New Research in Physics

Modeling quantum effects in collisionless electrostatic plasmas and comparing results in two linear and nonlinear frameworks

Author

Zeynab Kiamehr

Tafresh University

Abstract
Classical plasma physics primarily addresses regimes characterized by high temperatures and low densities, where quantum mechanical effects are essentially negligible. In recent years, however, advances in technology—particularly in semiconductor engineering and the development of nanoscale structures—have opened the way for studying and forecasting practical applications of plasma physics under conditions where the quantum nature of particles plays a pivotal role. In this study, several approaches to modeling quantum effects in collisionless electrostatic plasmas are explored. The most comprehensive kinetic description of these phenomena is based on the Wigner equation, the quantum analogue of the classical Vlasov equation. The Wigner formalism is remarkable in that it frames quantum theory within the familiar classical phase space, though it introduces the challenge of distribution functions that may take on negative values. From an equivalent standpoint, the Wigner model can be reformulated in terms of N single-particle Schrödinger equations coupled with the Poisson equation—an approach known as the Hartree formalism—which closely parallels the multi-flow method in classical plasma physics. To manage the complexity inherent in these kinetic frameworks, a quantum fluid model can be obtained by taking velocity-space moments of the Wigner equation. These reduced models facilitate the investigation of collective particle dynamics with reasonable accuracy and within a more tractable theoretical structure. Furthermore, in specific regimes characterized by high excitation energies, semiclassical kinetic models of the Vlasov–Poisson type can be utilized, provided that the initial ground state is determined according to quantum mechanical principles. The models discussed herein have been validated and cross-compared in both linear and nonlinear scenarios. Their outcomes indicate that integrating quantum and classical formulations yields a richer and more nuanced understanding of plasma behavior at microscopic scales and under extreme physical conditions.

Keywords

Plasma physics
quantum effects
modeling in plasmas
collisionless plasma
kinetic model
F. Calvayrac, P.-G. Reinhard, E. Suraud, and C. Ullrich, Nonlinear electron dynamics in metal clusters. Phys. Rep. 337, 493-578, (2000). [DOI:10.1016/S0370-1573(00)00043-0]
F. Calvayrac, P.-G. Reinhard, E. Suraud, and C. Ullrich, Nonlinear electron dynamics in metal clusters. Phys. Rep. 337, 493-578, (2000). [DOI:10.1016/S0370-1573(00)00043-0]
J. Heimerl, A. Rasputnyi, J. Pölloth, S. Meier, M. Chekhova, P. Hommelhoff, Quantum light drives electrons strongly at metal needle tips. Nature Physics. 21, 1899-1904, (2025). [DOI:10.1038/s41567-025-03087-1] [PMID]
J. Heimerl, A. Rasputnyi, J. Pölloth, S. Meier, M. Chekhova, P. Hommelhoff, Quantum light drives electrons strongly at metal needle tips. Nature Physics. 21, 1899-1904, (2025). [DOI:10.1038/s41567-025-03087-1] [PMID]
S. D. Brorson, J. G. Fujimoto, and E. P. Ippen, Femtosecond electronic heat-transport dynamics in thin gold films. Phys. Rev. Lett. 59, 1962-1965, (1987). [DOI:10.1103/PhysRevLett.59.1962] [PMID]
S. D. Brorson, J. G. Fujimoto, and E. P. Ippen, Femtosecond electronic heat-transport dynamics in thin gold films. Phys. Rev. Lett. 59, 1962-1965, (1987). [DOI:10.1103/PhysRevLett.59.1962] [PMID]
V.A. Cimmelli, D. Jou, A. Sellitto, Hydrodynamic, electronic and optic analogies with heat transport in extended thermodynamics. Journal of Non-Equilibrium Thermodynamics. 49, 2, (2024). [DOI:10.1515/jnet-2023-0096]
V.A. Cimmelli, D. Jou, A. Sellitto, Hydrodynamic, electronic and optic analogies with heat transport in extended thermodynamics. Journal of Non-Equilibrium Thermodynamics. 49, 2, (2024). [DOI:10.1515/jnet-2023-0096]
C. Su'arez, W. E. Bron, and T. Juhasz, Dynamics and transport of electronic carriers in thin gold films. Phys. Rev. Lett. 75, 4536-4539, (1995). [DOI:10.1103/PhysRevLett.75.4536] [PMID]
C. Su'arez, W. E. Bron, and T. Juhasz, Dynamics and transport of electronic carriers in thin gold films. Phys. Rev. Lett. 75, 4536-4539, (1995). [DOI:10.1103/PhysRevLett.75.4536] [PMID]
S. Giannini, J. Blumberger, Charge Transport in Organic Semiconductors: The Perspective from Nonadiabatic Molecular Dynamics. Accounts of Chemical Research. 55 (2022). [DOI:10.1021/acs.accounts.1c00675] [PMID]
S. Giannini, J. Blumberger, Charge Transport in Organic Semiconductors: The Perspective from Nonadiabatic Molecular Dynamics. Accounts of Chemical Research. 55 (2022). [DOI:10.1021/acs.accounts.1c00675] [PMID]
J.-Y. Bigot, V. Halt'e, J.-C. Merle, and A. Daunois, Electron dynamics in metallic nanoparticles. Chem. Phys. 251, 181-203, (2000). [DOI:10.1016/S0301-0104(99)00298-0]
J.-Y. Bigot, V. Halt'e, J.-C. Merle, and A. Daunois, Electron dynamics in metallic nanoparticles. Chem. Phys. 251, 181-203, (2000). [DOI:10.1016/S0301-0104(99)00298-0]
J. Zhang, B. Zhu, L. Zhanga, J. Yu, Femtosecond transient absorption spectroscopy investigation into the electron transfer mechanism in photocatalysis. Chemical communications. 6 (2023) [DOI:10.1039/D2CC06300J] [PMID]
J. Zhang, B. Zhu, L. Zhanga, J. Yu, Femtosecond transient absorption spectroscopy investigation into the electron transfer mechanism in photocatalysis. Chemical communications. 6 (2023) [DOI:10.1039/D2CC06300J] [PMID]
A. Domps, P.-G. Reinhard, and E. Suraud, Fermionic Vlasov propagation for Coulomb interacting systems. Ann. Phys. (N.Y.) 260, 171-190, (1997). [DOI:10.1006/aphy.1997.5710]
A. Domps, P.-G. Reinhard, and E. Suraud, Fermionic Vlasov propagation for Coulomb interacting systems. Ann. Phys. (N.Y.) 260, 171-190, (1997). [DOI:10.1006/aphy.1997.5710]
S. Jin, J. Jang, Non-relativistic limit of fermion ground states for the relativistic Vlasov-Poisson system. Journal of Mathematical Physics. 66, 011504, (2025). [DOI:10.1063/5.0225340]
S. Jin, J. Jang, Non-relativistic limit of fermion ground states for the relativistic Vlasov-Poisson system. Journal of Mathematical Physics. 66, 011504, (2025). [DOI:10.1063/5.0225340]
N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, Self-consistent study of the resonant-tunneling diode. Phys. Rev. B 39, 7720-7735, (1989). [DOI:10.1103/PhysRevB.39.7720] [PMID]
N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, Self-consistent study of the resonant-tunneling diode. Phys. Rev. B 39, 7720-7735, (1989). [DOI:10.1103/PhysRevB.39.7720] [PMID]
S.Y. Zhong, S.S. Zhang, X.X. Sun, M.S. Smith, Study of the deformed halo nucleus 31Ne with Glauber model based on microscopic self-consistent structures. Science China Physics, Mechanics & Astronomy. 65, 262011, (2022). [DOI:10.1007/s11433-022-1894-6]
S.Y. Zhong, S.S. Zhang, X.X. Sun, M.S. Smith, Study of the deformed halo nucleus 31Ne with Glauber model based on microscopic self-consistent structures. Science China Physics, Mechanics & Astronomy. 65, 262011, (2022). [DOI:10.1007/s11433-022-1894-6]
P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer, Vienna, 1990. [DOI:10.1007/978-3-7091-6961-2]
P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer, Vienna, 1990. [DOI:10.1007/978-3-7091-6961-2]
S. Mondal, A. Sur, A novel unified modeling of heat and moisture transport in a semiconductor containing spherical cavity. Acta Mechanica. 6 (2026). [DOI:10.1007/s00707-025-04618-w]
S. Mondal, A. Sur, A novel unified modeling of heat and moisture transport in a semiconductor containing spherical cavity. Acta Mechanica. 6 (2026). [DOI:10.1007/s00707-025-04618-w]
M. C. Yalabik, G. Neofotistos, K. Diff, H. Guo and J. D. Gunton, Quantum mechanical simulation of charge transport in very small semiconductor structures. IEEE Trans. Electron Devices 36, 1009-1013, (2002). [DOI:10.1109/16.24341]
M. C. Yalabik, G. Neofotistos, K. Diff, H. Guo and J. D. Gunton, Quantum mechanical simulation of charge transport in very small semiconductor structures. IEEE Trans. Electron Devices 36, 1009-1013, (2002). [DOI:10.1109/16.24341]
P.R. Bueno, Quantum Rate Dynamics for Coherent Electron Transport at Material/Electrolyte Interfaces. ACS Applied Materials & Interfaces. 13 (2026). [DOI:10.1021/acsami.5c25018] [PMID]
P.R. Bueno, Quantum Rate Dynamics for Coherent Electron Transport at Material/Electrolyte Interfaces. ACS Applied Materials & Interfaces. 13 (2026). [DOI:10.1021/acsami.5c25018] [PMID]
J. H. Luscombe, A. M. Bouchard and M. Luban, Electron confinement in quantum nanostructures: Self-consistent Poisson-Schr¨odinger theory. Phys. Rev. B 46, 10262-10268, (1992). [DOI:10.1103/PhysRevB.46.10262] [PMID]
J. H. Luscombe, A. M. Bouchard and M. Luban, Electron confinement in quantum nanostructures: Self-consistent Poisson-Schr¨odinger theory. Phys. Rev. B 46, 10262-10268, (1992). [DOI:10.1103/PhysRevB.46.10262] [PMID]
H. Jubair, Electronic and optical coupling in semiconductor nanostructures: A systematic review of quantum dots, nanowires, and nanoplatelets for high-efficiency solar cells. Journal of Power Sources. 669, 239394, (2026). [DOI:10.1016/j.jpowsour.2026.239394]
H. Jubair, Electronic and optical coupling in semiconductor nanostructures: A systematic review of quantum dots, nanowires, and nanoplatelets for high-efficiency solar cells. Journal of Power Sources. 669, 239394, (2026). [DOI:10.1016/j.jpowsour.2026.239394]
A. Arnold and H. Steinr¨uck, The 'electromagnetic' Wigner equation for an electron with spin. Z. Angew. Math. Phys. 40, 793-815, (1989). [DOI:10.1007/BF00945803]
A. Arnold and H. Steinr¨uck, The 'electromagnetic' Wigner equation for an electron with spin. Z. Angew. Math. Phys. 40, 793-815, (1989). [DOI:10.1007/BF00945803]
C. Etl, M. Ballicchia, M. Nedjalkov, H. Kosina, Gauge-invariant Wigner equation for electromagnetic fields:Strong and weak formulation. Physics Letters A. 565, 5, 131127, (2026). [DOI:10.1016/j.physleta.2025.131127]
C. Etl, M. Ballicchia, M. Nedjalkov, H. Kosina, Gauge-invariant Wigner equation for electromagnetic fields:Strong and weak formulation. Physics Letters A. 565, 5, 131127, (2026). [DOI:10.1016/j.physleta.2025.131127]
S. Balberg and S. L. Shapiro, The properties of matter in white dwarfs and neutron stars, astro-ph/0004317.
S. Balberg and S. L. Shapiro, The properties of matter in white dwarfs and neutron stars, astro-ph/0004317.
F. F. Chen, Introduction to plasma physics and controlled fusion, Plenum Press, New York, 1984.
F. F. Chen, Introduction to plasma physics and controlled fusion, Plenum Press, New York, 1984.
L. D. Landau and E. M. Lifshitz, Statistical Physics, part 1, Butterworth-Heinemann, Oxford, 1980. [DOI:10.1016/B978-0-08-057046-4.50008-7]
L. D. Landau and E. M. Lifshitz, Statistical Physics, part 1, Butterworth-Heinemann, Oxford, 1980. [DOI:10.1016/B978-0-08-057046-4.50008-7]
N. W. Ashcroft and N. D. Mermin, Solid state physics, Saunders College Publishing, Orlando, 1976.
N. W. Ashcroft and N. D. Mermin, Solid state physics, Saunders College Publishing, Orlando, 1976.
M. Bonitz et al., Theory and simulation of strong correlations in quantum Coulomb systems. J. Phys. A: Math. Gen. 36, 5921-5930, (2003). [DOI:10.1088/0305-4470/36/22/313]
M. Bonitz et al., Theory and simulation of strong correlations in quantum Coulomb systems. J. Phys. A: Math. Gen. 36, 5921-5930, (2003). [DOI:10.1088/0305-4470/36/22/313]
E. P. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749-759, (1932). [DOI:10.1103/PhysRev.40.749]
E. P. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749-759, (1932). [DOI:10.1103/PhysRev.40.749]
J. E. Moyal, Quantum mechanics as a statistical theory. Proc. Cambridge Phil. Soc. 45 (1949), 99-124.24 Giovanni Manfredi V. I. Tatarskii, The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26 (1983), 311-327 [Usp. Fis. Nauk. 139, 587], (1983). [DOI:10.1070/PU1983v026n04ABEH004345]
J. E. Moyal, Quantum mechanics as a statistical theory. Proc. Cambridge Phil. Soc. 45 (1949), 99-124.24 Giovanni Manfredi V. I. Tatarskii, The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26 (1983), 311-327 [Usp. Fis. Nauk. 139, 587], (1983). [DOI:10.1070/PU1983v026n04ABEH004345]
M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Distribution functions in physics: Fundamentals. Phys. Rep. 106, 121-167 , (1984). [DOI:10.1016/0370-1573(84)90160-1]
M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Distribution functions in physics: Fundamentals. Phys. Rep. 106, 121-167 , (1984). [DOI:10.1016/0370-1573(84)90160-1]
G. Manfredi and M. R. Feix, Entropy and Wigner functions. Phys. Rev. E 53, 6460-6470, (2000). [DOI:10.1103/PhysRevE.53.6460] [PMID]
G. Manfredi and M. R. Feix, Entropy and Wigner functions. Phys. Rev. E 53, 6460-6470, (2000). [DOI:10.1103/PhysRevE.53.6460] [PMID]
J.E. Drummond, Plasma Physics, McGraw- Hill, New York, 1961.
J.E. Drummond, Plasma Physics, McGraw- Hill, New York, 1961.
F. Haas, G. Manfredi, J. Goedert, Nyquist method for Wigner-Poisson quantum plasmas. Phys. Rev. E 64, 026413, (2001). [DOI:10.1103/PhysRevE.64.026413] [PMID]
F. Haas, G. Manfredi, J. Goedert, Nyquist method for Wigner-Poisson quantum plasmas. Phys. Rev. E 64, 026413, (2001). [DOI:10.1103/PhysRevE.64.026413] [PMID]
M. Bonitz, D. C. Scott, R. Binder, and S. W. Scott, Nonlinear carrier-plasmon interaction in a one-dimensional quantum plasma. Phys. Rev. B 50, 15095-15098, (1994). [DOI:10.1103/PhysRevB.50.15095] [PMID]
M. Bonitz, D. C. Scott, R. Binder, and S. W. Scott, Nonlinear carrier-plasmon interaction in a one-dimensional quantum plasma. Phys. Rev. B 50, 15095-15098, (1994). [DOI:10.1103/PhysRevB.50.15095] [PMID]
E. K. U. Gross, J. F. Dobson, and M. Petersilka, Density functional theory of time-dependent phenomena, Topics in Current Chemistry, vol. 181, Springer, Berlin, pp. 81-172, 1996.
E. K. U. Gross, J. F. Dobson, and M. Petersilka, Density functional theory of time-dependent phenomena, Topics in Current Chemistry, vol. 181, Springer, Berlin, pp. 81-172, 1996.
D. Pines, Classical and quantum plasmas. J. Nucl. Energy C 2, 5-17, (1961). [DOI:10.1088/0368-3281/2/1/301]
D. Pines, Classical and quantum plasmas. J. Nucl. Energy C 2, 5-17, (1961). [DOI:10.1088/0368-3281/2/1/301]
J. Dawson, On Landau damping. Phys. Fluids 4 , 869-874, (1961). [DOI:10.1063/1.1706419]
J. Dawson, On Landau damping. Phys. Fluids 4 , 869-874, (1961). [DOI:10.1063/1.1706419]
F. Haas, G. Manfredi and M. R. Feix, Multistream model for quantum plasmas. Phys. Rev. E 62, 2763-2772, (2000). [DOI:10.1103/PhysRevE.62.2763] [PMID]
F. Haas, G. Manfredi and M. R. Feix, Multistream model for quantum plasmas. Phys. Rev. E 62, 2763-2772, (2000). [DOI:10.1103/PhysRevE.62.2763] [PMID]
D. Anderson, B. Hall, M. Lisak, and M. Marklund, Statistical effects in the multistream model for quantum plasmas. Phys. Rev. E 65, 046417, (2002). [DOI:10.1103/PhysRevE.65.046417] [PMID]
D. Anderson, B. Hall, M. Lisak, and M. Marklund, Statistical effects in the multistream model for quantum plasmas. Phys. Rev. E 65, 046417, (2002). [DOI:10.1103/PhysRevE.65.046417] [PMID]
G. Manfredi and F. Haas, Self-consistent fluid model for a quantum electron gas. Phys. Rev. B 64, 075316, (2001). [DOI:10.1103/PhysRevB.64.075316]
G. Manfredi and F. Haas, Self-consistent fluid model for a quantum electron gas. Phys. Rev. B 64, 075316, (2001). [DOI:10.1103/PhysRevB.64.075316]
D. Deutsch. Quantum theory, the church{turing principle and the universal quantum computer. Proc. R. Soc. Lond., 1985 [DOI:10.1098/rspa.1985.0070]
D. Deutsch. Quantum theory, the church{turing principle and the universal quantum computer. Proc. R. Soc. Lond., 1985 [DOI:10.1098/rspa.1985.0070]
C. L. Gardner, Quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54, 409-427, (1994). [DOI:10.1137/S0036139992240425]
C. L. Gardner, Quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54, 409-427, (1994). [DOI:10.1137/S0036139992240425]
G. Manfredi and P.-A. Hervieux, Vlasov simulations of ultrafast electron dynamics and transport in thin metal films. Phys. Rev. B 70 (2004), 201402(R). [DOI:10.1103/PhysRevB.70.201402]
G. Manfredi and P.-A. Hervieux, Vlasov simulations of ultrafast electron dynamics and transport in thin metal films. Phys. Rev. B 70 (2004), 201402(R). [DOI:10.1103/PhysRevB.70.201402]
W. R. Frensley, Boundary conditions for open quantum systems driven far from equilibrium. Rev. Mod. Phys. 62, 745-791, (1990). [DOI:10.1103/RevModPhys.62.745]
W. R. Frensley, Boundary conditions for open quantum systems driven far from equilibrium. Rev. Mod. Phys. 62, 745-791, (1990). [DOI:10.1103/RevModPhys.62.745]
W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133-A1138, (1965). [DOI:10.1103/PhysRev.140.A1133]
W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133-A1138, (1965). [DOI:10.1103/PhysRev.140.A1133]
W. Ekardt, Work function of small metal particles: Self-consistent spherical jellium-background model. Phys. Rev. B 29, 1558-1564, (1984). [DOI:10.1103/PhysRevB.29.1558]
W. Ekardt, Work function of small metal particles: Self-consistent spherical jellium-background model. Phys. Rev. B 29, 1558-1564, (1984). [DOI:10.1103/PhysRevB.29.1558]
R. G. Parr and W. Young, Density functional theory of atoms and molecules, Oxford University Press, New York, 1989.
R. G. Parr and W. Young, Density functional theory of atoms and molecules, Oxford University Press, New York, 1989.
R. Balescu, Statistical mechanics of charged particles, John Wiley, London, 1963.
R. Balescu, Statistical mechanics of charged particles, John Wiley, London, 1963.
M. Bonitz, R. Binder, D. C. Scott, S. W. Koch, and D. Kremp, Theory of plasmons in quasi-onedimensional degenerate plasmas. Phys. Rev. E 49, 5535-5545, (1994). [DOI:10.1103/PhysRevE.49.5535] [PMID]
M. Bonitz, R. Binder, D. C. Scott, S. W. Koch, and D. Kremp, Theory of plasmons in quasi-onedimensional degenerate plasmas. Phys. Rev. E 49, 5535-5545, (1994). [DOI:10.1103/PhysRevE.49.5535] [PMID]
P. Bertrand and M. R. Feix, Nonlinear electron plasma oscillation: the" water bag model". Phys. Lett. A 28, 68-69, (1968). [DOI:10.1016/0375-9601(68)90606-3]
P. Bertrand and M. R. Feix, Nonlinear electron plasma oscillation: the" water bag model". Phys. Lett. A 28, 68-69, (1968). [DOI:10.1016/0375-9601(68)90606-3]
D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. Proceedings: Mathematical and Physical Sciences. 439, 553-558, (1992). [DOI:10.1098/rspa.1992.0167]
D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. Proceedings: Mathematical and Physical Sciences. 439, 553-558, (1992). [DOI:10.1098/rspa.1992.0167]
N. Suh, M. R. Feix, and P. Bertrand, Numerical simulation of the quantum Liouville-Poisson system. J. Comput. Phys. 94, 403-418, (1991). [DOI:10.1016/0021-9991(91)90227-C]
N. Suh, M. R. Feix, and P. Bertrand, Numerical simulation of the quantum Liouville-Poisson system. J. Comput. Phys. 94, 403-418, (1991). [DOI:10.1016/0021-9991(91)90227-C]
G. Manfredi, Long-time behavior of nonlinear landau damping. Phys. Rev. Lett. 79, 2815-2818, (1997). [DOI:10.1103/PhysRevLett.79.2815]
G. Manfredi, Long-time behavior of nonlinear landau damping. Phys. Rev. Lett. 79, 2815-2818, (1997). [DOI:10.1103/PhysRevLett.79.2815]
C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330-351, (1976). [DOI:10.1016/0021-9991(76)90053-X]
C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330-351, (1976). [DOI:10.1016/0021-9991(76)90053-X]
F. Filbet, E. Sonnendrucker, and P. Bertrand, Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172, 166-187, (2001). [DOI:10.1006/jcph.2001.6818]
F. Filbet, E. Sonnendrucker, and P. Bertrand, Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172, 166-187, (2001). [DOI:10.1006/jcph.2001.6818]
J. L. Lopez, Nonlinear Ginzburg-Landau-type approach to quantum dissipation. Phys. Rev. E 69, 026110, (2004). [DOI:10.1103/PhysRevE.69.026110] [PMID]
J. L. Lopez, Nonlinear Ginzburg-Landau-type approach to quantum dissipation. Phys. Rev. E 69, 026110, (2004). [DOI:10.1103/PhysRevE.69.026110] [PMID]
A. O. Caldeira and A. J. Leggett, Path integral approach to quantum Brownian motion. Physica A 121, 587-616, (1983). [DOI:10.1016/0378-4371(83)90013-4]
A. O. Caldeira and A. J. Leggett, Path integral approach to quantum Brownian motion. Physica A 121, 587-616, (1983). [DOI:10.1016/0378-4371(83)90013-4]
L. Diosi, Calderia-Leggett master equation and medium temperatures. Physica A 199, 517-526, (1993). [DOI:10.1016/0378-4371(93)90065-C]
L. Diosi, Calderia-Leggett master equation and medium temperatures. Physica A 199, 517-526, (1993). [DOI:10.1016/0378-4371(93)90065-C]
W. H. Zurek, S. Habib, and J. P. Paz, Coherent states via decoherence. Phys. Rev. Lett. 70, 1187-1190, (1993). [DOI:10.1103/PhysRevLett.70.1187] [PMID]
W. H. Zurek, S. Habib, and J. P. Paz, Coherent states via decoherence. Phys. Rev. Lett. 70, 1187-1190, (1993). [DOI:10.1103/PhysRevLett.70.1187] [PMID]
W. Ebeling, A. F¨orstery, H. Hessz and M. Yu. Romanovsky, Thermodynamic and kinetic properties of hot nonideal plasmas. Plasma Phys. Control. Fusion 38, A31-A47, (1996). [DOI:10.1088/0741-3335/38/12A/004]
W. Ebeling, A. F¨orstery, H. Hessz and M. Yu. Romanovsky, Thermodynamic and kinetic properties of hot nonideal plasmas. Plasma Phys. Control. Fusion 38, A31-A47, (1996). [DOI:10.1088/0741-3335/38/12A/004]
S. Lloyd. Universal quantum simulators. Science. 273 , 1073-1078, (1996). [DOI:10.1126/science.273.5278.1073] [PMID]
S. Lloyd. Universal quantum simulators. Science. 273 , 1073-1078, (1996). [DOI:10.1126/science.273.5278.1073] [PMID]
B. Shokri and A. A. Rukhadze, Quantum surface wave on a thin plasma layer. Phys. Plasmas 6, 3450-3454, (1999). [DOI:10.1063/1.873604]
B. Shokri and A. A. Rukhadze, Quantum surface wave on a thin plasma layer. Phys. Plasmas 6, 3450-3454, (1999). [DOI:10.1063/1.873604]
B. Shokri and A. A. Rukhadze, Quantum drift waves. Phys. Plasmas 6, 4467-4471, (1999). [DOI:10.1063/1.873733]
B. Shokri and A. A. Rukhadze, Quantum drift waves. Phys. Plasmas 6, 4467-4471, (1999). [DOI:10.1063/1.873733]
L. Tian, R. Blatt, and P. Zoller. Scalable ion trap quantum computing without moving ions. The European Physical Journal D. 32, 201-208, (2004). [DOI:10.1140/epjd/e2004-00172-5]
L. Tian, R. Blatt, and P. Zoller. Scalable ion trap quantum computing without moving ions. The European Physical Journal D. 32, 201-208, (2004). [DOI:10.1140/epjd/e2004-00172-5]
N. Topaloglu, A. Aksekili, B. Yegin, a displacement dependent damper for a vibration-based energy harvester. Conference: SET2011, 10th International Conference on Sustainable Energy TechnologiesAt: Istanbul, Turkey.
N. Topaloglu, A. Aksekili, B. Yegin, a displacement dependent damper for a vibration-based energy harvester. Conference: SET2011, 10th International Conference on Sustainable Energy TechnologiesAt: Istanbul, Turkey.

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