We first analyze the relations between the work of Planck and Klimontovich and show that Klimontovich's kinetic theory of non-ideal plasmas confirms Planck's partition function for the hydrogen plasma. We discuss and check the Planck-Brillouin-Larkin (PBL) expressions for Hydrogen with PIMC calculations and derive new partition functions and mass action constants for Helium and Lithium. The new partition functions consider Planck's compensation effects between discrete and scattering states (Levinson effects). We show that any calculation of partition functions without taking compensation effects into account is quantum-statistically incorrect. Our new expressions, which are given as series in beta=$$ \tilde{\beta}= $$ Ry/kBT$$ \mathrm{Ry}/{k}_{\mathrm{B}}T $$, are of particular interest for the temperature range beyond the ionization temperatures. There, the new partition functions disappear faster with temperature than the standard ones, which corresponds to a faster destruction of atoms in plasmas. The partition functions and the associated mass action constants are defined as the convergent contributions to the corresponding quantum-statistical virial coefficients of pressure expansions. We estimate the third and fourth Coulomb virial coefficients and the mass action functions based on the new Slater functions. Based on the new results, which imply changes in the low orders of the parameter beta$$ \tilde{\beta} $$, we predict lower concentrations for the formation of Hydrogen, Helium, and Lithium atoms and ions at higher temperatures than those obtained with standard methods.

The OCP plasma model which has been the favourite plasma model of Gabor Kalman is simple but on the other side connected with some principal difficulties, and gave rise to some controversies. We discuss here the three main problems of Coulomb systems, the limit cases of the parameter xi similar to (e(2)/(h) over bar root T) : xi -> +/-infinity and xi -> 0. We show first that Taylor expansions in (1/xi) similar to (h) over bar are in general divergent and have asymptotic character and expansions in xi similar to e(2) are convergent. We study the analytic properties of the partition functions and the thermodynamic functions. Assuming analytizity with respect to the relevant physical parameter for pair interactions xi similar to (e(i)e(j)/(h) over barT) we can show that the analyticity with respect to this parameter allows to extend several OCP-properties, except the exchange functions, to many component systems by analytic continuation of the case delta < 0 to xi > 0. In particular follows that the Taylor coefficients of analytic OCP functions may be extended to any multi-component Coulomb system. Further, we discuss also the most difficult case xi -> 0 and the problem with contributions linear in the interaction, the so-called Hartree terms.

We summarize the method of hydrodynamic approximation for weakly ionized plasmas developed with Klimontovich in 1962 and give a generalization to many-component systems using Onsagers matrix theory and including dispersion effects. We develop the conductivity theory of complex plasma and electrolyte mixtures based on the model of charged hard spheres with given non-additive contact distances, including frequency-dependent electric fields. These generalizations are made with the aim to allow applications to complex natural systems as atmospheric plasmas and seawater. Finally, we give as an example a numerical calculation of the single ion conductivities of a six-component seawater model.

We study the effective interactions and the mass action constants for pair and triple associations in classical and quantum plasmas. Avoiding double counting, we derive new expressions for the mass action constants. The calculations resulted in values that were substantially smaller than the standard ones in relevant temperature ranges by up to 50 percent. On this basis, we determine the pressure of H, He and Li plasmas and the osmotic coefficient of electrolytes with higher charges such as, e.g., seawater. Classical and quantum Coulomb systems show strong similarities. The contributions in low orders with respect to the interaction e2 are suppressed by thermal and screening effects. The contributions of weakly bound states, near the continuum edge, to the mass action constants are reduced, replacing the exponential functions with cropped exponentials. The new mass action constants are consistent with well-known extended limiting cases of screening effects. We analyze classical examples including the salts CaCl2 and LaCl3, and a model of seawater including multiple associations. In the case of quantum systems, we follow the work of Planck–Brillouin–Larkin for H plasmas and study He and Li plasmas. The equation of state (EoS) for wide-density regions is obtained through the concatenation of the EoS for the low-density region of partial ionization with the EoS of degenerate plasmas, where all bound states are dissolved and Fermi, Hartree–Fock and Wigner contributions dominate.

Günter Kelbg did remarkable early work in the field of quantum statistical physics, in particular for dense quantum plasmas. In 2022 we celebrated his 100th birthday. On this occasion, we give a brief overview of his main scientific achievements in the field of quantum plasmas, complemented by some biographical background of his research and teaching environment at Rostock University. Kelbg's main achievement is the derivation of a regularized quantum pair potential. We demonstrate that it is still of high relevance for molecular dynamics and quantum Monte Carlo simulations of dense plasmas.

In previous work we developed a new statistical method for calculating the individual activities of ions including the association of ions. Here we study multi-particle electrostatic interactions connected within higher cluster integrals and identify the ionization constants of the mass action law of associating ion clusters. In contrast to Bjerrum and Fuoss, our concept of association is not based on spatial criteria, but instead on the strength of interaction measured in powers of the Bjerrum parameter (e2/D0kBTa; a is contact) and defined by asymptotic properties of the cluster integrals. For ion pair formation our mass action constant is the classical counterpart of Planck's famous hydrogenic partition function. As a rule, the new association constants are smaller than traditional expressions, e.g., by Fuoss and Kraus, in the interesting regions of interaction parameters about fifty percent. Several examples including CaCl2, MgCl2, Na2SO4, K2SO4, LaCl3 and a model of seawater are studied. For several associating electrolytes and seawater, reasonable agreement with experiments and Monte Carlo results is achieved.

We discuss several problems of batteries and fuel cells from the point of view of nonlinear dynamics and review some earlier work on modeling this class of systems as well as new developments. We consider batteries and fuel cells as active nonlinear electrochemical circuits with properties depending on many factors as load, age, load history etc. We show that most satisfactory battery regimes are reached by coupling of an odd number of circuits in opposite phases. Specific points of discussion are types of dynamic regimes and the efficiency of the conversion of chemical into electrical energy in dependence on the work load, the life cycles of batteries, including installation, work under load, aging and decay. Further we discuss specific properties of managing battery networks, in particular the cycle of replacing of old batteries by fresh ones including the optimization of this cycle. The last part of this work is merely a list of open tasks to be elaborated.

A few salient soliton-like wave evolutionary features of one-and two-dimensional lattices of interacting active units are provided here. In the latter case, particular attention is given to the crystal-like triangular lattice. On the one hand, the units are coupled with nearest neighbors anharmonic forces (Morse potential). On the other hand the units are endowed with the possibility of an input-output energy balance that permits evolution to a steady state and the appearance of metastable states which on occasion can be quite long lasting. The lifetimes of such metastable states depend on the lattice parameter values and the wave front width. Eventually, all metastable states evolve to steady translational modes especially under influence of noise. (c) 2021 Elsevier Ltd. All rights reserved.

The equation of state (EOS) for plasmas of the two lightest elements H and He, and mixtures as typical for the plasmas in the sun are calculated. The contributions of deep bound states are included by using inverted fugacity expansions. The inversion of fugacities to densities is reduced to solvable algebraic problems and expressed by rational polynomials. The calculation of relative pressures is carried out separately for low and high densities. Near the crossing point, in between, the separate solutions are connected to each other by smooth concatenation. Applications to hydrogen–helium plasmas in the sun including estimates for the isentropic EOS are discussed.

We consider the dynamics of electrons and holes moving in two-dimensional lattice layers and bilayers. As an example, we study triangular lattices with units interacting via anharmonic Morse potentials and investigate the dynamics of excess electrons and electron-hole pairs according to the Schrödinger equation in the tight binding approximation. We show that when single-site lattice solitons or M-solitons are excited in one of the layers, those lattice deformations are capable of trapping excess electrons or electron-hole pairs, thus forming quasiparticle compounds moving approximately with the velocity of the solitons. We study the temporal and spatial nonlinear dynamical evolution of localized excitations on coupled triangular double layers. Furthermore, we find that the motion of electrons or electron-hole pairs on a bilayer is slaved by solitons. By case studies of the dynamics of charges bound to solitons, we demonstrate that the slaving effect may be exploited for controlling the motion of the electrons and holes in lattice layers, including also bosonic electron-hole-soliton compounds in lattice bilayers, which represent a novel form of quasiparticles.

The equations of state (EoS) and other thermodynamic properties of plasmas of the light elements H, He, and Li, are calculated using inverted fugacity expansions. Fugacity expansions are known as an alternative to density expansions but show often an inferior convergence. If, however, the inversion can be solved, the fugacity representations may be very efficient. In particular, the contributions of deeply bound states are included in the fugacity expansion in a very effective way. The mathematical problems on nonlinearity connected with the inversion of fugacities to densities are reduced to solvable algebraic problems. The inversion of fugacities to densities is solved separately for two density regions: (i) In the low density, non-degenerate region we consider ring contributions describing screening effects and ladder contributions describing bound state formation. (ii) In the high density, degenerate region the electrons are described by the Fermi–Dirac distribution. Hartree–Fock contributions and Pauli blocking have to be taken into account. The ions are considered as classical, strongly correlated subsystem eventually forming a Wigner lattice. We solve the inversion problem for each of the regions. Near the crossing point, the separate solutions are connected to each other, either by smooth concatenation at the crossing point or by Padé approximations.

Provided in this paper is a theory of long-range electron transfer with near sound (supersonic or subsonic) velocity along one-dimensional crystal lattices. The theory represents the development of an earlier work by introducing Marcus formulation. To illustrate its application to a realistic case, the theory is used to offer an explanation of two puzzling observations made by Donovan and Wilson in transient photoconduction experiments with non-dopable perfectly crystalline polydiacetylene crystals in the presence of an electric field: transport velocity value close to sound velocity being independent of field for four orders of magnitude of field (10(2) V/m-10(6) V/m) and, in the low field values, an ultra-high mobility greater than 20 m(2)/V s. We also study factors eventually leading to lowering of the transport velocity.

Matter appears on our planet, in the solar system and in the rest of Universe in rather different forms.

The development of a systematic statistical theory for systems with Coulomb interactions Coulomb interaction divergence is related to characteristic problems:

From the classical kinetic theory of gases we know the equation of state of the ideal gas, βp = n (see Chap. 1 ). For real gases, the interaction forces between the molecules lead to corrections to the ideal gas equation of state. We mention the classical theory by van der Waals and the systematic expansions with respect to density, called virial expansions.

In this chapter, we will introduce useful tools of Quantum Statistics. Most of them will be used in later chapters of this book to solve concrete problems. Our survey covers, of course, the most prominent methods such as density operators introduced by von Neumann and Landau, Wigner’s phase-space functions method, and Bogolyubov’s method of reduced density operators. Matsubara’s thermodynamical Green’s functions and real-time Green’s functions are important methods in the field of quantum plasmas but are discussed here only rather briefly.

The pioneers of the theory of classical kinetic equations are Rudolf Clausius (1822–1888), James Clerk Maxwell (1831–1879) and Ludwig Boltzmann (1844–1906). Their theories are based on the classical dynamics of particles in the phase space according to Liouville and on detailled studies of the classical mechanics of collisions between neutral molecules.

Quantum statistics is a many body theory describing macroscopic matter. Let us first summarize concepts of classical many body theory and subsequently concepts of many body quantum theory, just what we need in the following. After this we will proceed to the simplest quantum statistical ensembles.