44,231 research outputs found
Towards Rigorous Derivation of Quantum Kinetic Equations
We develop a rigorous formalism for the description of the evolution of
states of quantum many-particle systems in terms of a one-particle density
operator. For initial states which are specified in terms of a one-particle
density operator the equivalence of the description of the evolution of quantum
many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and
by the Cauchy problem of the generalized quantum kinetic equation together with
a sequence of explicitly defined functionals of a solution of stated kinetic
equation is established in the space of trace class operators. The links of the
specific quantum kinetic equations with the generalized quantum kinetic
equation are discussed.Comment: 25 page
Backward Clusters, Hierarchy and Wild Sums for a Hard Sphere System in a Low-Density Regime
We study the statistics of backward clusters in a gas of hard spheres at low
density. A backward cluster is defined as the group of particles involved
directly or indirectly in the backwards-in-time dynamics of a given tagged
sphere. We derive upper and lower bounds on the average size of clusters by
using the theory of the homogeneous Boltzmann equation combined with suitable
hierarchical expansions. These representations are known in the easier context
of Maxwellian molecules (Wild sums). We test our results with a numerical
experiment based on molecular dynamics simulations
Onsager's missing steps retraced
Onsager's paper on phase transition and phase coexistence in anisotropic
colloidal systems is a landmark in the theory of lyotropic liquid crystals.
However, an uncompromising scrutiny of Onsager's original derivation reveals
that it would be rigorously valid only for ludicrous values of the system's
number density (of the order of the reciprocal of the number of particles)
Based on Penrose's tree identity and an appropriate variant of the mean-field
approach for purely repulsive, hard-core interactions, our theory shows that
Onsager's theory is indeed valid for a reasonable range of densities
Continuous-time average-preserving opinion dynamics with opinion-dependent communications
We study a simple continuous-time multi-agent system related to Krause's
model of opinion dynamics: each agent holds a real value, and this value is
continuously attracted by every other value differing from it by less than 1,
with an intensity proportional to the difference.
We prove convergence to a set of clusters, with the agents in each cluster
sharing a common value, and provide a lower bound on the distance between
clusters at a stable equilibrium, under a suitable notion of multi-agent system
stability.
To better understand the behavior of the system for a large number of agents,
we introduce a variant involving a continuum of agents. We prove, under some
conditions, the existence of a solution to the system dynamics, convergence to
clusters, and a non-trivial lower bound on the distance between clusters.
Finally, we establish that the continuum model accurately represents the
asymptotic behavior of a system with a finite but large number of agents.Comment: 25 pages, 2 figures, 11 tex files and 2 eps file
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