Semi-classical Limit of Quantum Free Energy Minimizers for the Gravitational Hartree Equation
- PDF / 614,206 Bytes
- 47 Pages / 439.37 x 666.142 pts Page_size
- 46 Downloads / 152 Views
Semi-classical Limit of Quantum Free Energy Minimizers for the Gravitational Hartree Equation Woocheol Choi, Younghun Hong & Jinmyoung Seok Communicated by P. Rabinowitz
Abstract For the gravitational Vlasov–Poisson equation, Guo and Rein (Arch Rational Mech Anal 147(3):225–243, 1999) constructed a class of classical isotropic states as minimizers of free energies (or energy-Casimir functionals) under mass constraints. For the quantum counterpart, that is, the gravitational Hartree equation, isotropic states are constructed as free energy minimizers by Aki, Dolbeault and Sparber (Ann Henri Poincaré 12(6):1055–1079, 2011). In this paper, we are concerned with the correspondence between quantum and classical isotropic states. More precisely, we prove that as the Planck constant goes to zero, free energy minimizers for the Hartree equation converge to those for the Vlasov–Poisson equation in terms of potential functions as well as via the Wigner transform and the Töplitz quantization.
1. Introduction 1.1. Background We consider the gravitational Vlasov–Poisson equation 1 ∂t f + p · ∇q f + ∇q ( |·| ∗ ρ f ) · ∇ p f = 0,
(1.1)
where f = f (t, q, p) : R × R6 → [0, ∞) is a phase-space distribution, and ρf = f (·, p) d p R3
denotes the corresponding density function. This equation describes the mean-field dynamics of a large number of collisionless classical particles interacting each other by their mutual gravitational forces. In astrophysics, it provides a simpler but fairly accurate description of stellar dynamics whose size possibly ranges from 108 stars
W. Choi et al.
for dwarfs to 1014 stars for giants. Here, all pair interactions are approximated by 1 ∗ ρf. the self-generated potential − |·| The initial-value problem for (1.1) possesses a unique global classical solution [30,35,42,48]. The nonlinear evolution preserves the mass M( f ) := f (q, p) dqd p R6
and the energy 1 E( f ) := 2
1 | p| f (q, p) dqd p − 2 R6
2
R6
ρ f (x)ρ f (y) dxdy. |x − y|
Moreover, the measure preserving property of the characteristics derives conservation of a Casimir functional β f (q, p) dqd p, C( f ) := R6
with any given non-negative function β : [0, ∞) → [0, ∞) (see [22, Chapter 4]). Combining the energy and a Casimir functional, the energy-Casimir functional is defined by J ( f ) := E( f ) + C( f ). The quantity −C( f ) also can be considered as a (generalized) entropy, including the standard entropy functional ( f ln f )(q, p) dqd p, − R6
and thus the energy-Casimir functional J ( f ) may be called the (generalized) free energy. In many different physical contexts, energy-Casimir/free energy functionals have played a powerful role to construct stable steady states. As for the Vlasov– Poisson equation (1.1), Guo and Rein [26] established a large class of stable isotropic states by the energy-Casimir method. This important work is in fact the starting point of our discussion, but we state below a slightly clearer version in the survey article [45, Theorem 1.1]. We now assume that (A1) β ∈ C 1 ([0, ∞)) i
Data Loading...
