# Workshop: Presentation

### Steepest Entropy Ascent model for far-non-equilibrium dissipative evolution in quantum thermodynamics: a 1984 precursor of GENERIC (1997), gradient flows (1998), maximal entropy production (2001), and SEAQT (2014)

*Gian Paolo Beretta (Università di Brescia (Italy))*

Mon, 05 Mar 2018
• 11:00-11:40h
• Templergraben 55, Lecture Hall V

- Abstract
- We have shown in [1] that the dissipative (irreversible) structure of several mathematical frameworks for modeling nonequilibrium states and their dynamics can be unified by extending the idea of steepest entropy ascent (SEA) that we first introduced in [2] in quantum thermodynamics (QT) modeling. Such frameworks include: Small-Scale and Rarefied Gases Dynamics (Boltzmann equation and its kinetic models); Rational Extended Thermodynamics, Macroscopic Non-Equilibrium Thermodynamics, and Chemical Kinetics; Mesoscopic Non-Equilibrium Thermodynamics; Quantum Computing [3]. In [4] we proved that the dissipative part of the nonequilibrium formulation known as GENERIC is essentially an implementation of the SEA principle: whenever the GENERIC structure is equipped with an inner product, the SEA and GENERIC models of the irreversible component of the dynamics are essentially interchangeable, provided of course they assume the same kinematics. The talk will show as case study the use of SEA to develop models of the Boltzmann equation [5].

In [6] SEAQT has been used to model effectively quantum decoherence. By combining the SEA principle in the QT framework with a constrained-equilibrium (or hypoequilibrium) model reduction approach, Ref. [7] developed the SEAQT approach into a quantum statistical thermodynamic analysis of nonequilibrium evolution, especially effective in providing far-from-equilibrium time-dependent information also for macroscopic and mesoscopic systems. Rather than from a phenomenological description, the method starts from a more fundamental density-of-states representation of existing models of the system’s eigenstructure (see e.g. [8]).

- References (in the pdf version click on the title to get to the paper online)
- [1] G.P. Beretta,
*Steepest entropy ascent model for far-nonequilibrium thermodynamics: Unified implementation of the maximum entropy production principle*, Phys. Rev. E, Vol.**90**, 042113 (2014).

[2] G.P. Beretta,
*Steepest entropy ascent in quantum thermodynamics*,
in *The Physics of Phase Space*, Lecture Notes in Physics, (Springer-Verlag, Berlin),
edited by Y.S. Kim and W.W. Zachary, Vol. 278, 441-443 (1986).**See also:**

G.P. Beretta, E.P. Gyftopoulos, and J.L. Park,
*Quantum thermodynamics. A new equation of motion for a general quantum system*,
Nuovo Cimento B, Vol. **87**, 77-97 (1985).

John Maddox,
Nature,
Vol. 316, 11 (1985).

G.P. Beretta,
*Quantum thermodynamics of nonequilibrium. Onsager reciprocity and dispersion-dissipation relations*,
Found. Phys., Vol. **17**, 365-381 (1987).

G.P. Beretta,
*Nonlinear
quantum evolution equations to model irreversible adiabatic relaxation
with maximal entropy production and other nonunitary processes*,
Reps. Math. Phys., Vol. **64**, 139-168 (2009). http://dx.doi.org/10.1016/S0034-4877(09)90024-6

[3] F. Tabakin,
*Model dynamics for quantum computing*,
Ann. Phys., Vol. **383**, 33 (2017).

[4] A. Montefusco, F. Consonni, and G.P. Beretta,
*Essential equivalence of the GENERIC and Steepest Entropy Ascent models of dissipation for non-equilibrium thermodynamics*,
Phys. Rev. E, Vol. **91**, 042138 (2015).

[5] G.P. Beretta and N.G. Hadjiconstantinou,
*Steepest
entropy ascent models of the Boltzmann equation. Comparisons with
hard-sphere dynamics and relaxation-time models for homogeneous
relaxation from highly non-equilibrium states*,
Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition,
paper IMECE2013-64905, November 15-21, 2013, San Diego, CA, Proc. ASME 56352;
Vol. 8B: *Heat Transfer and Thermal Engineering*, V08BT09A050,
http://dx.doi.org/10.1115/IMECE2013-64905

[6] S. Cano-Andrade, G.P. Beretta, and M.R. von Spakovsky,
*Steepest-entropy-ascent quantum thermodynamic modeling of decoherence in two different microscopic composite systems*,
Phys. Rev. A, Vol. **91**, 013848 (2015).

[7] G. Li and M.R. von Spakovsky,
*Steepest-entropy-ascent
quantum thermodynamic modeling of the relaxation process of isolated
chemically reactive systems using density of states and the concept of
hypoequilibrium state*,
Phys. Rev. E, Vol. **93**, 012137 (2016).

[8] G. Li, M.R. von Spakovsky, and C. Hin,
*Steepest entropy ascent quantum thermodynamic model of electron and phonon transport*,
Phys. Rev. B, Vol. **97**, 024308 (2018).