Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The quantum drift-diffusion model: existence and exponential convergence
  to the equilibrium   This work is devoted to the analysis of the quantum drift-diffusion model
derived by Degond et al. The model is obtained as the diffusive limit of the
quantum Liouville-BGK equation, where the collision term is defined after a
local quantum statistical equilibrium. The corner stone of the model is the
closure relation between the density and the current, which is nonlinear and
nonlocal, and is the main source of the mathematical difficulties. The question
of the existence of solutions has been open since the derivation of the model,
and we provide here a first result in a one-dimensional periodic setting. The
proof is based on an approximation argument, and exploits some properties of
the minimizers of an appropriate quantum free energy. We investigate as well
the long time behavior, and show that the solutions converge exponentially fast
to the equilibrium. This is done by deriving a non-commutative logarithmic
Sobolev inequality for the local quantum statistical equilibrium.

7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con