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You are here: Home / Teams / Theoretical Physics / Research Topics / Mathematical physics / Quantum plasmas and Path integrals

Quantum plasmas and Path integrals

Angel Alastuey

For describing many terrestrial or astrophysical systems, it is sufficient to retain only Coulomb interaction between nuclei and elecrons. That interaction leads to atomic or molecular recombination, and it also provides effective interactions between recombined entities. Equilibrium properties of those systems have been first studied within the so-called chemical approach. In that framework, preformed entities, introduced at a phenomenological level, interact via suitable effective potentials. Since the sixties, many theoretical (diagrammatical methods, integral equations) and numerical (Monte Carlo, molecular dynamics) works based on that phenomenology, have been published in the literature.

Another approach, so-called physical, relies on the introduction of a quantum plasma made with nuclei and electrons interacting via the Coulomb potential. First papers were devoted to analytic calculations of perturbative expansions of thermodynamic quantities in various regimes, high density at zero temperature (using Feynman-like diagrammatics inspired from field theory) or low density at finite temperature (application by Ebeling of Morita effective-potential method). In such regimes, the plasma is almost fully ionized and recombination is weak. In order to account non-perturbatively for the formation of atoms or molecules, Rogers then proposed a semi-empirical treatment of both quantum effects and screening in low-fugacity expansions.

Our investigations are performed in the framework of the physical approach. Our basic tool is the Feynamn-Kac path integral representation of a quantum gas. Within that representation, Ginibre showed that an equivalent classical system made with loops (extended objects, analogous to polymers) can be introduced. By applying Mayer diagrammatics to the equivalent system made with charged loops, we derived various exact results. For instance, low-density expansions have been computed up to higher orders. Also, we showed that Debye exponential screening is destroyed by quantum fluctuations: quantum correlations only decay as 1/r^6 for large distances r. More recently, we built a representation of equilibrium quantities in terms of particle clusters, which can be viewed as a mathematical formalization of Rogers ideas. The application of that representation to the Hydrogen plasma in the atomic (Saha) regime, allowed us to compute and classify non-ideal leading contributions. This significantly improves the analytical knowledge of the corresponding phase diagram.

Now, we study various polarization phenomena in neutral or partially ionized phases, for instance screening of van der Waals interactions by free charges (interesting consequences for colloidal suspensions with salts), or the coupling between quantum and collective effects in a Clausius-Mossoti calculation of the dielectric constant for a neutral phase. Also, we elaborate numerical methods (fast and accurate at low temperatures) for estimating path integrals involving a few particles. Those estimations are useful for quantitative applications of our cluster representation. Moreover they provide reliable ingredients for quantum Monte Carlo methods or for phenomenological modelizations within the chemical approach.