Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism

Albert, D Z (1994a). The foundations of quantum mechanics and the approach to thermodynamic equilibrium. British Journal for the Philosophy of Science, 45(2), 669–677. Albert, D Z (1994b). The foundations of quantum mechanics and the approach to thermodynamic equilibrium. Erkenntnis, 41(2), 191–206. Albert, D Z. (2000). Time and chance. Harvard University Press. Anero, J G, Español, P, & Tarazona, P (2013). Functional thermo-dynamics: A generalization of dynamic density functional theory to non-isothermal situations. Journal of Chemical Physics, 139(3), 034106. Archer, A J, & Rauscher, M (2004). Dynamical density functional theory for interacting Brownian particles: Stochastic or deterministic? Journal of Physics A: Mathematical and General, 37(40), 9325. Balian, R, & Vénéroni, M (1985). Time-dependent variational principle for the expectation value of an observable: Mean-field applications. Annals of Physics, 164(2), 334–410. Bonetto, F, Lebowitz, J L, & Rey-Bellet, L. (2000). Fouriers law: A challenge to theorists. In A. Fokas, A. Grigoryan, T. Kibble, & B. Zegarlinski (Eds.), Mathematical physics 2000 (pp. 128150). Imperial College Press. Bricmont, J, & Kupiainen, A (2007). Towards a derivation of Fourier’s law for coupled anharmonic oscillators. Communications in Mathematical Physics, 274(3), 555–626. Brown, HR. (2017). Once and for all: The curious role of probability in the Past Hypothesis. In D. Bedingham, & O. Maroney (Eds.), Quantum foundations of statistical mechanics. Oxford University Press. http://philsci-archive.pitt.edu/13008/ Brown, H R, & Uffink, J (2001). The origins of time-asymmetry in thermodynamics: The minus first law. Studies in History and Philosophy of Modern Physics, 32(4), 525–538. Chandy, A J, & Frankel, S H (2010). The t-model as a large eddy simulation model for the Navier–Stokes equations. Multiscale Modeling & Simulation, 8(2), 445–462. Chen, E K (2021). Quantum mechanics in a time-asymmetric universe: On the nature of the initial quantum state. British Journal for the Philosophy of Science, 72(4), 1155–1183. Chorin, A J, Hald, O H, & Kupferman, R. (2002). Optimal prediction with memory. Physica D: Nonlinear Phenomena, 166(3-4), 239–257. Chua, E Y S (2021). Does von Neumann entropy correspond to thermodynamic entropy? Philosophy of Science, 88(1), 145–168. Clarkson, C, Ellis, G, Larena, J, & Umeh, O. (2011). Does the growth of structure affect our dynamical models of the universe? The averaging, backreaction, and fitting problems in cosmology. Reports on Progress in Physics, 74 (11), 112901. Das, S P (2004). Mode-coupling theory and the glass transition in supercooled liquids. Reviews of Modern Physics, 76(3), 785–851. Dhar, A, & Spohn, H (2019). Fourier’s law based on microscopic dynamics. Comptes Rendus Physique, 20(5), 393–401. Español, P, & Löwen, H (2009). Derivation of dynamical density functional theory using the projection operator technique. Journal of Chemical Physics, 131(24), 244101. Farr, M. (2021). What’s so special about initial conditions? Understanding the past hypothesis in directionless time. In Y. Ben-Menahem (Ed.), Rethinking laws of nature. Springer. http://philsci-archive.pitt.edu/19905/ Frigg, R. (2008). A field guide to recent work on the foundations of statistical mechanics. In D. Rickles (Ed.), The Ashgate companion to contemporary philosophy of physics (pp. 99–196). Ashgate . Frigg, R. (2016). Chance and determinism. In A. Hájek, & C. Hitchcock (Eds.), The Oxford handbook of probability and philosophy (pp. 460–474). Oxford University Press. Frigg, R, & Hoefer, C (2015). The best Humean system for statistical mechanics. Erkenntnis, 80(3), 551–574. Frisch, M (2005a). Counterfactuals and the past hypothesis. Philosophy of Science, 72(5), 739–750. Frisch, M. (2005b). Inconsistency, asymmetry, and non-locality: A philosophical investigation of classical electrodynamics. Oxford University Press. Frisch, M (2006). A tale of two arrows. Studies in History and Philosophy of Modern Physics, 37(3), 542–558. Fuchizaki, K, & Kawasaki, K (2002). Dynamical density functional theory for glassy behaviour. Journal of Physics: Condensed Matter, 14(46), 12203–12222. Fulde, P. (1995). Electron correlations in molecules and solids. Springer. Ghirardi, G C, Rimini, A, & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34(2), 470–491. Gibbs, J W. (1902). Elementary principles in statistical mechanics: Developed with especial reference to the rational foundation of thermodynamics. Scribner’s sons. Götze, W (1998). The essentials of the mode-coupling theory for glassy dynamics. Condensed Matter Physics, 1(4), 873–904. Grabert, H (1978). Nonlinear transport and dynamics of fluctuations. Journal of Statistical Physics, 19(5), 479–497. Grabert, H. (1982). Projection operator techniques in nonequilibrium statistical mechanics. Springer tracts in modern physics (vol. 95, 1st edn.). Springer. Hahn, E L (1950). Spin echoes. Physical Review, 80(4), 580. Hájek, A. (2019). Interpretations of probability. In E.N. Zalta (Ed.), The Stanford encyclopedia of philosophy (fall 2019 edn.). Metaphysics Research Lab: Stanford University. Han, M, Fruchart, M, Scheibner, C, Vaikuntanathan, S, de Pablo, J J, & Vitelli, V. (2021). Fluctuating hydrodynamics of chiral active fluids. Nature Physics, 17(11), 1260–1269. Haussmann, R. (2022). Microscopic density-functional approach to nonlinear elasticity theory. Journal of Statistical Mechanics: Theory and Experiment, 2022, 053210. Hemmo, M, & Shenker, O (2006). Von Neumann’s entropy does not correspond to thermodynamic entropy. Philosophy of Science, 73(2), 153–174. Henderson, L (2003). The von Neumann entropy: A reply to Shenker. British Journal for the Philosophy of Science, 54(2), 291–296. Huang, X, Kodama, T, Koide, T, & Rischke, D. H. (2011). Bulk viscosity and relaxation time of causal dissipative relativistic fluid dynamics. Physical Review C, 83(2), 024906. Janssen, L (2018). Mode-coupling theory of the glass transition: A primer. Frontiers in Physics, 6, 97. Jaynes, E T (1975a). Information theory and statistical mechanics. Physical Review, 106, 620–630. Jaynes, E T (1975b). Information theory and statistical mechanics. ii. Physical Review, 108, 171–190. Kawasaki, K (2006). Interpolation of stochastic and deterministic reduced dynamics. Physica A: Statistical Mechanics and its Applications, 362 (2), 249–260. Kawasaki, K (2009). A mini-review of structural glasses—a personal view—. Forma, 24(1), 3–9. Kawasaki, K, & Gunton, J D (1973). Theory of nonlinear transport processes: Nonlinear shear viscosity and normal stress effects. Physical Review A, 8, 2048–2064. Klippenstein, V, Tripathy, M, Jung, G, Schmid, F, & van der Vegt, N. F. A. (2021). Introducing memory in coarse-grained molecular simulations. Journal of Physical Chemistry B, 125(19), 4931–4954. La Caze, A. (2016). Frequentism. In A. Hájek & C. Hitchcock (Eds.), The Oxford handbook of probability and philosophy (pp. 341–359). Oxford University Press. Löwen, H (1994). Melting, freezing and colloidal suspensions. Physics Reports, 237(5), 249–324. Luczak, J (2016). On how to approach the approach to equilibrium. Philosophy of Science, 83(3), 393–411. Luczak, J (2018). How many aims are we aiming at? Analysis, 78(2), 244–254. Maeyama, S, & Watanabe, T H (2020). Extracting and modeling the effects of small-scale fluctuations on large-scale fluctuations by Mori–Zwanzig projection operator method. Journal of the Physical Society of Japan, 89(2), 024401. Marini Bettolo Marconi, U, & Tarazona, P (1999). Dynamic density functional theory of fluids. Journal of Chemical Physics, 110(16), 8032–8044. Menzel, A M, Saha, A, Hoell, C, & Löwen, H. (2016). Dynamical density functional theory for microswimmers. Journal of Chemical Physics, 144 (2), 024115. Meyer, H, Voigtmann, T, & Schilling, T. (2017). On the non-stationary generalized Langevin equation. Journal of Chemical Physics, 147(21), 214110. Meyer, H, Voigtmann, T, & Schilling, T. (2019). On the dynamics of reaction coordinates in classical, time-dependent, many-body processes. Journal of Chemical Physics, 150(17), 174118. Micadei, K, Peterson, J P S, Souza, A M, Sarthour, R S, Oliveira, I S, Landi, G T, Batalhão, T B, Serra, R M, & Lutz, E. (2019). Reversing the direction of heat flow using quantum correlations. Nature Communications, 10(1), 2456. Michel, M, Mahler, G, & Gemmer, J. (2005). Fourier’s law from Schrödinger dynamics. Physical Review Letters, 95(18), 180602. Michel, M, Gemmer, J, & Mahler, G. (2006). Microscopic quantum mechanical foundation of Fourier’s law. International Journal of Modern Physics B, 20(29), 4855–4883. Mori, H (1965). Transport, collective motion, and Brownian motion. Progress of Theoretical Physics, 33(3), 423–455. Myrvold, W. (2016). Probabilities in statistical mechanics. In A. Hajek, & C. Hitchcock (Eds.), The Oxford handbook of probability and philosophy (pp. 573–600). Oxford University Press. Myrvold, WC. (2011). Probabilities in statistical mechanics: Objective, subjective, or a bit of both? http://philsci-archive-dev.library.pitt.edu/8642/ Myrvold, W C. (2012). Deterministic laws and epistemic chances. In Y. Ben-Menahem, & M. Hemmo (Eds.), Probability in physics (pp. 73–85). Springer. Myrvold, W C (2020). Explaining thermodynamics: What remains to be done? In V. Allori (Ed.), Statistical mechanics and scientific explanation (pp. 113–143). World Scientific. Myrvold, W C. (2021). Beyond chance and credence: A theory of hybrid probabilities. Oxford University Press. Nakajima, S (1958). On quantum theory of transport phenomena: Steady diffusion. Progress of Theoretical Physics, 20(6), 948–959. Orlandini, S, Meloni, S, & Ciccotti, G. (2011). Hydrodynamics from Statistical mechanics: Combined dynamical-NEMD and conditional sampling to relax an interface between two immiscible liquids. Physical Chemistry Chemical Physics, 13(29), 13177–13181. Parish, E J, & Duraisamy, K (2017). Non-Markovian closure models for large eddy simulations using the mori-Zwanzig formalism. Physical Review Fluids, 2(1), 014604. Penrose, O. (1970). Foundations of statistical mechanics: A deductive treatment. Pergamon Press. Ras, T, Szafarczyk, M, & Fuchs, M. (2020). Elasticity of disordered binary crystals. Colloid and Polymer Science, 298, 803–818. Rau, J, & Müller, B (1996). From reversible quantum microdynamics to irreversible quantum transport. Physics Reports, 272(1), 1–59. Redhead, M L G. (1995). From physics to metaphysics. Cambridge University Press. Ridderbos, K (2002). The coarse-graining approach to statistical mechanics: How blissful is our ignorance? Studies in History and Philosophy of Modern Physics, 33(1), 65–77. Ridderbos, T M, & Redhead, M L G (1998). The spin-echo experiments and the second law of thermodynamics. Foundations of Physics, 28(8), 1237–1270. Robertson, B (1966). Equations of motion in nonequilibrium statistical mechanics. Physical Review, 144, 151–161. Robertson, K (2020). Asymmetry, abstraction, and autonomy: Justifying coarse-graining in statistical mechanics. British Journal for the Philosophy of Science, 71(2), 547–579. Robertson, K. (forthcoming). In search of the holy grail: How to reduce the second law of thermodynamics. British Journal for the Philosophy of Science. https://doi.org/10.1086/714795 Sagaut, P. (2006). Large eddy simulation for incompressible flows (3rd edn.). Springer. Schilling, T (2022). Coarse-grained modelling out of equilibrium. Physics Reports, 972, 1–45. Schmidt, M (2022). Power functional theory for many-body dynamics. Reviews of Modern Physics, 94(1), 015007. Shenker, O R (1999). Is-k Tr (ρlnρ) the entropy in quantum mechanics? British Journal for the Philosophy of Science, 50(1), 33–48. Sklar, L. (1995). Physics and chance: Philosophical issues in the foundations of statistical mechanics. Cambridge University Press. Spohn, H (1980). Kinetic equations from Hamiltonian dynamics: Markovian limits. Reviews of Modern Physics, 52(3), 569–615. Szamel, G, & Löwen, H (1991). Mode-coupling theory of the glass transition in colloidal systems. Physical Review A, 44(12), 8215–8219. te Vrugt, M (2021). The five problems of irreversibility. Studies in History and Philosophy of Science, 87, 136–146. te Vrugt, M. (forthcoming). How to distinguish between indistinguishable particles. British Journal for the Philosophy of Science. https://doi.org/10.1086/718495, arXiv:2112.00178 te Vrugt, M, & Wittkowski, R (2019). Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians. Physical Review E, 99, 062118. te Vrugt, M, & Wittkowski, R (2020a). Projection operators in statistical mechanics: A pedagogical approach. European Journal of Physics, 41 (4), 045101. te Vrugt, M, & Wittkowski, R (2020b). Relations between angular and Cartesian orientational expansions. AIP Advances, 10(3), 035106. te Vrugt, M, Löwen, H, & Wittkowski, R. (2020). Classical dynamical density functional theory: From fundamentals to applications. Advances in Physics, 69(2), 121–247. te Vrugt, M, Hossenfelder, S, & Wittkowski, R. (2021a). Mori-Zwanzig formalism for general relativity: A new approach to the averaging problem. Physical Review Letters, 127, 231101. te Vrugt, M, Tóth, GI, & Wittkowski, R. (2021b). Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility. Journal of Computational Electronics, 20, 2209–2231. Tóth, G.I. (2022). Emergent pseudo time-irreversibility in the classical many-body system of pair interacting particles. Physica D: Nonlinear Phenomena, 437, 133336. Von Kutschera, F (1969). Zur Problematik der naturwissenschaftlichen Verwendung des subjektiven Wahrscheinlichkeitsbegriffs. Synthese, 20, 84–103. Wallace, D. (2011). The Logic of the Past Hypothesis. In B. Loewer, E. Winsberg, & B. Weslake (Eds.), Time’s arrows and the probability structure of the world, Harvard. http://philsci-archive.pitt.edu/8894/ Wallace, D. (2012). The emergent multiverse: Quantum theory according to the Everett interpretation. Oxford University Press. Wallace, D (2015). The quantitative content of statistical mechanics. Studies in History and Philosophy of Modern Physics, 52, 285–293. Wallace, D. (2021). Probability and irreversibility in modern statistical mechanics: Classical and quantum. In D. Bedingham, O. Maroney, & C. Timpson (Eds.), Quantum foundations of statistical mechanics. Oxford University Press. arXiv:2104.11223 Walz, C, & Fuchs, M (2010). Displacement field and elastic constants in nonideal crystals. Physical Review B, 81(13), 134110. Wittkowski, R, Löwen, H, & Brand, H. R. (2012). Extended dynamical density functional theory for colloidal mixtures with temperature gradients. Journal of Chemical Physics, 137(22), 224904. Wittkowski, R, Löwen, H, & Brand, H. R. (2013). Microscopic approach to entropy production. Journal of Physics A: Mathematical and Theoretical, 46(35), 355003. Zeh, H. D. (2007). The physical basis of the direction of time (5th edn.). Springer Verlag. Zwanzig, R (1960). Ensemble method in the theory of irreversibility. Journal of Chemical Physics, 33(5), 1338–1341. Zwanzig, R (1973). Nonlinear generalized Langevin equations. Journal of Statistical Physics, 9(3), 215–220. Zwanzig, R. (2001). Nonequilibrium statistical mechanics. Oxford University Press.