12,243 research outputs found
Random equations in nilpotent groups
In this paper we study satisfiability of random equations in an infinite
finitely generated nilpotent group G. We show that the set SAT(G,k) of all
equations in k > 1 variables over G which are satisfiable in G has an
intermediate asymptotic density in the space of all equations in k variables
over G. When G is a free abelian group of finite rank, we compute this density
precisely; otherwise we give some non-trivial upper and lower bounds. For k = 1
the set SAT(G,k) is negligible. Usually the asymptotic densities of interesting
sets in groups are either zero or one. The results of this paper provide new
examples of algebraically significant sets of intermediate asymptotic density.Comment: 25 page
On the Proof by Reductio ad Absurdum of the Hohenberg-Kohn Theorem for Ensembles of Fractionally Occupied States of Coulomb Systems
It is demonstrated that the original reductio ad absurdum proof of the
generalization of the Hohenberg-Kohn theorem for ensembles of fractionally
occupied states for isolated many-electron Coulomb systems with Coulomb-type
external potentials by Gross et al. [Phys. Rev. A 37, 2809 (1988)] is
self-contradictory since the to-be-refuted assumption (negation) regarding the
ensemble one-electron densities and the assumption about the external
potentials are logically incompatible to each other due to the Kato
electron-nuclear cusp theorem. It is however proved that the Kato theorem
itself provides a satisfactory proof of this theorem.Comment: 9 pages. Int. J. Quantum Chem., to appea
"Quantum phase transitions" in classical nonequilibrium processes
Diffusion limited reaction of the Lotka-Volterra type is analyzed taking into
account the discrete nature of the reactants. In the continuum approximation,
the dynamics is dominated by an elliptic fixed-point. This fixed-point becomes
unstable due to discretization effects, a scenario similar to quantum phase
transitions. As a result, the long-time asymptotic behavior of the system
changes and the dynamics flows into a limit cycle.
The results are verified by numerical simulations.Comment: 9 pages, 3 figures include
Ballistic transport in the one-dimensional Hubbard model: the hydrodynamic approach
We outline a general formalism of hydrodynamics for quantum systems with
multiple particle species which undergo completely elastic scattering. In the
thermodynamic limit, the complete kinematic data of the problem consists of the
particle content, the dispersion relations, and a universal dressing
transformation which accounts for interparticle interactions. We consider
quantum integrable models and we focus on the one-dimensional fermionic Hubbard
model. By linearizing hydrodynamic equations, we provide exact closed-form
expressions for Drude weights, generalized static charge susceptibilities and
charge-current correlators valid on hydrodynamic scale, represented as integral
kernels operating diagonally in the space of mode numbers of thermodynamic
excitations. We find that, on hydrodynamic scales, Drude weights manifestly
display Onsager reciprocal relations even for generic (i.e. non-canonical)
equilibrium states, and establish a generalized detailed balance condition for
a general quantum integrable model. We present the first exact analytic
expressions for the general Drude weights in the Hubbard model, and explain how
to reconcile different approaches for computing Drude weights from the previous
literature.Comment: 4 pages + supplemental materia
Potential functionals versus density functionals
Potential functional approximations are an intriguing alternative to density
functional approximations. The potential functional that is dual to the Lieb
density functional is defined and properties given. The relationship between
Thomas-Fermi theory as a density functional and as a potential functional is
derived. The properties of several recent semiclassical potential functionals
are explored, especially in their approach to the large particle number and
classical continuum limits. The lack of ambiguity in the energy density of
potential functional approximations is demonstrated. The density-density
response function of the semiclassical approximation is calculated and shown to
violate a key symmetry condition
- …