The properties of plasmas in the low-density limit are described by virial expansions. Analytical expressions are known for the lowest virial coefficients from Green's function approaches. Recently, accurate path-integral Monte Carlo (PIMC) simulations were performed for the hydrogen plasma at low densities by Filinov and Bonitz (Phys. Rev. E 2023, 108, 055212), which made a comparison of the virial expansions and the derivation of interpolation formulas possible. The exact expression for the second virial coefficient is used to test the accuracy of the PIMC simulations and the range of application of the virial expansions. To describe plasmas in a wider range of density and temperature, the concept of quasiparticles is considered. Medium modifications of free and bound states are obtained from the spectral function. Mean-field effects are presented, such as exchange terms, Pauli blocking and screening. The density expansions of the quasiparticle shifts is considered. The combination of PIMC simulations with benchmarks from exact virial expansion results allows us to obtain precise results for the EoS in the low-density range. At low densities, the results are compared with the Saha equation to introduce the medium-dependent ionization potential. The relation to the Beth-Uhlenbeck formula and concepts such as the Mott effect, ionization potential depression (IPD), and ionization degree are discussed. The limits of current PIMC results for hydrogen plasmas are shown. Further improvements of the PIMC simulations are required to compare with analytical benchmarks.

Expressions for the thermodynamic and transport properties of plasmas are derived from quantum statistics in the form of equilibrium correlation functions. These can be evaluated using analytical methods or numerical approaches such as DFT-MD and PIMC simulations. Virial expansions are obtained using the Green's function method. They provide benchmarks for numerical simulations and are useful in the low-density range. The results for the equation of state are discussed for the uniform electron gas and the hydrogen plasma. Transport properties such as the dielectric function are also of interest. Virial expansions are considered for the electrical direct current conductivity as a special case of the dielectric function. Examples are given, and it is explained where further work is needed to obtain a consistent description of the properties of hot and dense plasmas.

We derive a generalized Beth-Uhlenbeck formula for the entropy of a dense fermion system with strong two-particle correlations, including scattering states and bound states. We work within the Phi-derivable approach to the thermodynamic potential. The formula takes the form of an energy-momentum integral over a statistical distribution function times a unique spectral density. In the near mass-shell limit, the spectral density reduces, contrary to na & iuml;ve expectations, not to a Lorentzian but rather to a "squared Lorentzian" shape. The relation of the Beth-Uhlenbeck formula to the Phi-derivable approach is exact at the two-loop level for Phi. The formalism we develop, which extends the Beth-Uhlenbeck approach beyond the low-density limit, includes Mott dissociation of bound states, in accordance with Levinson's theorem, and the self-consistent back reaction of correlations in the fermion propagation. We discuss applications to further systems, such as quark matter and nuclear matter.

The composition of partially ionised plasmas is investigated for densities and temperatures at which the free electrons are degenerate. Based on a quantum statistical approach, the effect of Pauli blocking is addressed. Specifically, one- and two-electron ions are studied. Approximations for deriving an in-medium Schrödinger equation for the ionization potential are indicated. New results regarding the degree of ionisation and the Mott effect are presented. Standard codes for plasma properties do not take Pauli blocking effects into account and are therefore unable to explain the experiments in the high-density regime, where the electrons are degenerate.

This roadmap presents the state-of-the-art, current challenges and near future developments anticipated in the thriving field of warm dense matter physics. Originating from strongly coupled plasma physics, high pressure physics and high energy density science, the warm dense matter physics community has recently taken a giant leap forward. This is due to spectacular developments in laser technology, diagnostic capabilities, and computer simulation techniques. Only in the last decade has it become possible to perform accurate enough simulations & experiments to truly verify theoretical results as well as to reliably design experiments based on predictions. Consequently, this roadmap discusses recent developments and contemporary challenges that are faced by theoretical methods, and experimental techniques needed to create and diagnose warm dense matter. A large part of this roadmap is dedicated to specific warm dense matter systems and applications in astrophysics, inertial confinement fusion and novel material synthesis.

We present a freeze-out approach for describing the formation of heavy elements in expanding nuclear matter. Applying concepts used in modeling heavy-ion collisions or ternary fission, we determine the abundances of heavy elements taking into account in-medium effects such as Pauli blocking and the Mott effect, which describes the dissolution of nuclei at high densities of nuclear matter. With this approach, we search for a universal initial distribution in a quasi-equilibrium state from which the coarse-grained pattern of the solar abundances of heavy elements freezes out and evolves by radioactive decay of the excited states. The universal initial state is characterized by the Lagrange parameters, which are related to temperature and chemical potentials of neutrons and protons. We show that such a state exists and determine a temperature of 5.266 MeV, a neutron chemical potential of 940.317 MeV and a proton chemical potential of 845.069 MeV, with a baryon number density of 0.013 fm−3 and a proton fraction of 0.13. Heavy neutron-rich nuclei such as the hypothetical double-magic nucleus 358Sn appear in the initial distribution and contribute to the observed abundances after fission. We discuss astrophysical scenarios for the realization of this universal initial distribution for heavy-element nucleosynthesis, including supernova explosions, neutron star mergers and the inhomogeneous Big Bang. The latter scenario may be of interest in the light of early massive objects observed with the James Webb Space Telescope and opens new perspectives on the universality of the observed r-process patterns and the lack of observations of population III stars.

Based on a generalized Beth-Uhlenbeck approach to thermodynamics of QCD motivated by cluster decomposition we present a unified equation of state of hot strongly interacting matter and analyze its properties in a wide range of temperatures. The hadrons are treated as color singlet multiquark clusters in medium with a background gluon field in the Polyakov gauge. The confining aspect of QCD is accounted for by the Polyakov loop mechanism and by a large vacuum quark mass motivated by a confining density functional approach. We demonstrate that an abrupt switching between hadronic and partonic degrees of freedom, which is one of striking manifestations of dynamical restoration of chiral symmetry, is accompanied by a smooth behavior of entropy density at chiral crossover. Individual contributions of different components of strongly interacting matter to its speed of sound are analyzed for the first time. It is shown that restoration of chiral symmetry drives speed of sound of hadron gas to negative values, manifesting its mechanical instability and being in a strike disagreements with the lattice QCD data. Accounting for the partonic excitations naturally resolves this contradiction.

We employ the Zubarev approach of the non-equilibrium statistical operator to investigate the enhancement of the low-$p_T$ region of pion spectra, introducing an effective pion chemical potential to describe the overpopulation of low-energy pion states. We test a corresponding freeze-out approach by analyzing the transverse-momentum spectra of identified particles measured recently with high precision by the ALICE Collaboration in Pb+Pb collisions at CERN LHC. A blast-wave model and a blast-wave-based particle generator, coupled to a hadronic transport model, are utilized. Bayesian inference methods are applied to extract the most probable sets of thermodynamic parameters at the chemical freeze-out hypersurface. Both models for the overpopulated pion states, the hadronic transport model and the thermal model with a nonzero pion chemical potential, provide a satisfactory description of the observed pion spectra. However, both approaches contain approximations which can be improved within a systematic nonequilibrium approach. We demonstrate that the introduction of a nonequilibrium pion chemical potential offers an efficient alternative to the conventional explanation of the low-$p_T$ enhancement, typically attributed to resonance decays with subsequent thermalization. A similar discussion holds also for the kaon spectra.

We review the role of primordial black holes for illuminating the dark ages of the cosmological evolution and as dark matter candidates. We elucidate the role of phase transitions for primordial black hole formation in the early Universe and focus our attention to the cosmological QCD phase transition within a recent microscopical model. We explore the impact of physics beyond the Standard Model on the cosmic equation of state and the probability distribution for the formation of primordial black holes which serve as dark-matter candidates. We argue that besides primordial black holes also droplet-like quark-gluon plasma inhomogeneities may become gravitationally stabilized for a sufficiently long epoch to distill baryon number and form nuclear matter droplets which upon their evaporation may enrich the cosmos locally with heavy r-process elements already in the early Universe.

We present a unified approach to the transition from hadronic matter to quark matter where hadrons are treated as bound states of quarks which dissociate at high densities due to quark Pauli blocking. The newly developed approach makes use of a cluster virial expansion formulated in terms of a generalized Φ-derivable approach to multi-quark correlations with bound and continuum states in their spectrum encoded in hadron phase shifts. Our model can be used to obtain thermodynamic functions not only at zero and small chemical potentials, where they are consistent with lattice QCD simulations, but also at large chemical potentials where lattice QCD simulations have the sign problem. By applying a reaction-kinetic criterion for the chemical freeze-out of multi-quark clusters in heavy-ion collisions, we demonstrate that the chemical freeze-out coincides with their Mott transition. The approach can be applied to study the effects of the QCD transition on primordial black hole formation in the early Universe and on hybrid neutron star formation in supernova explosions and binary neutron star mergers.

A striking feature in the observed chemical composition of the majority of stars is the universality of the relative abundances of the heavy elements, although some outliers exist. We demonstrate that a nonequilibrium freeze-out approach provides a natural way of accounting for the typical abundance pattern and its variation. Here, we use a phenomenological method to characterize the coarse-grained distribution of heavy r-process elements in several astrophysical objects. The Lagrange parameters show only minor fluctuations when comparing different stars. Larger deviations are observed in stars with low metallicity. The variations in the Lagrange parameters for these stars are presented. The determination of the Lagrange parameters can be instrumental in identifying possible sources for the formation of heavy elements. In particular, density fluctuations are considered as a source for the production of heavy elements in the early Universe.

A nonequilibrium state does not relax to thermodynamic equilibrium but to a state which takes into account long-living fluctuations as quasi-conserved quantities. This state is described by the relevant statistical operator within the Zubarev method to derive the nonequilibrium statistical operator. We apply this approach to the spectra of particles produced in ultrarelativistic heavy ion collisions at the LHC experiments at CERN. We show that controversal explanations of the low-momentum part of the spectrum given by an extended hydrodynamic-like Blast-Wave approach [with mesonic chemical potentials] and the reaction-kinetic description of the hadron resonance gas can be considered as special approximation of a more general nonequilibrium approach which takes mesonic chemical potentials into account to describe quasi-conserved particle numbers, but takes also continuum correlations, in particular resonances, into account using the Beth-Uhlenbeck [or Dashen-Ma-Bernstein] virial expansion. We present results for the spectra of pions, kaons, and protons and explain why different approaches can explain the data obtained from the ALICE experiments.

We describe multiquark clusters in quark matter within a Beth-Uhlenbeck approach in a background gluon field coupled to the underlying chiral quark dynamics using the Polyakov gauge which establishes the center symmetry of color SU(3) that suppresses colored states as an aspect of confinement. Quark confinement is modeled by a large quark mass in vacuum motivated by a confining density functional approach. A multiquark cluster containing $n$ quarks and antiquarks is described as a binary composite of smaller subclusters $n_1$ and $n_2$ ($n_1+n_2=n$). It has a spectrum consisting of a bound state and a scattering state continuum. For the corresponding cluster-cluster phase shifts we discuss simple ans\"atze that capture the Mott dissociation of clusters as a function of temperature and chemical potential. We go beyond the simple "step-up-step-down" model that ignores continuum correlations and introduce an improved model that includes them in a generic form. In order to explain the model, we restrict ourselves here to the cases where the cluster size is $1 \le n \le 6$. A striking result is the suppression of the abundance of colored multiquark clusters at low temperatures by the coupling to the Polyakov loop and their importance for a quantitative description of lattice QCD thermodynamics at non-vanishing baryochemical potentials. An important ingredient are Polyakov-loop generalized distribution functions of $n$-quark clusters which are derived here for the first time. Within our approach we calculate thermodynamic properties such as baryon density and entropy. We demonstrate that the limits of a hadron resonance gas at low temperatures and $\mathcal{O}(g^2)$ perturbative QCD at high temperatures are correctly reproduced. A comparison with lattice calculations shows that our model is able to give a unified, systematic approach to describe properties of the quark-gluon-hadron system.

We investigate the ground -state structures of 12 - 14 C isotopes, which exhibit the coexistence of the cluster and shell-like states, using the cluster breaking Tohsaki-Horiuchi-Schuck-R & ouml;pke (CB-THSR) wave function. This approach incorporates cluster breaking nucleon -nucleon pairs, making it particularly suitable for describing the cluster states that feature mixed configurations and dynamic cluster motion. Through variational optimization of THSR bases, we find the coupling between the motions of clusters and valence neutrons in different configurations of 12 - 14 C isotopes, e.g., dumbbell -like molecular orbits of valence neutrons coupled with an oblate 3- alpha core, and a prolate 3- alpha core surrounded by ringlike valence neutron distributions. For describing the coexistence of shell-like structures and cluster formation in the compact cores of 12 - 14 C isotopes, we introduce CB pairs into the THSR approach to formulate the CB-THSR wave function, which allows a further optimization of the variational wave function and the binding energy. The results suggest oblate and shell-like structures for the ground states of 12 C and 13 - 14 C, respectively. As compared to the original THSR calculation, our new model yields relatively more accurate spectra for 12 - 14 C isotopes, as well as their matter rms radii. The improved results show that alpha clusters are broken by the CB pairs, and the CB-THSR wave functions describe a more compact spatial extension of cluster motion, while such strong overlaps between alpha clusters are not energetically favored in pure cluster models. Finally, we discuss the breaking of alpha clusters in 12 - 14 C isotopes via the calculations of momentum distributions and form factors, and it is found that both the particle exchange and the paired excitation should be included for describing the compact cluster states.

This Topical Collection of the European Physics Journal A is devoted to recent progress in the nuclear many-body problem. In particular, it aims at a comprehensive compilation of developments related to the work of a pioneer in that field, Peter Schuck, who passed away in 2022. Together with Peter Ring, he co-authored the book on “The Nuclear Many-Body Problem”. Different concepts presented in this seminal book have been elaborated further within a broad international collaboration. For instance, the quasi-particle approaches in connection with nuclear superfluidity and cluster formation in nuclear systems, in particular alpha-particle condensation and quartetting at subsaturation densities, have been put forward inspired by Peter Schuck. These advances obtained in the nuclear many-body problem can also be applied to other systems, for instance solid state physics. This Topical Collection is considered as addendum and continuation of the textbook of P. Ring and P. Schuck.

We report the results of the second charged-particle transport coefficient code comparison workshop, which was held in Livermore, California on 24–27 July 2023. This workshop gathered theoretical, computational, and experimental scientists to assess the state of computational and experimental techniques for understanding charged-particle transport coefficients relevant to high-energy-density plasma science. Data for electronic and ionic transport coefficients, namely, the direct current electrical conductivity, electron thermal conductivity, ion shear viscosity, and ion thermal conductivity were computed and compared for multiple plasma conditions. Additional comparisons were carried out for electron–ion properties such as the electron–ion equilibration time and alpha particle stopping power. Overall, 39 participants submitted calculated results from 18 independent approaches, spanning methods from parameterized semi-empirical models to time-dependent density functional theory. In the cases studied here, we find significant differences—several orders of magnitude—between approaches, particularly at lower temperatures, and smaller differences—roughly a factor of five—among first-principles models. We investigate the origins of these differences through comparisons of underlying predictions of ionic and electronic structure. The results of this workshop help to identify plasma conditions where computationally inexpensive approaches are accurate, where computationally expensive models are required, and where experimental measurements will have high impact.

^20 Ne can be considered as a double-magic ^16 O core nucleus surrounded by four nucleons, the constituents of an α -like quartet. Similar to other nuclei ( ^212 Po, ^104 Ti, etc.) with a quartet on top of a double-magic core nucleus, significant α -like correlations are expected. Correlations in the ground state of ^20 Ne are investigated using different approaches. The quartetting wave function approach (QWFA) predicts a large α -like cluster contribution near the surface of the nuclei. The Tohsaki-Horiuchi-Schuck-Röpke (THSR) approach describes α -like clustering in nuclear systems. The results of the QWFA in the Thomas-Fermi and shell-model approximation are compared with THSR calculations for the container model. Results for the α formation probability and the rms radii are shown.

An improved virial expansion for the low-density limit of the electrical conductivity σ(T,n) of hydrogen as the simplest ionic plasma is presented. Quantum statistical methods provide exact values for the lowest virial coefficients, which serve as a benchmark for analytical approaches to electrical conductivity as well as for numerical results from density functional theory-based molecular dynamics simulations (DFT-MD) or path-integral Monte Carlo simulations. The correction factor introduced by Reinholz et al. [Phys. Rev. E 91, 043105 (2015)] is applied to describe the inclusion of electron–electron collisions in DFT-based calculations of transport coefficients. As a benchmark, the first virial coefficient is correctly described with this approach. The value of the second virial coefficient is discussed, and questions about its value according to DFT-MD simulations are addressed.

Physical properties of plasmas, such as equations of state (EoS) and transport coefficients, are expressed in terms of correlation functions, which can be calculated using various approaches (analytical theory and numerical simulations). The method of Green's functions provides benchmark values for these properties in the low‐density limit. For the EoS and electrical conductivity, expansions with respect to density (virial expansions) are considered. The comparison of analytical results with numerical simulations is used to verify theory, to prove the accuracy of simulations, and establish interpolation formulas.