Bayesian estimation of the specific shear and bulk viscosity of quark–gluon plasma


Ultrarelativistic collisions of heavy atomic nuclei produce an extremely hot and dense phase of matter, known as quark–gluon plasma (QGP), which behaves like a near-perfect fluid with the smallest specific shear viscosity—the ratio of the shear viscosity to the entropy density—of any known substance1. Due to its transience (lifetime ~ 10−23 s) and microscopic size (10−14 m), the QGP cannot be observed directly, but only through the particles it emits; however, its characteristics can be inferred by matching the output of computational collision models to experimental observations. Previous work, using viscous relativistic hydrodynamics to simulate QGP, has achieved semiquantitative constraints on key physical properties, such as its specific shear and bulk viscosity, but with large, poorly defined uncertainties2,3,4,5,6,7,8. Here, we present the most precise estimates so far of QGP properties, including their quantitative uncertainties. By applying established Bayesian parameter estimation methods9 to a dynamical collision model and a wide variety of experimental data, we extract estimates of the temperature-dependent specific shear and bulk viscosity simultaneously with related initial-condition properties. The method is extensible to other collision models and experimental data and may be used to characterize additional aspects of high-energy nuclear collisions.

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Fig. 1: Estimated temperature-dependent specific shear viscosity of the QGP compared with common fluids.
Fig. 2: Estimated temperature dependence of the specific shear and bulk viscosity.
Fig. 3: Posterior distribution for the initial energy deposition parameter.
Fig. 4: Model calculations using the best-fit MAP parameters compared to experimental data.

Data availability

All data necessary to reproduce the present results are available at This includes the parameter design points, the results of all model calculations, and the posterior distribution generated by MCMC sampling.

Code availability

The computational collision model and related tools are available at The code for performing the Bayesian analysis is available at


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This research was completed using 3 million CPU hours provided by the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy Office of Science User Facility operated under contract no. DE-AC02-05CH11231. J.S.M. and S.A.B. are supported by the US Department of Energy grant no. DE-FG02-05ER41367 and J.E.B. by NSF grant no. NSF-ACI-1550225. J.S.M. also acknowledges former support by the DOE/NNSA Stockpile Stewardship Graduate Fellowship under grant no. DE-FC52-08NA28752 for research contributing to this work. We thank U. Heinz and The Ohio State University group for general discussions and assistance, including the use of their viscous relativistic hydrodynamics code, and S. Pratt for insights into the parameter estimation procedure and particlization model.

Author information

J.E.B. performed the Bayesian analysis, developed several components of the nuclear collision model, and is the primary author of the manuscript. J.S.M. codeveloped the initial-condition model and contributed to multiple aspects of the collision model and computer experiment design. S.A.B. (principal investigator) conceived the project and directed its overall execution.

Correspondence to Jonah E. Bernhard.

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Peer review information: Nature Physics thanks Kari J. Eskola, Johanna Stachel and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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