Journal article
Swetamber Das, Jason R. Green
Phys. Rev. E, vol. 106(5), American Physical Society, 2022 Nov, p. 054135
Phys. Rev. E, vol. 106(5), American Physical Society, 2022 Nov, p. 054135
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arXiv
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APA
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Das, S., & Green, J. R. (2022). Density matrix formulation of dynamical systems. Phys. Rev. E, 106(5), 054135. https://doi.org/10.1103/PhysRevE.106.054135
Chicago/Turabian
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Das, Swetamber, and Jason R. Green. “Density Matrix Formulation of Dynamical Systems.” Phys. Rev. E 106, no. 5 (November 2022): 054135.
MLA
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Das, Swetamber, and Jason R. Green. “Density Matrix Formulation of Dynamical Systems.” Phys. Rev. E, vol. 106, no. 5, American Physical Society, Nov. 2022, p. 054135, doi:10.1103/PhysRevE.106.054135.
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@article{das2022a,
title = {Density matrix formulation of dynamical systems},
year = {2022},
month = nov,
issue = {5},
journal = {Phys. Rev. E},
pages = {054135},
publisher = {American Physical Society},
volume = {106},
doi = {10.1103/PhysRevE.106.054135},
author = {Das, Swetamber and Green, Jason R.},
month_numeric = {11}
}
Abstract
Physical systems that are dissipating, mixing, and developing turbulence also irreversibly transport statistical density. However, predicting the evolution of density from atomic and molecular scale dynamics is challenging for nonsteady, open, and driven nonequilibrium processes. Here, we establish a theory to address this challenge for classical dynamical systems that is analogous to the density matrix formulation of quantum mechanics. We show that a classical density matrix is similar to the phase-space metric and evolves in time according to generalizations of Liouville's theorem and Liouville's equation for non-Hamiltonian systems. The traditional Liouvillian forms are recovered in the absence of dissipation or driving by imposing trace preservation or by considering Hamiltonian dynamics. Local measures of dynamical instability and chaos are embedded in classical commutators and anticommutators and directly related to Poisson brackets when the dynamics are Hamiltonian. Because the classical density matrix is built from the Lyapunov vectors that underlie classical chaos, it offers an alternative computationally tractable basis for the statistical mechanics of nonequilibrium processes that applies to systems that are driven, transient, dissipative, regular, and chaotic.
Physical systems that are dissipating, mixing, and developing turbulence also irreversibly transport statistical density. However, predicting the evolution of density from atomic and molecular scale dynamics is challenging for nonsteady, open, and driven nonequilibrium processes. Here, we establish a theory to address this challenge for classical dynamical systems that is analogous to the density matrix formulation of quantum mechanics. We show that a classical density matrix is similar to the phase-space metric and evolves in time according to generalizations of Liouville's theorem and Liouville's equation for non-Hamiltonian systems. The traditional Liouvillian forms are recovered in the absence of dissipation or driving by imposing trace preservation or by considering Hamiltonian dynamics. Local measures of dynamical instability and chaos are embedded in classical commutators and anticommutators and directly related to Poisson brackets when the dynamics are Hamiltonian. Because the classical density matrix is built from the Lyapunov vectors that underlie classical chaos, it offers an alternative computationally tractable basis for the statistical mechanics of nonequilibrium processes that applies to systems that are driven, transient, dissipative, regular, and chaotic.
