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The nuclear equation of state (EOS) is at the center of numerous theoretical and experimental efforts in nuclear physics. With advances in microscopic theories for nuclear interactions, the availability of experiments probing nuclear matter under conditions not reached before, endeavors to develop sophisticated and reliable transport simulations to interpret these experiments, and the advent of multi-messenger astronomy, the next decade will bring new opportunities for determining the nuclear matter EOS, elucidating its dependence on density, temperature, and isospin asymmetry. Among controlled terrestrial experiments, collisions of heavy nuclei at intermediate beam energies (from a few tens of MeV/nucleon to about 25 GeV/nucleon in the fixed-target frame) probe the widest ranges of baryon density and temperature, enabling studies of nuclear matter from a few tenths to about 5 times the nuclear saturation density and for temperatures from a few to well above a hundred MeV, respectively. Collisions of neutron-rich isotopes further bring the opportunity to probe effects due to the isospin asymmetry. However, capitalizing on the enormous scientific effort aimed at uncovering the dense nuclear matter EOS, both at RHIC and at FRIB as well as at other international facilities, depends on the continued development of state-of-the-art hadronic transport simulations. This white paper highlights the essential role that heavy-ion collision experiments and hadronic transport simulations play in understanding strong interactions in dense nuclear matter, with an emphasis on how these efforts can be used together with microscopic approaches and neutron star studies to uncover the nuclear EOS., Comment: White paper prepared for the 2023 Long Range Plan. v3: Updated version as published in Progress in Particle and Nuclear Physics. Note: the published version does not include the executive summary; in the updated arXiv version, the executive summary is included as an appendix. v4: Corrected list of authors

Depending on the energy regime, the dynamics of heavy-ion collisions reveals a variety of different mechanisms which are attributed to the combination of collective and dissipative effects. Semi-classical approaches have been successful in describing chaotic regimes at Fermi-energies but they gradually lose precision when extending to collective behaviour and in general when low-energy features become more determinant in the dynamics. To improve on this description, we propose a theoretical approach starting from the TDHF scheme. A quantum representation with a moving basis function has been worked out with a double aim. Firstly, achieving a simplified solution to handle the evolution in time. Secondly, introducing beyond-mean-field extensions and stochastic contributions. Applications to nuclear collisions at incident energies around low to Fermi energy are presented., Comment: This is a proceeding to the International Workshop on Multi-facets of EoS and Clustering IWM-EC conference

Transport models are the main method to obtain physics information from low to relativistic-energy heavy-ion collisions. The Transport Model Evaluation Project (TMEP) has been pursued to test the robustness of transport model predictions in reaching consistent conclusions from the same type of physical model. Calculations under controlled conditions of physical input and set-up were performed with various participating codes. These included both calculations of nuclear matter in a box with periodic boundary conditions, and more realistic calculations of heavy-ion collisions. In this intermediate review, we summarize and discuss the present status of the project. We also provide condensed descriptions of the 26 participating codes, which contributed to some part of the project. These include the major codes in use today. We review the main results of the studies completed so far. They show, that in box calculations the differences between the codes can be well understood and a convergence of the results can be reached. These studies also highlight the systematic differences between the two families of transport codes, known as BUU and QMD type codes. However, when the codes were compared in full heavy-ion collisions using different physical models, as recently for pion production, they still yielded substantially different results. This calls for further comparisons of heavy-ion collisions with controlled models and of box comparisons of important ingredients, like momentum-dependent fields, which are currently underway. We often indicate improved strategies in performing transport simulations and thus provide guidance to code developers. Results of transport simulations of heavy-ion collisions from a given code will have more significance if the code can be validated against benchmark calculations such as the ones summarized in this review., Comment: 114 pages, 14 figures, 479 references, accepted for publication in Progress of Particle and Nuclear Phsics

Violent nuclear collisions are open systems which require a non-equilibrium description when the process should be followed from the first instants. The heated system produced in the collision, can no more be treated within an independent-particle picture and additional correlations should be taken into account: they rely to in-medium dissipation and phase-space fluctuations. Their interplay with the one-body collective behaviour activates the transport dynamics: large-amplitude fluctuations and bifurcations in a variety of mechanisms appear, from fusion to neck formation till eventually freezing out the system into several intermediate-mass clusters. Starting from fundamental concepts tested on nuclear matter, a microscopic description is built up to address violent processes occurring in heavy-ion collisions at Fermi energies and in spallation reactions, and it is applied to experimental observables., Comment: HDR report. Review written in order to be conferred the French accreditation to supervise research (Habilitation \`a diriger des recherches)

Transport simulations are very valuable for extracting physics information from heavy-ion collision experiments. With the emergence of many different transport codes in recent years, it becomes important to estimate their robustness in extracting physics information from experiments. We report on the results of a transport code comparison project. 18 commonly used transport codes were included in this comparison: 9 Boltzmann-Uehling-Uhlenbeck-type codes and 9 Quantum-Molecular-Dynamics-type codes. These codes have been required to simulate Au+Au collisions using the same physics input for mean fields and for in-medium nucleon-nucleon cross sections, as well as the same initialization set-up, the impact parameter, and other calculational parameters at 100 and 400 AMeV incident energy. Among the codes we compare one-body observables such as rapidity and transverse flow distributions. We also monitor non-observables such as the initialization of the internal states of colliding nuclei and their stability, the collision rates and the Pauli blocking. We find that not completely identical initializations constitute partly for different evolutions. Different strategies to determine the collision probabilities, and to enforce the Pauli blocking, also produce considerably different results. There is a substantial spread in the predictions for the observables, which is much smaller at the higher incident energy. We quantify the uncertainties in the collective flow resulting from the simulation alone as about $30\%$ at 100 AMeV and $13\%$ at 400 AMeV, respectively. We propose further steps within the code comparison project to test the different aspects of transport simulations in a box calculation of infinite nuclear matter. This should, in particular, improve the robustness of transport model predictions at lower incident energies where abundant amounts of data are available., Comment: 26 pages, 13 figures, a manuscript on the heavy-ion collision part of transport code comparison project