The modeling of irreversible systems is a fundamental issue in each physical field. When con- sidering irreversibility due to dissipation, classical mechanics boasts many tools and established theories. Among them, the study of metriplectic structures unveils a simple theory based on linear algebraic tools that have immediate thermodynamical translation, not only in classical [1] but also in quantum [2] mechanics. A dynamical system defined by a metriplectic structure is a dissipative model characterized by a specific pair of tensors, which defines the Leibniz brackets. Generally, these tensors give the Poisson brackets and a symmetric metric that models purely dissipative dynamics. In literature, several examples of metriplectic systems representing irreversible dynamics can be found [3{6], but none of these models the process of the two-photon absorbtion (TPA) [7]. This work applies the metriplectic theory and Leibniz algebrae to a dissipative nonlinear optical phenomenon: the TPA by a two-level atom with negligible spontaneous emission. Such a scenario is described by the extension of the symplectic algebra of the Hamiltonian system to a metriplectic algebra of brackets [6], where the Hamiltonian component of the motion is still given by the original Poisson brackets, while a suitable semi-defined metric bracket generates the non-Hamiltonian component. An extension of the Hamiltonian, namely the free energy of the system, represents the metriplectic generator of the motion. The foregoing program interprets the dynamics of classical dissipative systems as ows generated by a new kind of Leibniz algebrae of brackets [8], namely the metriplectic bracket. In details, first we formulate the physical problem in terms of the conservative part H0 of the total Hamiltonian H, the metric tensor G and the metriplectic brackets << *;* >>; then we find the mathematical expression of H as function of the dynamical variables q; p; n, with q; p the real and the imaginary part of the complex electromagnetic field amplitude, respectively, and n the population of the second level. In particular, we find the dissipative part U of H, which depends only on n. To conclude, we also find the free energy F and the equilibrium states, varying with the definition of entropy, as expected. We believe that this manuscript opens the way to explore irreversibility in nonlinear optics. [1] L. A. Turski, Physics Letters A 125, 461 (1987). [2] L. A. Turski, in From Quantum Mechanics to Technology (Springer Berlin Heidelberg, Berlin, Heidelberg, 1996). [3] P. J. Morrison, Physics Letters A 100, 423 (1984). [4] M. Materassi and E. Tassi, Physica D 241, 729 (2012). [5] M. Materassi, Entropy 17, 1329 (2015). [6] P. J. Morrison, Physica D 18, 410 (1986). [7] J. Moloney and A. Newell, Nonlinear Optics, Advanced Book Program (Avalon Publishing, 2004). [8] P. Guha, Journal of Mathematical Analysis and Applica- tions 326, 121 (2007).
Metriplectic Structure of a Radiation-Matter Interaction Toy Model
Massimo Materassi
2018
Abstract
The modeling of irreversible systems is a fundamental issue in each physical field. When con- sidering irreversibility due to dissipation, classical mechanics boasts many tools and established theories. Among them, the study of metriplectic structures unveils a simple theory based on linear algebraic tools that have immediate thermodynamical translation, not only in classical [1] but also in quantum [2] mechanics. A dynamical system defined by a metriplectic structure is a dissipative model characterized by a specific pair of tensors, which defines the Leibniz brackets. Generally, these tensors give the Poisson brackets and a symmetric metric that models purely dissipative dynamics. In literature, several examples of metriplectic systems representing irreversible dynamics can be found [3{6], but none of these models the process of the two-photon absorbtion (TPA) [7]. This work applies the metriplectic theory and Leibniz algebrae to a dissipative nonlinear optical phenomenon: the TPA by a two-level atom with negligible spontaneous emission. Such a scenario is described by the extension of the symplectic algebra of the Hamiltonian system to a metriplectic algebra of brackets [6], where the Hamiltonian component of the motion is still given by the original Poisson brackets, while a suitable semi-defined metric bracket generates the non-Hamiltonian component. An extension of the Hamiltonian, namely the free energy of the system, represents the metriplectic generator of the motion. The foregoing program interprets the dynamics of classical dissipative systems as ows generated by a new kind of Leibniz algebrae of brackets [8], namely the metriplectic bracket. In details, first we formulate the physical problem in terms of the conservative part H0 of the total Hamiltonian H, the metric tensor G and the metriplectic brackets << *;* >>; then we find the mathematical expression of H as function of the dynamical variables q; p; n, with q; p the real and the imaginary part of the complex electromagnetic field amplitude, respectively, and n the population of the second level. In particular, we find the dissipative part U of H, which depends only on n. To conclude, we also find the free energy F and the equilibrium states, varying with the definition of entropy, as expected. We believe that this manuscript opens the way to explore irreversibility in nonlinear optics. [1] L. A. Turski, Physics Letters A 125, 461 (1987). [2] L. A. Turski, in From Quantum Mechanics to Technology (Springer Berlin Heidelberg, Berlin, Heidelberg, 1996). [3] P. J. Morrison, Physics Letters A 100, 423 (1984). [4] M. Materassi and E. Tassi, Physica D 241, 729 (2012). [5] M. Materassi, Entropy 17, 1329 (2015). [6] P. J. Morrison, Physica D 18, 410 (1986). [7] J. Moloney and A. Newell, Nonlinear Optics, Advanced Book Program (Avalon Publishing, 2004). [8] P. Guha, Journal of Mathematical Analysis and Applica- tions 326, 121 (2007).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
