Relativistic lattice Boltzmann methods: theory and applications. (English) Zbl 1479.76075
Summary: We present a systematic account of recent developments of the relativistic Lattice Boltzmann method (RLBM) for dissipative hydrodynamics. We describe in full detail a unified, compact and dimension-independent procedure to design relativistic LB schemes capable of bridging the gap between the ultra-relativistic regime, \(k_{\mathrm{B}} T \gg m c^2\), and the non-relativistic one, \(k_{\mathrm{B}} T \ll m c^2\). We further develop a systematic derivation of the transport coefficients as a function of the kinetic relaxation time in \(d =1, 2, 3\) spatial dimensions. The latter step allows to establish a quantitative bridge between the parameters of the kinetic model and the macroscopic transport coefficients. This leads to accurate calibrations of simulation parameters and is also relevant at the theoretical level, as it provides neat numerical evidence of the correctness of the Chapman-Enskog procedure. We present an extended set of validation tests, in which simulation results based on the RLBMs are compared with existing analytic or semi-analytic results in the mildly-relativistic (\(k_{\mathrm{B}} T \sim m c^2\)) regime for the case of shock propagations in quark-gluon plasmas and laminar electronic flows in ultra-clean graphene samples. It is hoped and expected that the material collected in this paper may allow the interested readers to reproduce the present results and generate new applications of the RLBM scheme.
MSC:
| 76M28 | Particle methods and lattice-gas methods |
| 76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 35Q20 | Boltzmann equations |
Keywords:
transport coefficient; kinetic relaxation time; ultra-relativistic regime; Chapman-Enskog procedure; shock propagationReferences:
| [1] | Cattaneo, C., Sulla conduzione del Calore, Atti Semin. Mat. Fis. Univ. Modena, 3, 83-101 (1948) · Zbl 0035.26203 |
| [2] | Lichnerowicz, A., Relativistic hydrodynamics and magnetohydrodynamics: Lectures on the existence of solutions, (Mathematical Physics Monograph Series (1967), Benjamin), URL https://books.google.it/books?id=oqZAAAAAIAAJ · Zbl 0193.55401 |
| [3] | Eckart, C., The thermodynamics of irreversible processes. III. Relativistic theory of the simple fluid, Phys. Rev., 58, 919-924 (1940) · JFM 66.1077.02 |
| [4] | Muller, I., Zum paradoxon der warmeleitungstheorie, Z. Phys., 198, 329-344 (1967) · Zbl 0143.23602 |
| [5] | Landau, L.; Lifshitz, E., Fluid Mechanics (1987), Elsevier Science, URL https://books.google.it/books?id=eVKbCgAAQBAJ · Zbl 0655.76001 |
| [6] | Israel, W., Nonstationary irreversible thermodynamics: A causal relativistic theory, Ann. Physics, 100, 310-331 (1976) |
| [7] | Israel, W.; Stewart, J. M., On transient relativistic thermodynamics and kinetic theory. II, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365, 1720, 43-52 (1979) |
| [8] | De Groot, S. R., (Van Leeuwen, W. A.; Van Weert, C. G., Relativistic Kinetic Theory. Principles and Applications (1980), North-Holland: North-Holland Amsterdam, Netherlands) |
| [9] | Rezzolla, L.; Zanotti, O., Relativistic Hydrodynamics (2013), Oxford University Press, URL https://books.google.de/books?id=aS1oAgAAQBAJ · Zbl 1297.76002 |
| [10] | Romatschke, P.; Romatschke, U., Relativistic Fluid Dynamics In and Out of Equilibrium: And Applications to Relativistic Nuclear Collisions (2019), Cambridge University Press · Zbl 07056608 |
| [11] | Florkowski, W.; Heller, M. P.; Spaliński, M., New theories of relativistic hydrodynamics in the LHC era, Rep. Progr. Phys., 81, 4, 046001 (2018) |
| [12] | Aidala, C., Creation of quark-gluon plasma droplets with three distinct geometries, Nat. Phys., 15, 214-220 (2019) |
| [13] | Lucas, A.; Fong, K. C., Hydrodynamics of electrons in graphene, J. Phys.: Condens. Matter, 30, 053001 (2018) |
| [14] | Maldacena, J., The large-n limit of superconformal field theories and supergravity, Internat. J. Theoret. Phys., 38, 4, 1113-1133 (1999) · Zbl 0969.81047 |
| [15] | Nagle, J. L.; Zajc, W. A., Small system collectivity in relativistic hadronic and nuclear collisions, Annu. Rev. Nucl. Part. Sci., 68, 1, 211-235 (2018) |
| [16] | Succi, S., Lattice Boltzmann 2038, Europhys. Lett., 109, 50001 (2015) |
| [17] | Succi, S., The Lattice Boltzmann Equation: For Complex States of Flowing Matter (2018), OUP Oxford · Zbl 1485.76003 |
| [18] | Mendoza, M.; Boghosian, B. M.; Herrmann, H. J.; Succi, S., Fast lattice Boltzmann solver for relativistic hydrodynamics, Phys. Rev. Lett., 105, 014502 (2010) |
| [19] | Denicol, G. S.; Niemi, H.; Molnár, E.; Rischke, D. H., Derivation of transient relativistic fluid dynamics from the Boltzmann equation, Phys. Rev. D, 85, 114047 (2012) |
| [20] | Huovinen, P.; Molnar, D., Applicability of causal dissipative hydrodynamics to relativistic heavy ion collisions, Phys. Rev. C, 79, 014906 (2009) |
| [21] | Bouras, I.; Molnár, E.; Niemi, H.; Xu, Z.; El, A.; Fochler, O.; Greiner, C.; Rischke, D. H., Investigation of shock waves in the relativistic Riemann problem: A comparison of viscous fluid dynamics to kinetic theory, Phys. Rev. C, 82, 024910 (2010) |
| [22] | Florkowski, W.; Ryblewski, R.; Strickland, M., Testing viscous and anisotropic hydrodynamics in an exactly solvable case, Phys. Rev. C, 88, 024903 (2013) |
| [23] | Muronga, A., Relativistic dynamics of nonideal fluids: Viscous and heat-conducting fluids. I. General aspects and \(3 + 1\) formulation for nuclear collisions, Phys. Rev. C, 76, 014909 (2007) |
| [24] | Muronga, A., Relativistic dynamics of non-ideal fluids: Viscous and heat-conducting fluids. II. Transport properties and microscopic description of relativistic nuclear matter, Phys. Rev. C, 76, 014910 (2007) |
| [25] | Betz, B.; Henkel, D.; Rischke, D. H., Complete second-order dissipative fluid dynamics, J. Phys. G: Nucl. Part. Phys., 36, 6, 064029 (2009) |
| [26] | El, A.; Xu, Z.; Greiner, C., Extension of relativistic dissipative hydrodynamics to third order, Phys. Rev. C, 81, 041901 (2010) |
| [27] | Denicol, G. S.; Koide, T.; Rischke, D. H., Dissipative relativistic fluid dynamics: A new way to derive the equations of motion from kinetic theory, Phys. Rev. Lett., 105, 162501 (2010) |
| [28] | Betz, B.; Denicol, G. S.; Koide, T.; Molnár, E.; Niemi, H.; Rischke, D., Second order dissipative fluid dynamics from kinetic theory, EPJ Web Conf., 13, 07005 (2011) |
| [29] | Jaiswal, A.; Bhalerao, R. S.; Pal, S., Complete relativistic second-order dissipative hydrodynamics from the entropy principle, Phys. Rev. C, 87, 021901 (2013) |
| [30] | Jaiswal, A., Relativistic dissipative hydrodynamics from kinetic theory with relaxation-time approximation, Phys. Rev. C, 87, 051901 (2013) |
| [31] | Jaiswal, A., Relativistic third-order dissipative fluid dynamics from kinetic theory, Phys. Rev. C, 88, 021903 (2013) |
| [32] | Bhalerao, R. S.; Jaiswal, A.; Pal, S.; Sreekanth, V., Particle production in relativistic heavy-ion collisions: A consistent hydrodynamic approach, Phys. Rev. C, 88, 044911 (2013) |
| [33] | Bhalerao, R. S.; Jaiswal, A.; Pal, S.; Sreekanth, V., Relativistic viscous hydrodynamics for heavy-ion collisions: A comparison between the Chapman-Enskog and Grad methods, Phys. Rev. C, 89, 054903 (2014) |
| [34] | Chattopadhyay, C.; Jaiswal, A.; Pal, S.; Ryblewski, R., Relativistic third-order viscous corrections to the entropy four-current from kinetic theory, Phys. Rev. C, 91, 024917 (2015) |
| [35] | Mendoza, M.; Boghosian, B. M.; Herrmann, H. J.; Succi, S., Derivation of the lattice Boltzmann model for relativistic hydrodynamics, Phys. Rev. D, 82, 105008 (2010) |
| [36] | Romatschke, P.; Mendoza, M.; Succi, S., Fully relativistic lattice Boltzmann algorithm, Phys. Rev. C, 84, 034903 (2011) |
| [37] | Romatschke, P., Relativistic (lattice) Boltzmann equation with nonideal equation of state, Phys. Rev. D, 85, 065012 (2012) |
| [38] | Li, Q.; Luo, K. H.; Li, X. J., Lattice Boltzmann method for relativistic hydrodynamics: Issues on conservation law of particle number and discontinuities, Phys. Rev. D, 86, 085044 (2012) |
| [39] | Schwarz, D., The first second of the universe, Ann. Phys., 12, 4, 220-270 (2003) · Zbl 1026.83501 |
| [40] | Mohseni, F.; Mendoza, M.; Succi, S.; Herrmann, H. J., Lattice Boltzmann model for ultrarelativistic flows, Phys. Rev. D, 87, 083003 (2013) |
| [41] | Mendoza, M.; Karlin, I.; Succi, S.; Herrmann, H. J., Relativistic lattice Boltzmann model with improved dissipation, Phys. Rev. D, 87, 065027 (2013) |
| [42] | Gabbana, A.; Mendoza, M.; Succi, S.; Tripiccione, R., Towards a unified lattice kinetic scheme for relativistic hydrodynamics, Phys. Rev. E, 95, 053304 (2017) |
| [43] | Ambruş, V. E.; Blaga, R., High-order quadrature-based lattice Boltzmann models for the flow of ultrarelativistic rarefied gases, Phys. Rev. C, 98, 035201 (2018) |
| [44] | Hupp, D.; Mendoza, M.; Bouras, I.; Succi, S.; Herrmann, H. J., Relativistic lattice Boltzmann method for quark-gluon plasma simulations, Phys. Rev. D, 84, 125015 (2011) |
| [45] | Gabbana, A.; Mendoza, M.; Succi, S.; Tripiccione, R., Kinetic approach to relativistic dissipation, Phys. Rev. E, 96, 023305 (2017) |
| [46] | Ambruş, V. E.; Guga-Roşian, C., Lattice Boltzmann study of the one-dimensional boost-invariant expansion with anisotropic initial conditions, AIP Conf. Proc., 2071, 1, 020014 (2019) |
| [47] | Coelho, R. C.; Mendoza, M.; Doria, M. M.; Herrmann, H. J., Fully dissipative relativistic lattice Boltzmann method in two dimensions, Comput. & Fluids, 172, 318-331 (2018) · Zbl 1410.76346 |
| [48] | Ambruş, V. E., Transport coefficients in ultrarelativistic kinetic theory, Phys. Rev. C, 97, 024914 (2018) |
| [49] | Gabbana, A.; Simeoni, D.; Succi, S.; Tripiccione, R., Relativistic dissipation obeys Chapman-Enskog asymptotics: Analytical and numerical evidence as a basis for accurate kinetic simulations, Phys. Rev. E, 99, 052126 (2019) |
| [50] | Oettinger, D.; Mendoza, M.; Herrmann, H. J., GaussIan quadrature and lattice discretization of the Fermi-Dirac distribution for graphene, Phys. Rev. E, 88, 013302 (2013) |
| [51] | Furtmaier, O.; Mendoza, M.; Karlin, I.; Succi, S.; Herrmann, H. J., Rayleigh-Bénard instability in graphene, Phys. Rev. B, 91, 085401 (2015) |
| [52] | Coelho, R. C.V.; Mendoza, M.; Doria, M. M.; Herrmann, H. J., Kelvin-Helmholtz instability of the Dirac fluid of charge carriers on graphene, Phys. Rev. B, 96, 184307 (2017) |
| [53] | Gabbana, A.; Mendoza, M.; Succi, S.; Tripiccione, R., Numerical evidence of electron hydrodynamic whirlpools in graphene samples, Comput. & Fluids, 172, 644-650 (2018) · Zbl 1410.76478 |
| [54] | Mendoza, M.; Herrmann, H. J.; Succi, S., Preturbulent regimes in graphene flow, Phys. Rev. Lett., 106, 156601 (2011) |
| [55] | Gabbana, A.; Polini, M.; Succi, S.; Tripiccione, R.; Pellegrino, F. M.D., Prospects for the detection of electronic preturbulence in graphene, Phys. Rev. Lett., 121, 236602 (2018) |
| [56] | Pasechnik, R.; Sumbera, M., Phenomenological review on quark-gluon plasma: Concepts vs. observations, Universe, 3, 1 (2017) |
| [57] | Scardina, F.; Das, S. K.; Minissale, V.; Plumari, S.; Greco, V., Estimating the charm quark diffusion coefficient and thermalization time from D meson spectra at energies available at the BNL relativistic heavy ion collider and the CERN large hadron collider, Phys. Rev. C, C96, 4, 044905 (2017) |
| [58] | Cercignani, C.; Kremer, G. M., The Relativistic Boltzmann Equation: Theory and Applications (2002), Birkhäuser Basel · Zbl 1011.76001 |
| [59] | Anderson, J.; Witting, H., A relativistic relaxation-time model for the Boltzmann equation, Physica, 74, 3, 466-488 (1974) |
| [60] | Anderson, J.; Witting, H., Relativistic quantum transport coefficients, Physica, 74, 3, 489-495 (1974) |
| [61] | Karsch, F.; Miller, D. E., Exact equation of state for ideal relativistic quantum gases, Phys. Rev. A, 22, 1210-1219 (1980) |
| [62] | Grad, H., On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2, 4, 331-407 (1949) · Zbl 0037.13104 |
| [63] | S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, third ed., Cambridge University Press, 197, http://dx.doi.org/10.1119/1.1942035. · Zbl 0063.00782 |
| [64] | Molnár, E.; Niemi, H.; Denicol, G. S.; Rischke, D. H., Relative importance of second-order terms in relativistic dissipative fluid dynamics, Phys. Rev. D, 89, 074010 (2014) |
| [65] | Tsumura, K.; Kunihiro, T., Derivation of relativistic hydrodynamic equations consistent with relativistic Boltzmann equation by renormalization-group method, Eur. Phys. J. A, 48, 11, 162 (2012) |
| [66] | Mendoza, M.; Karlin, I.; Succi, S.; Herrmann, H. J., Ultrarelativistic transport coefficients in two dimensions, J. Stat. Mech. Theory Exp., 2013, P02036 (2013) · Zbl 1456.76163 |
| [67] | Florkowski, W.; Jaiswal, A.; Maksymiuk, E.; Ryblewski, R.; Strickland, M., Relativistic quantum transport coefficients for second-order viscous hydrodynamics, Phys. Rev. C, 91, 054907 (2015) |
| [68] | Tsumura, K.; Kikuchi, Y.; Kunihiro, T., Relativistic causal hydrodynamics derived from Boltzmann equation: A novel reduction theoretical approach, Phys. Rev. D, 92, 085048 (2015) |
| [69] | Kikuchi, Y.; Tsumura, K.; Kunihiro, T., Derivation of second-order relativistic hydrodynamics for reactive multicomponent systems, Phys. Rev. C, 92, 064909 (2015) |
| [70] | Kikuchi, Y.; Tsumura, K.; Kunihiro, T., Mesoscopic dynamics of fermionic cold atoms — Quantitative analysis of transport coefficients and relaxation times, Phys. Lett. A, 380, 24, 2075-2080 (2016) |
| [71] | García-Perciante, A. L.; Méndez, A. R.; Escobar-Aguilar, E., Heat flux for a relativistic dilute bidimensional gas, J. Stat. Phys., 167, 123-134 (2017) · Zbl 1375.35549 |
| [72] | Higuera, F. J.; Succi, S.; Benzi, R., Lattice gas dynamics with enhanced collisions, Europhys. Lett., 9, 345 (1989) |
| [73] | He, X.; Luo, L.-S., Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E, 56, 6811-6817 (1997) |
| [74] | Shan, X.; He, X., Discretization of the velocity space in the solution of the Boltzmann equation, Phys. Rev. Lett., 80, 65-68 (1998) |
| [75] | Martys, N. S.; Shan, X.; Chen, H., Evaluation of the external force term in the discrete Boltzmann equation, Phys. Rev. E, 58, 6855-6857 (1998) · Zbl 1073.81571 |
| [76] | See Supplemental Material at https://doi.org/10.1016/j.physrep.2020.03.004. |
| [77] | Philippi, P. C.; Hegele, L. A.; dos Santos, L. O.E.; Surmas, R., From the continuous to the lattice Boltzmann equation: The discretization problem and thermal models, Phys. Rev. E, 73, 056702 (2006) |
| [78] | Shan, X., General solution of lattices for Cartesian lattice Bhatanagar-Gross-Krook models, Phys. Rev. E, 81, 036702 (2010) |
| [79] | Shan, X., The mathematical structure of the lattices of the lattice Boltzmann method, J. Comput. Sci., 17, 475-481 (2016) |
| [80] | Blaga, R.; Ambruş, V. E., Quadrature-based lattice Boltzmann model for relativistic flows, AIP Conf. Proc., 1796, 020010 (2017) |
| [81] | Shan, X.; Yuan, X.-F.; Chen, H., Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation, J. Fluid Mech., 550, 413-441 (2006) · Zbl 1097.76061 |
| [82] | Shan, X.; Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E, 47, 1815-1819 (1993) |
| [83] | Shan, X.; Chen, H., Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation, Phys. Rev. E, 49, 2941-2948 (1994) |
| [84] | Guo, Z.; Zheng, C.; Shi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65, 046308 (2002) · Zbl 1244.76102 |
| [85] | Sbragaglia, M.; Benzi, R.; Biferale, L.; Chen, H.; Shan, X.; Succi, S., Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria, J. Fluid Mech., 628, 299-309 (2009) · Zbl 1181.76112 |
| [86] | Succi, S.; Amati, G.; Bernaschi, M.; Falcucci, G.; Lauricella, M.; Montessori, A., Towards exascale lattice Boltzmann computing, Comput. & Fluids, 181, 107-115 (2019) · Zbl 1410.76380 |
| [87] | Shet, A. G.; Sorathiya, S. H.; Krithivasan, S.; Deshpande, A. M.; Kaul, B.; Sherlekar, S. D.; Ansumali, S., Data structure and movement for lattice-based simulations, Phys. Rev. E, 88, 013314 (2013) |
| [88] | Shet, A. G.; Sorathiya, S. H.; Krithivasan, S.; Deshpande, A. M.; Kaul, B.; Sherlekar, S. D.; Ansumali, S., On vectorization for lattice based simulations, Internat. J. Modern Phys. C, 24, 12, 1340011 (2013) |
| [89] | Calore, E.; Gabbana, A.; Schifano, S.; Tripiccione, R., Optimization of lattice Boltzmann simulations on heterogeneous comput., Int. J. High Perform. Comput. Applications, 33, 1, 124-139 (2019) |
| [90] | Gabbana, A.; Simeoni, D.; Succi, S.; Tripiccione, R., Probing bulk viscosity in relativistic flows (2019), arXiv:1910.04275 |
| [91] | Taylor, G. I.; Green, A. E., Mechanism of the production of small eddies from large ones, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 158, 895, 499-521 (1937) · JFM 63.1358.03 |
| [92] | Greif, M.; Bouras, I.; Greiner, C.; Xu, Z., Electric conductivity of the quark-gluon plasma investigated using a perturbative QCD based parton cascade, Phys. Rev. D, 90, 094014 (2014) |
| [93] | Mohamad, A. A.; Succi, S., A note on equilibrium boundary conditions in lattice Boltzmann fluid dynamic simulations, Eur. Phys. J. Spec. Top., 171, 1, 213-221 (2009) |
| [94] | Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31 (1978) · Zbl 0387.76063 |
| [95] | Thompson, K. W., The special relativistic shock tube, J. Fluid Mech., 171, 365-375 (1986) · Zbl 0609.76133 |
| [96] | Xu, Z.; Greiner, C., Transport rates and momentum isotropization of gluon matter in ultrarelativistic heavy-ion collisions, Phys. Rev. C, 76, 024911 (2007) |
| [97] | Bouras, I.; Molnár, E.; Niemi, H.; Xu, Z.; El, A.; Fochler, O.; Greiner, C.; Rischke, D. H., Relativistic shock waves in viscous gluon matter, Phys. Rev. Lett., 103, 032301 (2009) |
| [98] | El, A.; Muronga, A.; Xu, Z.; Greiner, C., Shear viscosity and out of equilibrium dynamics, Phys. Rev. C, 79, 044914 (2009) |
| [99] | Plumari, S.; Puglisi, A.; Scardina, F.; Greco, V., Shear viscosity of a strongly interacting system: Green-kubo correlator versus Chapman-Enskog and relaxation-time approximations, Phys. Rev. C, 86, 054902 (2012) |
| [100] | Ruggieri, M.; Scardina, F.; Plumari, S.; Greco, V., Elliptic flow from non-equilibrium initial condition with a saturation scale, Phys. Lett. B, 727, 1, 177-181 (2013) |
| [101] | Plumari, S.; Guardo, G. L.; Scardina, F.; Greco, V., Initial-state fluctuations from midperipheral to ultracentral collisions in an event-by-event transport approach, Phys. Rev. C, 92, 054902 (2015) |
| [102] | Plumari, S., Anisotropic flows and the shear viscosity of the QGP within an event-by-event massive parton transport approach, Eur. Phys. J. C, 79, 1, 2 (2019) |
| [103] | Florkowski, W.; Friman, B.; Jaiswal, A.; Ryblewski, R.; Speranza, E., Relativistic fluid dynamics of spin-polarized systems of particles (2019), arXiv:1901.00352 |
| [104] | Torre, I.; Tomadin, A.; Geim, A. K.; Polini, M., Nonlocal transport and the hydrodynamic shear viscosity in graphene, Phys. Rev. B, 92, 165433 (2015) |
| [105] | Pellegrino, F. M.D.; Torre, I.; Geim, A. K.; Polini, M., Electron hydrodynamics dilemma: Whirlpools or no whirlpools, Phys. Rev. B, 94, 155414 (2016) |
| [106] | Berdyugin, A. I.; Xu, S. G.; Pellegrino, F. M.D.; Krishna Kumar, R.; Principi, A.; Torre, I.; Ben Shalom, M.; Taniguchi, T.; Watanabe, K.; Grigorieva, I. V.; Polini, M.; Geim, A. K.; Bandurin, D. A., Measuring hall viscosity of graphene’s electron fluid, Science, 364, 162-165 (2019) |
| [107] | Tomadin, A.; Polini, M., Theory of the plasma-wave photoresponse of a gated graphene sheet, Phys. Rev. B, 88, 205426 (2013) |
| [108] | Bandurin, D. A.; Torre, I.; Kumar, R. K.; Ben Shalom, M.; Tomadin, A.; Principi, A.; Auton, G. H.; Khestanova, E.; Novoselov, K. S.; Grigorieva, I. V.; Ponomarenko, L. A.; Geim, A. K.; Polini, M., Negative local resistance caused by viscous electron backflow in graphene, Science, 351, 6277, 1055-1058 (2016) |
| [109] | Krishna Kumar, R.; Bandurin, D. A.; Pellegrino, F.; Cao, Y.; Principi, A.; Guo, H.; Auton, G.; Ben Shalom, M.; Ponomarenko, L. A.; Falkovich, G.; Grigorieva, I.; Levitov, L. S.; Polini, M.; Geim, A. K., Super-ballistic flow of viscous electron fluid through graphene constrictions, Nat. Phys., 13 (2017) |
| [110] | Bandurin, D. A.; Shytov, A. V.; Levitov, L. S.; Kumar, R. K.; Berdyugin, A. I.; Shalom, M. B.; Grigorieva, I. V.; Geim, A. K.; Falkovich, G., Fluidity onset in graphene, Nature Commun., 9 (2018) |
| [111] | Abramowitz, M.; Stegun, I. A.; Miller, D., Handbook of mathematical functions with formulas, graphs and mathematical tables (national bureau of standards applied mathematics series no. 55), J. Appl. Mech., 32, 239 (1965) |
| [112] | Taub, A. H., Relativistic Rankine-Hugoniot equations, Phys. Rev., 74, 328-334 (1948) · Zbl 0035.12103 |
| [113] | Grosswald, E., Bessel Polynomials (Lecture Notes in Mathematics) (1979), Springer |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
