Phương trình chính cho các hàm Wigner với sự sụp đổ ngẫu nhiên và mối quan hệ của chúng với tính không thể đảo ngược nhiệt động lực học

Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932) Weinbub, J., Ferry, D.K.: Recent advances in Wigner function approaches. Appl. Phys. Rev. 5, 041104 (2018) Ferry, D.K., Nedjalkov, M., Weinbub, J., Ballicchia, M., Welland, I., Selberherr, S.: Complex systems in phase space. Entropy 22, 1103 (2020) Ferry, D.K., Nedjalkov, M.: The Wigner Function in Science and Technology. IOP Publishing, Bristol (2018) Rundle, R.P., Everitt, M.J.: Overview of the phase space formulation of quantum mechanics with application to quantum technologies. Adv. Quantum Technol. 4, 2100016 (2021) Weiss, C., Cornish, S.L., Gardiner, S.A., Breuer, H.-P.: Superballistic center-of-mass motion in one-dimensional attractive Bose gases: decoherence-induced Gaussian random walks in velocity space. Phys. Rev. A 93, 013605 (2016) Schulte, C.H.H., Hansom, J., Jones, A.E., Matthiesen, C., Le Gall, C., Atatüre, M.: Quadrature squeezed photons from a two-level system. Nature 525, 222–225 (2015) Schleich, W.P.: Quantum Optics in Phase Space. Wiley, Berlin (2011) Olivares, S.: Quantum optics in the phase space. Eur. Phys. J. Spec. Top. 203, 3–24 (2012) Rundle, R.P., Davies, B.I., Dwyer, V.M., Tilma, T., Everitt, M.J.: Visualization of correlations in hybrid discrete-continuous variable quantum systems. J. Phys. Commun. 4, 025002 (2020) Ferraro, D., Feller, A., Ghibaudo, A., Thibierge, E., Bocquillon, E., Fève, G., Grenier, C., Degiovanni, P.: Wigner function approach to single electron coherence in quantum Hall edge channels. Phys. Rev. B 88, 205303 (2013) Kim, K.-Y., Kim, S., Tang, T.-W.: Accuracy balancing for the finite-difference-based solution of the discrete Wigner transport equation. J. Comput. Electron. 16, 148–154 (2017) Ferry, D.K., Welland, I.: Relativistic Wigner functions in transition metal dichalcogenides. J. Comput. Electron. 17, 110–117 (2018) Spisak, B.J., Wołoszyn, M., Szydłowski, D.: Dynamical localisation of conduction electrons in one-dimensional disordered systems. J. Comput. Electron. 14, 916–921 (2015) Groll, D., Hahn, T., Machnikowski, P., Wigger, D., Kuhn, T.: Controlling photoluminescence spectra of hBN color centers by selective phonon-assisted excitation: a theoretical proposal. Mater. Quantum Technol. 1, 015004 (2021) Hahn, T., Wigger, D., Kuhn, T.: Entropy dynamics of phonon quantum states generated by optical excitation of a two-level system. Entropy 22, 286 (2020) Hahn, T., Groll, D., Kuhn, T., Wigger, D.: Influence of excited state decay and dephasing on phonon quantum state preparation. Phys. Rev. B 100, 024306 (2019) Wigger, D., Gehring, H., Axt, V.M., Reiter, D.E., Kuhn, T.: Quantum dynamics of optical phonons generated by optical excitation of a quantum dot. J. Comput. Electron. 15, 1158–1169 (2016) Jacoboni, C., Bordone, P.: The Wigner-function approach to non-equilibrium electron transport. Rep. Prog. Phys. 67, 1033–1071 (2004) Zurek, W.H., Paz, J.P.: Quantum chaos: a decoherent definition. Physica D 83, 300–308 (1995) Cancellieri, E., Bordone, P., Jacoboni, C.: Effect of symmetry in the many-particle Wigner function. Phys. Rev. B 76, 214301 (2007) Rundle, R.P., Tilma, T., Samson, J.H., Dwyer, V.M., Bishop, R.F., Everitt, M.J.: General approach to quantum mechanics as a statistical theory. Phys. Rev. A 99, 012115 (2019) Nedjalkov, M., Weinbub, J., Ballicchia, M., Selberherr, S., Dimov, I., Ferry, D.: Wigner equation for general electromagnetic fields: the Weyl–Stratonovich transform. Phys. Rev. B 99, 014423 (2019) Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124, 1041–1104 (2006) Sels, D., Brosens, F.: Wigner distribution functions for complex dynamical systems: the emergence of the Wigner-Boltzmann equation. Phys. Rev. E 88, 042101 (2013) Bondarev, B.V.: Quantum Markovian master equation theory of particle migration in a stochastic medium. Phys. A 183, 159–174 (1992) Jüngel, A., López, J.L., Montejo-Gàmez, J.: A new derivation of the quantum Navier-Stokes equations in the Wigner–Fokker–Planck-approach. J. Stat. Phys. 145, 1661–1673 (2011) Arnold, A., López, J.L., Markowich, P.A., Soler, J.: An analysis of quantum Fokker-Planck models: a Wigner function approach. Rev. Mat. Iberoam. 20, 771–814 (2004) Wallace, D.: Probability and irreversibility in modern statistical mechanics: classical and quantum. arXiv:2104.11223 (2021), forthcoming in D. Bedingham, O. Maroney and C. Timpson (eds.), Quantum Foundations of Statistical Mechanics (Oxford University Press, Oxford) Habib, S., Jacobs, K., Mabuchi, H., Ryne, R., Shizume, K., Sundaram, B.: Quantum-classical transition in nonlinear dynamical systems. Phys. Rev. Lett. 88, 040402 (2002) Jasiak, R., Manfredi, G., Hervieux, P.-A., Haefele, M.: Quantum-classical transition in the electron dynamics of thin metal films. New J. Phys. 11, 063042 (2009) te Vrugt, M., Wittkowski, R.: Orientational order parameters for arbitrary quantum systems. Ann. Phys. (Berl.) 532, 2000266 (2020) te Vrugt, M., Löwen, H., Wittkowski, R.: Classical dynamical density functional theory: from fundamentals to applications. Adv. Phys. 69, 121–247 (2020) Burghardt, I., Parlant, G.: On the dynamics of coupled Bohmian and phase-space variables: a new hybrid quantum-classical approach. J. Chem. Phys. 120, 3055–3058 (2004) Burghardt, I., Bagchi, B.: On the non-adiabatic dynamics of solvation: a molecular hydrodynamic formulation. Chem. Phys. 329, 343–356 (2006) Schrödinger, E.: Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften 23, 807–812 (1935) Friebe, C., Kuhlmann, M., Lyre, H., Näger, P.M., Passon, O., Stöckler, M.: The Philosophy of Quantum Physics. Springer, Wiesbaden (2018) Ghirardi, G.C., Rimini, A., Weber, T.: Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34, 470–491 (1986) Ghirardi, G.C., Pearle, P., Rimini, A.: Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles. Phys. Rev. A 42, 78–89 (1990) Ghirardi, G.C., Bassi, A.: Collapse theories. In: Zalta E.N. (eds.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University (2020) Ghirardi, G.C., Grassi, R., Benatti, F.: Describing the macroscopic world: closing the circle within the dynamical reduction program. Found. Phys. 25, 5–38 (1995) Bassi, A., Lochan, K., Satin, S., Singh, T.P., Ulbricht, H.: Models of wave-function collapse, underlying theories, and experimental tests. Rev. Mod. Phys. 85, 471–527 (2013) Adler, S.L.: Lower and upper bounds on CSL parameters from latent image formation and IGM heating. J. Phys. A: Math. Theor. 40, 2935 (2007) Vinante, A., Carlesso, M., Bassi, A., Chiasera, A., Varas, S., Falferi, P., Margesin, B., Mezzena, R., Ulbricht, H.: Narrowing the parameter space of collapse models with ultracold layered force sensors. Phys. Rev. Lett. 125, 100404 (2020) Donadi, S., Piscicchia, K., Curceanu, C., Diosi, L., Laubenstein, M., Bassi, A.: Underground test of gravity-related wave function collapse. Nat. Phys. 17, 74–78 (2020) Zheng, D., Leng, Y., Kong, X., Li, R., Wang, Z., Luo, X., Zhao, J., Duan, C.-K., Huang, P., Du, J., Carlesso, M., Bassi, A.: Room temperature test of the continuous spontaneous localization model using a levitated micro-oscillator. Phys. Rev. Res. 2, 013057 (2020) Schrinski, B., Stickler, B.A., Hornberger, K.: Collapse-induced orientational localization of rigid rotors. J. Opt. Soc. Am. B 34, C1–C7 (2017) Das, S., Lochan, K., Sahu, S., Singh, T.P.: Quantum to classical transition of inflationary perturbations: continuous spontaneous localization as a possible mechanism. Phys. Rev. D 88, 085020 (2013) Garraway, B.M., Knight, P.L.: Comparison of quantum-state diffusion and quantum-jump simulations of two-photon processes in a dissipative environment. Phys. Rev. A 49, 1266–1274 (1994) Buffa, M., Nicrosini, O., Rimini, A.: Dissipation and reduction effects of spontaneous localization on superconducting states. Found. Phys. Lett. 8, 105–125 (1995) Rae, A.I.M.: Can GRW theory be tested by experiments on SQUIDS? J. Phys. A: Math. Gen. 23, L57–L60 (1990) Tóth, G. I.: Time-irreversibility in the classical many-body system in the hydrodynamic limit. arXiv:2009.03089v4 (2021) te Vrugt, M.: The five problems of irreversibility. Stud. Hist. Philos. Sci. 87, 136–146 (2021) Albert, D.Z.: The foundations of quantum mechanics and the approach to thermodynamic equilibrium. Br. J. Philos. Sci. 45, 669–677 (1994) Albert, D.Z.: The foundations of quantum mechanics and the approach to thermodynamic equilibrium. Erkenntnis 41, 191–206 (1994) Albert, D.Z.: Time and Chance. Harvard University Press, Cambridge (2000) Hemmo, M., Shenker, O.: Can we explain thermodynamics by quantum decoherence? Stud. Hist. Philos. Mod. Phys. 32, 555–568 (2001) Hemmo, M., Shenker, O.: Quantum decoherence and the approach to equilibrium. Philos. Sci. 70, 330–358 (2003) Hemmo, M., Shenker, O.: Quantum decoherence and the approach to equilibrium (II). Stud. Hist. Philos. Mod. Phys. 36, 626–648 (2005) Price, H.: Boltzmann’s time bomb. Br. J. Philos. Sci. 53, 83–119 (2002) Monton, B.: The problem of ontology for spontaneous collapse theories. Stud. Hist. Philos. Mod. Phys. 35, 407–421 (2004) North, J.: Time in thermodynamics. In: Callender, C. (ed.) The Oxford Handbook of Philosophy of Time, pp. 312–350. Oxford University Press, Oxford (2011) Pearle, P.: Combining stochastic dynamical state-vector reduction with spontaneous localization. Phys. Rev. A 39, 2277–2289 (1989) Penrose, R.: On the gravitization of quantum mechanics 1: quantum state reduction. Found. Phys. 44, 557–575 (2014) Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relat. Gravit. 28, 581–600 (1996) Diósi, L.: A universal master equation for the gravitational violation of quantum mechanics. Phys. Lett. A 120, 377–381 (1987) Diósi, L.: Models for universal reduction of macroscopic quantum fluctuations. Phys. Rev. A 40, 1165–1174 (1989) Smirne, A., Vacchini, B., Bassi, A.: Dissipative extension of the Ghirardi–Rimini–Weber model. Phys. Rev. A 90, 062135 (2014) Smirne, A., Bassi, A.: Dissipative continuous spontaneous localization (CSL) model. Sci. Rep. 5, 12518 (2015) Tilma, T., Everitt, M.J., Samson, J.H., Munro, W.J., Nemoto, K.: Wigner functions for arbitrary quantum systems. Phys. Rev. Lett. 117, 180401 (2016) Case, W.B.: Wigner functions and Weyl transforms for pedestrians. Am. J. Phys. 76, 937–946 (2008) Gneiting, C., Fischer, T., Hornberger, K.: Quantum phase-space representation for curved configuration spaces. Phys. Rev. A 88, 062117 (2013) Hillery, M., O’Connell, R.F., Scully, M.O., Wigner, E.P.: Distribution functions in physics: fundamentals. Phys. Rep. 106, 121–167 (1984) Groenewold, H.J.: On the principles of elementary quantum mechanics. Physica 12, 405–460 (1946) Fairlie, D.B.: Moyal brackets, star products and the generalised Wigner function. Chaos Solitons Fractals 10, 365–371 (1999) Moyal, J.E.: Quantum mechanics as a statistical theory. Math. Proc. Camb. Philos. Soc. 45, 99–124 (1949) Jacquod, P., Petitjean, C.: Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom. Adv. Phys. 58, 671–96 (2009) Lee, H.-W.: Theory and application of the quantum phase-space distribution functions. Phys. Rep. 259, 147–211 (1995) Schlosshauer, M.: Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 76, 1267–1305 (2005) Wigner, E.P.: The problem of measurement. Am. J. Phys. 31, 6–15 (1963) Wallace, D.: The Emergent Multiverse: Quantum Theory According to the Everett Interpretation. Oxford University Press, Oxford (2012) Albert, D.Z.: Quantum Mechanics and Experience. Harvard University Press, Cambridge (1992) Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables I. Phys. Rev. 85, 166–179 (1952) Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables II. Phys. Rev. 85, 180–193 (1952) Hossenfelder, S., Palmer, T.: Rethinking superdeterminism. Front. Phys. 8, 139 (2020) Hossenfelder, S.: Superdeterminism: a guide for the perplexed. arXiv:2010.01324 (2020) Lombardi, O., Dieks, D.: Modal interpretations of quantum mechanics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University (2017) Everett, H., III.: “Relative state” formulation of quantum mechanics. Rev. Mod. Phys. 29, 454–462 (1957) Adler, S.L.: Why decoherence has not solved the measurement problem: a response to P.W Anderson. Stud. Hist. Philos. Mod. Phys. 34, 135–142 (2003) Bassi, A., Ghirardi, G.C.: Dynamical reduction models. Phys. Rep. 379, 257–426 (2003) Pearle, P., Squires, E.: Bound state excitation, nucleon decay experiments and models of wave function collapse. Phys. Rev. Lett. 73, 1–5 (1994) Feldmann, W., Tumulka, R.: Parameter diagrams of the GRW and CSL theories of wavefunction collapse. J. Phys. A: Math. Theor. 45, 065304 (2012) Bahrami, M., Smirne, A., Bassi, A.: Role of gravity in the collapse of a wave function: a probe into the Diósi-Penrose model. Phys. Rev. A 90, 062105 (2014) Bassi, A., Ghirardi, G.C.: Dynamical reduction models with general Gaussian noises. Phys. Rev. A 65, 042114 (2002) Tumulka, R.: A relativistic version of the Ghirardi Rimini–Weber model. J. Stat. Phys. 125, 821–840 (2006) Hanggi, P., Grabert, H., Talkner, P., Thomas, H.: Bistable systems: master equation versus Fokker–Planck modeling. Phys. Rev. A 29, 371–378 (1984) Gaveau, B., Moreau, M., Toth, J.: Master equation and Fokker–Planck equation: comparison of entropy and of rate constants. Lett. Math. Phys. 40, 101–115 (1997) Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715–775 (2003) Stickler, B.A., Schrinski, B., Hornberger, K.: Rotational friction and diffusion of quantum rotors. Phys. Rev. Lett. 121, 040401 (2018) Frensley, W.R.: Wigner-function model of a resonant-tunneling semiconductor device. Phys. Rev. B 36, 15701580 (1987) Pearle, P.: Collapse miscellany. In: Struppa, D.C., Tollaksen, J.M. (eds.) Quantum Theory: A Two-Time Success Story, pp. 131–156. Springer, Milan (2014) Näger, P. M.: Evidence for interactive common causes. Resuming the Cartwright–Hausman–Woodward debate, preprint (2021). Available at http://philsci-archive.pitt.edu/19531/ Ghirardi, G.C., Grassi, R., Rimini, A.: Continuous-spontaneous reduction model involving gravity. Phys. Rev. A 42, 1057–1064 (1990) Howl, R., Vedral, V., Naik, D., Christodoulou, M., Rovelli, C., Iyer, A.: Non-Gaussianity as a signature of a quantum theory of gravity. PRX Quantum 2, 010325 (2021) Toroš, M., Bassi, A.: Bounds on quantum collapse models from matter-wave interferometry: calculational details. J. Phys. A: Math. Theor. 51, 115302 (2018) Risken, H.: The Fokker–Planck Equation: Methods of Solution and Applications. Springer Series in Synergetics, vol. 18, 3rd edn. Springer, Berlin (1996) Frigg, R.: A field guide to recent work on the foundations of statistical mechanics. In: Rickles, D. (ed.) The Ashgate Companion to Contemporary Philosophy of Physics, pp. 99–196. Ashgate, Aldershot (2008) Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitz.-Ber. Akad. Wiss. Wien (II) 66, 275–370 (1872) Brown, H.R., Myrvold, W., Uffink, J.: Boltzmann’s H-theorem, its discontents, and the birth of statistical mechanics. Stud. Hist. Philos. Mod. Phys. 40, 174–191 (2009) Nedjalkov, M., Kosina, H., Selberherr, S., Ringhofer, C., Ferry, D.K.: Unified particle approach to Wigner–Boltzmann transport in small semiconductor devices. Phys. Rev. B 70, 115319 (2004) D. de las Heras, J. M. Brader, A. Fortini, and M. Schmidt,: Particle conservation in dynamical density functional theory. J. Phys. Condens. Matter 28, 244024 (2016) Tilloy, A., Stace, T.M.: Neutron star heating constraints on wave-function collapse models. Phys. Rev. Lett. 123, 080402 (2019) te Vrugt, M.: The mereology of thermodynamic equilibrium. Synthese (2021). https://doi.org/10.1007/s11229-021-03359-2 Archer, A.J., Rauscher, M.: Dynamical density functional theory for interacting Brownian particles: stochastic or deterministic? J. Phys. A: Math. Gen. 37, 9325–9333 (2004) Dean, D.S.: Langevin equation for the density of a system of interacting Langevin processes. J. Phys. A: Math. Gen. 29, L613–L617 (1996) Nakamura, T., Yoshimori, A.: Derivation of the nonlinear fluctuating hydrodynamic equation from the underdamped Langevin equation. J. Phys. A: Math. Theor. 42, 065001 (2009) Ridderbos, T.M., Redhead, M.L.G.: The spin-echo experiments and the second law of thermodynamics. Found. Phys. 28, 1237–1270 (1998) Micadei, K., Peterson, J.P., Souza, A.M., Sarthour, R.S., Oliveira, I.S., Landi, G.T., T. B. Batalh ao, R. M. Serra, E. Lutz: Reversing the direction of heat flow using quantum correlations. Nat. Commun. 10, 2456 (2019) Vacchini, B.: On the precise connection between the GRW master equation and master equations for the description of decoherence. J. Phys. A: Math. Theor. 40, 2463–2473 (2007) Narnhofer, H., Wreszinski, W.F.: On reduction of the wave-packet, decoherence, irreversibility and the second law of thermodynamics. Phys. Rep. 541, 249–278 (2014) Stamp, P.C.E.: Environmental decoherence versus intrinsic decoherence. Philos. Trans. R. Soc. A 370, 4429–4453 (2012) Linden, N., Popescu, S., Short, A.J., Winter, A.: Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009) Zurek, W.H., Paz, J.P.: Decoherence, chaos, and the second law. Phys. Rev. Lett. 72, 2508–2511 (1994) Ridderbos, K.: The coarse-graining approach to statistical mechanics: how blissful is our ignorance? Stud. Hist. Philos. Mod. Phys. 33, 65–77 (2002) Blatt, J.M.: An alternative approach to the ergodic problem. Prog. Theoret. Phys. 22, 745–756 (1959) Zurek, W.H., Habib, S., Paz, J.P.: Coherent states via decoherence. Phys. Rev. Lett. 70, 1187–1190 (1993) Bacciagaluppi, G., Dickson, M.: Dynamics for modal interpretations. Found. Phys. 29, 1165–1201 (1999) Caldeira, A.O., Leggett, A.J.: Path integral approach to quantum Brownian motion. Phys. A 121, 587–616 (1983) Halliwell, J.J.: Two derivations of the master equation of quantum Brownian motion. J. Phys. A: Math. Theor. 40, 3067 (2007) Tegmark, M.: Apparent wave function collapse caused by scattering. Found. Phys. Lett. 6, 571–590 (1993) Drossel, B.: What condensed matter physics and statistical physics teach us about the limits of unitary time evolution. Quantum Stud. Math. Found. 7, 217–231 (2020) Drossel, B.: Ten reasons why a thermalized system cannot be described by a many-particle wave function. Stud. Hist. Philos. Mod. Phys. 58, 12–21 (2017) Drossel, B..: On the relation between the second law of thermodynamics and classical and quantum mechanics. In: Falkenburg, B., Morrison, M. (eds.) Why More Is Different: Philosophical Issues in Condensed Matter Physics and Complex Systems, pp. 41–54. Springer, Berlin (2015) Drossel, B., Ellis, G.: Contextual wavefunction collapse: an integrated theory of quantum measurement. New J. Phys. 20, 113025 (2018) Hahn, E.L.: Spin echoes. Phys. Rev. 80, 580–594 (1950) Orban, J., Bellemans, A.: Velocity-inversion and irreversibility in a dilute gas of hard disks. Phys. Lett. A 24, 620–621 (1967) Komatsu, N., Abe, T.: Numerical irreversibility in time-reversible molecular dynamics simulation. Physica D 195, 391–397 (2004) Komatsu, N., Abe, T.: Noise-driven numerical irreversibility in molecular dynamics technique. Comput. Phys. Commun. 171, 187–196 (2005) Hammonds, K.D., Heyes, D.M.: Shadow Hamiltonian in classical NVE molecular dynamics simulations: a path to long time stability. J. Chem. Phys. 152, 024114 (2020) Hansen, J.-P., McDonald, I.R.: Theory of Simple Liquids, 3rd edn. Elsevier Academic Press, Oxford (2006) Lide, D.R.: CRC Handbook of Chemistry and Physics, vol. 84. CRC Press, Boca Raton (2004) Birdsall, C.K., Fuss, D.: Clouds-in-clouds, clouds-in-cells physics for many-body plasma simulation. J. Comput. Phys. 3, 494–511 (1969)