The classical-map hypernetted-chain method (CHNC method) is a method used in

many-body The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

for interacting uniform electron liquids in two and three dimensions, and for non-ideal

plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...

s. The method extends the famous hypernetted-chain method (HNC) introduced by J. M. J van Leeuwen et al. to

quantum fluid A quantum fluid refers to any system that exhibits quantum mechanical effects at the macroscopic level such as superfluids, superconductors, ultracold atoms, etc. Typically, quantum fluids arise in situations where both quantum mechanical effects an ...

s as well. The classical HNC, together with the

Percus–Yevick approximation In statistical mechanics the Percus–Yevick approximation is a closure relation to solve the Ornstein–Zernike equation. It is also referred to as the Percus–Yevick equation. It is commonly used in fluid theory to obtain e.g. expressions for t ...

, are the two pillars which bear the brunt of most calculations in the theory of interacting

classical fluid Classical fluidsR. Balescu, ''Equilibrium and Nonequilibrium Statistical Mechanics'', (John Wiley, 1975) are systems of particles which retain a definite volume, and are at sufficiently high temperatures (compared to their Fermi energy) that quantum ...

s. Also, HNC and PY have become important in providing basic reference schemes in the theory of fluids, and hence they are of great importance to the physics of many-particle systems. The HNC and PY integral equations provide the

pair distribution function The pair distribution function describes the distribution of distances between pairs of particles contained within a given volume. Mathematically, if ''a'' and ''b'' are two particles in a fluid, the pair distribution function of ''b'' with respect ...

s of the particles in a classical fluid, even for very high coupling strengths. The coupling strength is measured by the ratio of the potential energy to the kinetic energy. In a classical fluid, the kinetic energy is proportional to the temperature. In a quantum fluid, the situation is very complicated as one needs to deal with quantum operators, and matrix elements of such operators, which appear in various perturbation methods based on

Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...

s. The CHNC method provides an approximate "escape" from these difficulties, and applies to regimes beyond perturbation theory. In Robert B. Laughlin's famous Nobel Laureate work on the

fractional quantum Hall effect The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...

, an HNC equation was used within a classical plasma analogy. In the CHNC method, the pair-distributions of the interacting particles are calculated using a mapping which ensures that the quantum mechanically correct non-interacting pair distribution function is recovered when the Coulomb interactions are switched off. The value of the method lies in its ability to calculate the ''interacting'' pair distribution functions ''g''(''r'') at zero and finite temperatures. Comparison of the calculated ''g''(''r'') with results from Quantum Monte Carlo show remarkable agreement, even for very strongly correlated systems. The interacting pair-distribution functions obtained from CHNC have been used to calculate the exchange-correlation energies,

Landau parameter Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990) ...

s of

Fermi liquid Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interactions among the particles of the many-bod ...

s and other quantities of interest in many-body physics and

density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...

, as well as in the theory of hot plasmas.R. Bredow, Th. Bornath, W.-D. Kraeft, M.W.C. Dharma-wardana and R. Redmer, Contributions to Plasma Physics, ''55'', 222-229 (2015) DOI 10.1002/ctpp.201400080

See also

References

Further reading