-BE (1) (see Appendix) can be written as

(5) |

where , , and with . Note that we can reproduce in (5) by and are the components of . This system is -dimensional, rank 1, and level-1 vector differential equation in a neighborhood of an irregular singularity, . In the analysis of asymptotic solutions of (5), two terms play a crucial role: (i) the coefficient of order term, i.e. the diagonal matrix of Lyapunov exponents determining the rates at which exponential stability is reached at equilibrium in directions of the flow space, and (ii) the coefficient of order , i.e. the (diagonalized) mode-to-mode coupling matrix that shows how each mode is coupled to the neighboring modes. Knowing these two elements paves the way to recasting the dynamical system into the form (5) specifically prepared for asymptotic analysis shown first in generic nonlinear rank-1 systems of ODEs in Ref. Costin (1998). Since is not diagonal as it stands, the prepared form of the dynamical system was written in the basis of that diagonalizes it. Notice that in the RTA there is only one parameter and thus, the Lyapunov exponents are all identically maximal and given by .

In building the matrix it is appropriate to choose the eigenvectors of in such a way that for any . Doing so yields for where is the floor function and for odd . Also important is to state that rescaling the eigenvectors immediately corresponds to changing the expansion parameters in the trans-series ansatz defined below and for the sake of consistency, we will stick to the above condition for the eigenvectors from now on. Finally, let us write down the trans-series ansatz for :

(6) |

Here, where , are the product of expansion parameters , and are non-negative integers. The dot indicates the inner product, . It is better to exploit vector expressions for the nonperturbative sector number aka trans-monomial order , the anomalous dimension of the pseudo-modes , , and the Lyapunov exponents .

By substituting the ansatz (6) in the differential equation (5), one can obtain a recursive relations for the coefficients as

(7) |

where . With a little bit of algebra, Eq. (23) shows that the modes go like , resulting in for any . So physically, survives the longest to contribute to the observables in (4) at late times. The zeroth order term in the first nonperturbative sector () is always a constant normalizing the trans-series expansion parameters . For convenience, we choose to set for . Other trans-series coefficients can be recursively determined order by order in both and in (7).

In addition, at order Eq. (7) gives and . For example, at first two truncation orders the anomalous dimensions are given as , , respectively. It is notable that, physically, all must be real-valued but are in general complex, justifying the name ‘pseudo-modes’. The reality condition for is related to the choice of the initial values (expansion parameters) , which imposes the constraints for , and if is odd. Every flow line that evolves to the attractor of the equilibrium fixed point as , can be constructed from the set of asymptotic solutions to the dynamical system (5) for some . The allowed set of shapes the basin of attraction at any .

Initial data and UV information. Determining the -th expansion parameter is the most challenging task related to Eq. (6). This partly goes back to the fact that there are equations involved in the system of ODEs of (5), whereas the exact numerical expression of (3) is always prepared by only three initial values: and the parameter . Among these constants, the search for -dependence of is more difficult as the other two explicitly enter the trans-series via Écalle time itself. In order to obtain , we use the least squares optimization to fit the trans-series of all the modes involved at order simultaneously on the Stokes line. One, however, needs to be aware of “Stokes phenomenon”, and the UV data that eventually dictate the fate of flow lines.

First, we have to note that capturing the divergence of original asymptotic expansion () in (6) is done by initially Borel transforming the series. This yields some sort of singularity along the physically relevant Stokes line, that is, . Then, the Borel resummation of the original asymptotic series picks up a residue or discontinuity around the Stokes line (given more explicitly by Écalle’s bridge equation Dorigoni (2014)). Imposing the reality condition, one concludes that the overall constant appearing in the imaginary piece, aka “Stokes constant”, has to cancel the imaginary part of the . For , it is shown that this constant is with the overall fitting factor of .

Second, the flow lines heading to the IR should always take values in the basin of attraction, on the boundary of which lies the UV points Behtash et al. (2018). These are actually the RG flows that unveil the general behavior of the modes in kinetic theory and consequent renormalization of transport coefficients far from equilibrium. In the context of RTA Boltzmann theory, the RG time should appropriately be chosen to be with , where the initial relaxation time scale acts as a UV cutoff parameter. Then, for , we get the following autonomous system of equations describing the RG flows in the 3d Bjorken phase space of macroscopic variables:

(8a) | ||||

(8b) | ||||

(8c) |

Unless (while keeping finite), the theory does not have a UV completion since the limit cannot be achieved. At this limit, there are two UV fixed points: , an index-2 saddle point, and which is an index-1 saddle point (Fig. 1). The index represents the number of attracting directions that lie on the - plane while the orthogonal direction is obviously repelling. Considering the full theory , the UV points move into the physical domain and become and , of which the former gives the “free-streaming” limit. The latter explains a recently found solution discussed in Ref. Dash and Jaiswal (2018). At this point, the longitudinal pressure is maximum , which in turn causes a huge amount of pressure along the -direction due to , while there is a slowdown in the transverse direction because . This leads to an early-time escape to the IR fixed point. However, near free-streaming point, there is a local relaxation point appearing in around , an indication of more attracting directions in the UV, as is obvious from the larger index of the saddle point compared to that of the saddle point in Fig. 1. Thus, the initial anisotropy parameter simply measures the proximity of the initial point of flow lines to either one of the UV fixed points. The smaller (or larger) , the closer flow lines are to the saddle points (or ).

Non-perturbative contributions renormalize the transport coefficients. To find the renormalized coefficients, we straightforwardly do the sum over in (7) that leads to the master recursive relation

(9) |

where , , and the initial values of (9) are chosen to be consistent with the asymptotic solutions of (5). Notice that (9) is composed of nonlinear PDEs for every , and thus it is in general difficult to solve it analytically for . Nonetheless, it is exactly solvable for at any order . The exact results for are given by and respectively, where is the Lambert- function. Fig. 2 compares the bare and renormalized for different initial conditions for , and the agreement between the exact numerical and trans-series solutions is overall remarkable.

We may directly extract the renormalized transport coefficients from the near-IR asymptotic expansion of non-hydrodynamic modes found in Blaizot and Yan (2017) all the way into near-UV region by including the exponentially small contributions coming from the nonlinearities introduced in Eq. (5) by mode-to-mode coupling terms. Here, we have to pay attention to two important characteristics of an RG flow in connection with this simple picture: (1) The cutoff parameter for a fixed and nonzero , is merely an indicator of the distance from the critical line connecting the two UV fixed point. This means that in Fig. 1, the flows are initiated well inside the basin of attraction of the IR fixed point, farther away from this line along which a cutoff dependence would appear in the observables. (2) The anomalous dimensions and Lyapunov exponents are both the IR data appearing in the trans-series of each non-hydrodynamic mode. Along an RG flow initiated at some near-UV point when or in the basin of attraction, these two parameters never flow with the RG time. In other words, these data are invariant under an RG transformation of energy-momentum tensor governing the dynamical system (5), which pushes forward the flow lines starting near a fixed point and flowing to another fixed point.

It is, therefore, sought to find an approach to determine the far-from-equilibrium transport coefficients from a rheologic interpretation of the fluid. The deformation history of the fluid is traced in the transport coefficients by considering their nonlinear functional dependence on the velocity gradient tensor, e.g. , which is a situation encountered frequently in the study of non-Newtonian fluid dynamics that gives rise to some interesting physical phenomena known as shear thinning and thickening Larson and Larson (1999).

The second order gradient expansion of the distribution function Blaizot and Yan (2018, 2017); Teaney and Yan (2014); York and Moore (2009) determines the asymptotic form of to be of the form

(10) |

Thus, by comparing this result with the one obtained from the trans-series of (for ), we see that the first non-perturbatively corrected asymptotic term is given by , where . In a similar fashion, one can obtain from . The RG flow equation of these transport coefficients can therefore be derived from the RG flow of moments. Considering a generic observable constituting the resummed coefficients , we can write down its RG flow equation explicitly as

(11) |

where and . By treating as an effectively independent variable from Costin and Costin (2001), can be regarded as the coefficients of a power series in , i.e. . Hence, the piece in (11) is calculated by using the Cauchy integral formula to pick out the th coefficient of :

(12) |

where the beta function is defined through (5) as . Setting and renders the RG flow equation in terms of and , respectively. For different initial values of , Fig. 3 shows the -dependence of and . As , both observables asymptotically approach to their equilibrium values at the IR fixed point. There is, however, a shear thinning effect in the UV regime due to the early-time free streaming expansion, as seen in Fig. 3, until the interactions kick in and the flow initiates its relaxation onward in the shear thickening phase.

Conclusions. We outlined a new method of decomposing the distribution function in terms of non-hydrodynamic modes in a suitable basis, which cast the Boltzmann equation into a dynamical system for a non-equilibrium scale-invariant expanding plasma. We found the UV and IR fixed points of this system and examined its flow lines. It was shown that the exponentially small corrections tracing the nonlinear deformation history of the these flows eventually give a prescription for the renormalization of transport coefficients via trans-asymptotic matching Costin and Costin (2001). The short-lived non-hydrodynamic modes give rise to non-Newtonian characteristics of the renormalized transport coefficients, and as a result, hydrodynamics becomes valid in far-from-equilibrium regimes where and are large. Therefore, we conclude that the success of hydrodynamics in high energy nuclear collisions is intrinsically related with the transient rheological behavior of the fluid.

Acknowledgments. O. Costin has our sincere gratitude for his patience, extensive discussions and kind explanations of some concepts from his seminal paper. We thank G. Dunne, A. Kemper, J. M. Lynch, M. Spalinski, M. Heller, J. Casalderrey-Solana, T. Schäfer and M. Ünsal for useful discussions. We are grateful to T. Schäfer for pointing out the relation between our computations and non-Newtonian fluids. C.N.C.C thanks to C. Pinilla for her continuous support and encouragement. A. B., M. M. and S. K. are all supported in part by the US Department of Energy Grant No. DE-FG02-03ER41260, and M. M. is supported by the BEST (Beam Energy Scan Theory) DOE Topical Collaboration.

Appendix: Evolution equation of the moments . In this appendix we briefly expose the derivation of the evolution equation of the moments in (3). can be conveniently written as

(13) |

where the tensors and are respectively given by

(14a) | ||||

(14b) |

Next, by taking the time derivative of Eq. (13) one obtains

(15) |

where derivative is denoted by an overdot. The first term on the r.h.s. of (15) can be rewritten as

(16) | |||||

where use was made of together with the Boltzmann equation (1). After some algebra, the second term in Eq. (15) reads

(17) |

where the coefficients , and are

(18a) | |||

(18b) | |||

(18c) | |||

(18d) | |||

(18e) | |||

(18f) |

In Eq. (17) we employed the following properties of Laguerre and Legendre polynomials:

(19a) | |||

(19b) | |||

(19c) | |||

(19d) |

After equating Eqs. (16)-(17) with Eq. (15), one obtains the following evolution equation:

(20) |

where and . The energy-momentum conservation law gives the evolution equation of the temperature upon using the matching condition as well as the relations in (4):

(21) |

Let us consider only the case and choose to denote by below. Equations (Rheological behavior of a far-from-equilibrium expanding plasma) and (3) can be rewritten in terms of the variable as

(22) | |||

(23) |

where , , , and

(24) |

with . After solving (23) for , one gets the temperature from (22). Finally, the transformation leads us to Eq. (5).

## References

- Khachatryan et al. (2016) V. Khachatryan et al. (CMS), Phys. Rev. Lett. 116, 172302 (2016), arXiv:1510.03068 [nucl-ex] .
- Aad et al. (2016) G. Aad et al. (ATLAS), Phys. Rev. Lett. 116, 172301 (2016), arXiv:1509.04776 [hep-ex] .
- Weller and Romatschke (2017) R. D. Weller and P. Romatschke, Phys. Lett. B774, 351 (2017), arXiv:1701.07145 [nucl-th] .
- Werner et al. (2011) K. Werner, I. Karpenko, and T. Pierog, Phys. Rev. Lett. 106, 122004 (2011), arXiv:1011.0375 [hep-ph] .
- Bozek (2011) P. Bozek, Eur. Phys. J. C71, 1530 (2011), arXiv:1010.0405 [hep-ph] .
- Kurkela and Zhu (2015) A. Kurkela and Y. Zhu, Phys. Rev. Lett. 115, 182301 (2015), arXiv:1506.06647 [hep-ph] .
- Kurkela and Lu (2014) A. Kurkela and E. Lu, Phys. Rev. Lett. 113, 182301 (2014), arXiv:1405.6318 [hep-ph] .
- Critelli et al. (2017) R. Critelli, R. Rougemont, and J. Noronha, (2017), arXiv:1709.03131 [hep-th] .
- Denicol et al. (2014a) G. S. Denicol, U. W. Heinz, M. Martinez, J. Noronha, and M. Strickland, Phys.Rev.Lett. 113, 202301 (2014a), arXiv:1408.5646 [hep-ph] .
- Florkowski et al. (2013a) W. Florkowski, R. Ryblewski, and M. Strickland, Nucl.Phys. A916, 249 (2013a), arXiv:1304.0665 [nucl-th] .
- Florkowski et al. (2013b) W. Florkowski, R. Ryblewski, and M. Strickland, Phys. Rev. C88, 024903 (2013b), arXiv:1305.7234 [nucl-th] .
- Denicol et al. (2014b) G. S. Denicol, U. W. Heinz, M. Martinez, J. Noronha, and M. Strickland, Phys. Rev. D90, 125026 (2014b), arXiv:1408.7048 [hep-ph] .
- Chesler and Yaffe (2010) P. M. Chesler and L. G. Yaffe, Phys. Rev. D82, 026006 (2010), arXiv:0906.4426 [hep-th] .
- Heller et al. (2012) M. P. Heller, R. A. Janik, and P. Witaszczyk, Phys. Rev. Lett. 108, 201602 (2012), arXiv:1103.3452 [hep-th] .
- Wu and Romatschke (2011) B. Wu and P. Romatschke, Int. J. Mod. Phys. C22, 1317 (2011), arXiv:1108.3715 [hep-th] .
- van der Schee (2013) W. van der Schee, Phys. Rev. D87, 061901 (2013), arXiv:1211.2218 [hep-th] .
- Casalderrey-Solana et al. (2013) J. Casalderrey-Solana, M. P. Heller, D. Mateos, and W. van der Schee, Phys. Rev. Lett. 111, 181601 (2013), arXiv:1305.4919 [hep-th] .
- Behtash et al. (2018) A. Behtash, C. N. Cruz-Camacho, and M. Martinez, Phys. Rev. D97, 044041 (2018), arXiv:1711.01745 [hep-th] .
- Heller and Spalinski (2015) M. P. Heller and M. Spalinski, Phys. Rev. Lett. 115, 072501 (2015), arXiv:1503.07514 [hep-th] .
- Heller and Svensson (2018) M. P. Heller and V. Svensson, (2018), arXiv:1802.08225 [nucl-th] .
- Romatschke (2017a) P. Romatschke, Eur. Phys. J. C77, 21 (2017a), arXiv:1609.02820 [nucl-th] .
- Heller et al. (2016) M. P. Heller, A. Kurkela, and M. Spalinski, (2016), arXiv:1609.04803 [nucl-th] .
- Spaliński (2016) M. Spaliński, Phys. Rev. D94, 085002 (2016), arXiv:1607.06381 [nucl-th] .
- Casalderrey-Solana et al. (2017) J. Casalderrey-Solana, N. I. Gushterov, and B. Meiring, (2017), arXiv:1712.02772 [hep-th] .
- Denicol and Noronha (2018a) G. S. Denicol and J. Noronha, Phys. Rev. D97, 056021 (2018a), arXiv:1711.01657 [nucl-th] .
- Romatschke (2017b) P. Romatschke, (2017b), arXiv:1710.03234 [hep-th] .
- Strickland et al. (2018) M. Strickland, J. Noronha, and G. Denicol, Phys. Rev. D97, 036020 (2018), arXiv:1709.06644 [nucl-th] .
- Basar and Dunne (2015) G. Basar and G. V. Dunne, Phys. Rev. D92, 125011 (2015), arXiv:1509.05046 [hep-th] .
- Aniceto and Spaliński (2016) I. Aniceto and M. Spaliński, Phys. Rev. D93, 085008 (2016), arXiv:1511.06358 [hep-th] .
- Spaliński (2018) M. Spaliński, Phys. Lett. B776, 468 (2018), arXiv:1708.01921 [hep-th] .
- Romatschke (2018) P. Romatschke, Phys. Rev. Lett. 120, 012301 (2018), arXiv:1704.08699 [hep-th] .
- Lublinsky and Shuryak (2007) M. Lublinsky and E. Shuryak, Phys. Rev. C76, 021901 (2007), arXiv:0704.1647 [hep-ph] .
- Denicol and Noronha (2018b) G. S. Denicol and J. Noronha, (2018b), arXiv:1804.04771 [nucl-th] .
- Larson and Larson (1999) R. Larson and R. Larson, The Structure and Rheology of Complex Fluids, Topics in Chemical Engineering (OUP USA, 1999).
- Costin and Costin (2001) O. Costin and R. Costin, Inventiones mathematicae 145, 425 (2001).
- Molnar et al. (2016) E. Molnar, H. Niemi, and D. H. Rischke, Phys. Rev. D93, 114025 (2016), arXiv:1602.00573 [nucl-th] .
- Baym (1984) G. Baym, Phys. Lett. B138, 18 (1984).
- Denicol et al. (2011) G. S. Denicol, J. Noronha, H. Niemi, and D. H. Rischke, Phys. Rev. D83, 074019 (2011), arXiv:1102.4780 [hep-th] .
- Grad (1949) H. Grad, Commun.Pure Appl.Math. 2, 331 (1949).
- (40) A more general ansatz of this kind has been used in Lattice Boltzmann methods for relativistic systems Romatschke et al. (2011). Blaizot and Li Blaizot and Yan (2017, 2018) recently studied the thermalization of similar moments within the relaxation time and a more general nonlinear collisional kernel within the small angle approximation Blaizot and Yan (2017).
- (41) An extended version of this work including a thorough analysis of the dynamics of the low and high energy modes will be presented in an upcoming paper Behtash et al. ().
- Romatschke and Strickland (2003) P. Romatschke and M. Strickland, Phys. Rev. D68, 036004 (2003), hep-ph/0304092 .
- Martinez and Strickland (2010) M. Martinez and M. Strickland, Nucl. Phys. A848, 183 (2010), arXiv:1007.0889 [nucl-th] .
- Costin (1998) O. Costin, Duke Math. J. 93, 289 (1998).
- Dorigoni (2014) D. Dorigoni, (2014), arXiv:1411.3585 [hep-th] .
- Dash and Jaiswal (2018) A. Dash and A. Jaiswal, Phys. Rev. D97, 104005 (2018), arXiv:1711.07130 [gr-qc] .
- Blaizot and Yan (2017) J.-P. Blaizot and L. Yan, JHEP 11, 161 (2017), arXiv:1703.10694 [nucl-th] .
- Blaizot and Yan (2018) J.-P. Blaizot and L. Yan, Phys. Lett. B780, 283 (2018), arXiv:1712.03856 [nucl-th] .
- Teaney and Yan (2014) D. Teaney and L. Yan, Phys. Rev. C89, 014901 (2014), arXiv:1304.3753 [nucl-th] .
- York and Moore (2009) M. A. York and G. D. Moore, Phys.Rev. D79, 054011 (2009), arXiv:0811.0729 [hep-ph] .
- Romatschke et al. (2011) P. Romatschke, M. Mendoza, and S. Succi, Phys. Rev. C84, 034903 (2011), arXiv:1106.1093 [nucl-th] .
- (52) A. Behtash, C. N. Cruz-Camacho, S. Kamata, and M. Martinez, Work in progress.