Quantum thermodynamics. A new equation of motion for a general quantum system

Beretta, Gian Paolo; Gyftopoulos, Elias P.; Park, J. L.

doi:10.1007/bf02729244

Cited by 79 publications

(100 citation statements)

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“…In our formulation of Quantum Thermodynamics [2,3,4,5,6], the mathematical framework is the same as that just summarized for QSM, but the fundamental difference is in the physical meaning of the density operator. Indeed, QT assumes that the true individual quantum state of a system isolated and uncorrelated from the rest of the universe is represented by a density operator ρ which are not necessarily idempotent.…”

Section: Framework D: Quantum Thermodynamicsmentioning

confidence: 99%

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Nonlinear Dynamical Equation for Irreversible, Steepest-Entropy-Ascent Relaxation to Stable Equilibrium

Beretta¹

2007

AIP Conference Proceedings

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Abstract. We discuss the structure and main features of the nonlinear evolution equation proposed by this author as the fundamental dynamical law within the framework of Quantum Thermodynamics. The nonlinear equation generates a dynamical group providing a unique deterministic description of irreversible, conservative relaxation towards equilibrium from any non-equilibrium state, and satisfies a very restrictive stability requirement equivalent to Hatsopoulos-Keenan statement of the second law of thermodynamics. Here, we emphasize its mathematical structure and its applicability also within other contexts, such as Classical and Quantum Statistical Mechanics, and Information Theory.

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Section: Framework D: Quantum Thermodynamicsmentioning

confidence: 99%

“…With the above background, the nonlinear evolution equation proposed by the author [3,4,5,6,12] to meet our design specifications takes the compact form:…”

Section: Stepeest-s-ascent Nonlinear Evolution Equationmentioning

confidence: 99%

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Nonlinear Dynamical Equation for Irreversible, Steepest-Entropy-Ascent Relaxation to Stable Equilibrium

Beretta¹

2007

AIP Conference Proceedings

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“…But if ρ evolves according to Eq. (24) in [1], it can be verified thatNρ =ρN = νρ, which means that this property is preserved in time implicitly, and any additional constraint on the average of N is superfluous.Also, it appears that Beretta's approach to multicomponent systems and separability [11] falls outside the ansatz used in [1], and the associated mathematical difficulties remain to be solved. This is due to the occurrence of products of partial traces of the total density matrix ρ over various subsystems, which prevent the reduction of the equation of motion for ρ to an equation of motion for γ .…”

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confidence: 99%

“…Shortly after this work was approved for publication, it was brought to my attention [2] that a closely related undertaking on the subject was reported about two decades ago by G.P.Beretta et al [3]- [8], and later also by Korsch et al [10,11]. The present Addendum is intended to acknowledge and correct this oversight, and to provide a few brief observations on the relation between these works and the results presented in [1].…”

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confidence: 99%

Addendum to “Nonlinear quantum evolution with maximal entropy production”

Gheorghiu-Svirschevski

2001

Phys. Rev. A

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The author calls attention to previous work with related results, which has escaped scrutiny before the publication of the article "Nonlinear quantum evolution with maximal entropy production", Phys.Rev. A63, 022105 (2001).In a recently published paper [1], I propose a nonlinear extension of the nonrelativistic Liouville-von Neumann equation, which incorporates both the unitary propagation of pure quantum states and the second principle of thermodynamics. Shortly after this work was approved for publication, it was brought to my attention [2] that a closely related undertaking on the subject was reported about two decades ago by G.P. , and later also by Korsch et al. [10,11]. The present Addendum is intended to acknowledge and correct this oversight, and to provide a few brief observations on the relation between these works and the results presented in [1].In ref.[3], Beretta et al. proposed that the dynamical principle of quantum theory be replaced by a postulated nonlinear equation of motion which is, algebraically, a generalization of Eq.(24) in [1] (see also Eq.(91)). General properties of this equation are also presented in an axiomatic framework, particularly with regard to the nature of nondissipative and equilibrium states (see also [4]). Unfortunately, the applicability of the theory was hindered by a number of mathematical difficulties, e.g., by lack of a definitive proof of the positivity of the evolution. A well-behaved example was given later for the two-level system in interaction with an external field [5]. Such problems notwithstanding, it was argued [6,7] that the proposed nonlinear evolution drives the density matrix along a direction of steepest entropy ascent under given constraints, and ref.[7] provides a notable theorem on exact, generalized Onsager reciprocity, not restricted to the near-equilibrium regime. The stability of thermodynamic equilibrium states has also been discussed [8], with reference to (but not limited to the context of) the postulated nonlinear dynamics. More recently, Korsch et al. studied a family of closely related dissipative dynamics [9], and also derived explicit solutions for a driven harmonic oscillator by Lie-algebra techniques [10].Beretta's confidence in the physicality of his construction seems to find vindication after all. In ref.[1] the theory is formulated in terms of state operators γ associated to the density matrix ρ [ρ = γγ + ] and the equation of motion is derived from a variational principle which observes the principles of quantum mechanics and the fundamental laws of thermodynamics. The existence and uniqueness of the solutions for ρ follow from the equation of motion for γ, and the positivity of the evolution is guaranteed by construction. The reader can also find alternative proofs for the fundamental properties of the equation of motion, a derivation of a near-equilibrium limit, exact to first-order deviations of γ from the equilibrium state, as well as a proof of the equivalence between symmetry covariance, conservation laws and associated com...

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The Big Game of Energy and Entropy

Wanderlingh

1995

Thinking Physics for Teaching

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Quantum thermodynamics. A new equation of motion for a general quantum system

Cited by 79 publications

References 4 publications

Nonlinear Dynamical Equation for Irreversible, Steepest-Entropy-Ascent Relaxation to Stable Equilibrium

Nonlinear Dynamical Equation for Irreversible, Steepest-Entropy-Ascent Relaxation to Stable Equilibrium

Addendum to “Nonlinear quantum evolution with maximal entropy production”

The Big Game of Energy and Entropy

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