# Classical-map hypernetted-chain method

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For the Canadian radio station, see [CHNC-FM](/source/CHNC-FM).

The **classical-map hypernetted-chain method** (**CHNC method**) is a method used in [many-body](/source/Many-body_problem) [theoretical physics](/source/Theoretical_physics) for interacting uniform electron liquids in two and three dimensions, and for non-ideal [plasmas](/source/Plasma_(physics)). The method extends the famous [hypernetted-chain method](/source/Hypernetted-chain_equation) (HNC) introduced by [J.M.J. van Leeuwen](/source/J.M.J._van_Leeuwen) et al.[1] to [quantum fluids](/source/Quantum_fluid) as well. The classical HNC, together with the [Percus–Yevick approximation](/source/Percus%E2%80%93Yevick_approximation), are the two pillars which bear the brunt of most calculations in the theory of interacting [classical fluids](/source/Classical_fluid). Also, HNC and PY have become important in providing basic reference schemes in the theory of fluids,[2] and hence they are of great importance to the physics of many-particle systems.

The HNC and PY integral equations provide the [pair distribution functions](/source/Pair_distribution_function) 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 diagrams](/source/Feynman_diagram). The CHNC method provides an approximate "escape" from these difficulties, and applies to regimes beyond perturbation theory. In [Robert B. Laughlin](/source/Robert_B._Laughlin)'s famous Nobel Laureate work on the [fractional quantum Hall effect](/source/Fractional_quantum_Hall_effect), 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.[3] 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 parameters](https://en.wikipedia.org/w/index.php?title=Landau_parameter&action=edit&redlink=1) of [Fermi liquids](/source/Fermi_liquid) and other quantities of interest in many-body physics and [density functional theory](/source/Density_functional_theory), as well as in the theory of hot plasmas.[4][5]

## See also

- [Fermi liquid](/source/Fermi_liquid)

- [Many-body theory](/source/Many-body_theory)

- [Quantum fluid](/source/Quantum_fluid)

- [Radial distribution function](/source/Radial_distribution_function)

## References

1. **[^](#cite_ref-1)** J.M.J. van Leeuwen; J. Groenveld; J. de Boer (1959). "New method for the calculation of the pair correlation function I". *[Physica](/source/Physica_(journal))*. **25** (7–12): 792. [Bibcode](/source/Bibcode_(identifier)):[1959Phy....25..792V](https://ui.adsabs.harvard.edu/abs/1959Phy....25..792V). [doi](/source/Doi_(identifier)):[10.1016/0031-8914(59)90004-7](https://doi.org/10.1016%2F0031-8914%2859%2990004-7).

1. **[^](#cite_ref-2)** R. Balescu (1975). *Equilibrium and Non-equilibrium Statistical Mechanics*. [Wiley](/source/John_Wiley_%26_Sons). pp. 257–277.

1. **[^](#cite_ref-3)** M.W.C. Dharma-wardana; F. Perrot (2000). "Simple Classical Mapping of the Spin-Polarized Quantum Electron Gas: Distribution Functions and Local-Field Corrections". *[Physical Review Letters](/source/Physical_Review_Letters)*. **84** (5): 959–962. [arXiv](/source/ArXiv_(identifier)):[cond-mat/9909056](https://arxiv.org/abs/cond-mat/9909056). [Bibcode](/source/Bibcode_(identifier)):[2000PhRvL..84..959D](https://ui.adsabs.harvard.edu/abs/2000PhRvL..84..959D). [doi](/source/Doi_(identifier)):[10.1103/PhysRevLett.84.959](https://doi.org/10.1103%2FPhysRevLett.84.959). [PMID](/source/PMID_(identifier)) [11017415](https://pubmed.ncbi.nlm.nih.gov/11017415).

1. **[^](#cite_ref-4)** M. W. C. Dharma-wardana, M. W. C.; and François Perrot, Phys. Rev. B **66**, 014110 (2002)

1. **[^](#cite_ref-5)** 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

## Further reading

- C. Bulutay; B. Tanatar (2002). ["Spin-dependent analysis of two-dimensional electron liquids"](http://repository.bilkent.edu.tr/bitstream/11693/24708/1/Spin-dependent%20analysis%20of%20two-dimensional%20electron%20liquids.pdf) (PDF). *[Physical Review B](/source/Physical_Review_B)*. **65** (19) 195116. [Bibcode](/source/Bibcode_(identifier)):[2002PhRvB..65s5116B](https://ui.adsabs.harvard.edu/abs/2002PhRvB..65s5116B). [doi](/source/Doi_(identifier)):[10.1103/PhysRevB.65.195116](https://doi.org/10.1103%2FPhysRevB.65.195116). [hdl](/source/Hdl_(identifier)):[11693/24708](https://hdl.handle.net/11693%2F24708).

- M.W.C. Dharma-wardana; F. Perrot (2002). "Equation of state and the Hugoniot of laser shock-compressed deuterium: Demonstration of a basis-function-free method for quantum calculations". *[Physical Review B](/source/Physical_Review_B)*. **66** (1) 014110. [arXiv](/source/ArXiv_(identifier)):[cond-mat/0112324](https://arxiv.org/abs/cond-mat/0112324). [Bibcode](/source/Bibcode_(identifier)):[2002PhRvB..66a4110D](https://ui.adsabs.harvard.edu/abs/2002PhRvB..66a4110D). [doi](/source/Doi_(identifier)):[10.1103/PhysRevB.66.014110](https://doi.org/10.1103%2FPhysRevB.66.014110).

- N.Q. Khanh; H. Totsuji (2004). "Electron correlation in two-dimensional systems: CHNC approach to finite-temperature and spin-polarization effects". *[Solid State Communications](/source/Solid_State_Communications)*. **129** (1): 37–42. [Bibcode](/source/Bibcode_(identifier)):[2004SSCom.129...37K](https://ui.adsabs.harvard.edu/abs/2004SSCom.129...37K). [doi](/source/Doi_(identifier)):[10.1016/j.ssc.2003.09.010](https://doi.org/10.1016%2Fj.ssc.2003.09.010).

- M.W.C. Dharma-wardana (2005). "Spin and temperature dependent study of exchange and correlation in thick two-dimensional electron layers". *[Physical Review B](/source/Physical_Review_B)*. **72** (12) 125339. [arXiv](/source/ArXiv_(identifier)):[cond-mat/0506804](https://arxiv.org/abs/cond-mat/0506804). [Bibcode](/source/Bibcode_(identifier)):[2005PhRvB..72l5339D](https://ui.adsabs.harvard.edu/abs/2005PhRvB..72l5339D). [doi](/source/Doi_(identifier)):[10.1103/PhysRevB.72.125339](https://doi.org/10.1103%2FPhysRevB.72.125339).

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