Constrained minimizers of the von Neumann entropy and their characterization
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Calculus of Variations
Constrained minimizers of the von Neumann entropy and their characterization Romain Duboscq1 · Olivier Pinaud2 Received: 31 October 2019 / Accepted: 31 March 2020 / Published online: 11 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider in this work the problem of minimizing the von Neumann entropy under the constraints that the density of particles, the current, and the kinetic energy of the system is fixed at each point of space. The unique minimizer is a self-adjoint positive trace class operator, and our objective is to characterize its form. We will show that this minimizer is solution to a self-consistent nonlinear eigenvalue problem. One of the main difficulties in the proof is to parametrize the feasible set in order to derive the Euler–Lagrange equation, and we will proceed by constructing an appropriate form of perturbations of the minimizer. The question of deriving quantum statistical equilibria is at the heart of the quantum hydrodynamical models introduced by Degond and Ringhofer (J Statist Phys 112:(3–4), 587-628, 2003). An original feature of the problem is the local nature of constraints, i.e. they depend on position, while more classical models consider the total number of particles, the total current and the total energy in the system to be fixed. Mathematics Subject Classification Primary 35Q40; Secondary 82B10
1 Introduction This work is concerned with the study of minimizers of quantum entropies, which are solutions to problems of the form min Tr s() , A
(1)
Communicated by J. Ball.
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Romain Duboscq [email protected] Olivier Pinaud [email protected]
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Institut de Mathématiques de Toulouse, UMR5219, CNRS, INSA, Université de Toulouse, 31077 Toulouse, France
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Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
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R. Duboscq, O. Pinaud
where is a density operator (i.e. a self-adjoint trace class positive operator), s an entropy function, typically the Boltzmann entropy s(x) = x log(x) − x for x ≥ 0, and Tr(·) denotes operator trace. The feasible set A includes linear constraints on involving particles density, current, and energy. This problem is motivated by a series of papers by Degond and Ringhofer on the derivation of quantum hydrodynamical models from first principles, see [1–4]. It is also a problem arising in the work of Nachtergaele and Yau in their derivation of the Euler equation from quantum dynamics [17]. In [5], Degond and Ringhofer main idea is to transpose to the quantum setting the entropy closure strategy that Levermore used for kinetic equations [12]. The kinetic formulation starts with the transport equation (all physical constants are set to one), ∂t f + {H , f } = Q( f ),
f ≡ f (t, x, p), (x, p) ∈ Rd × Rd ,
(2)
where f ≥ 0 is the particle distribution function, H is a classical Hamiltonian, e.g. H (x, p) = | p|2 /2 + V (x) for some potential V , {H , f } = ∇ p H · ∇x f − ∇x H · ∇ p f is the Poisson bracket, an
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