26,514 research outputs found
Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics
A new method of deriving reduced models of Hamiltonian dynamical systems is
developed using techniques from optimization and statistical estimation. Given
a set of resolved variables that define a model reduction, the
quasi-equilibrium ensembles associated with the resolved variables are employed
as a family of trial probability densities on phase space. The residual that
results from submitting these trial densities to the Liouville equation is
quantified by an ensemble-averaged cost function related to the information
loss rate of the reduction. From an initial nonequilibrium state, the
statistical state of the system at any later time is estimated by minimizing
the time integral of the cost function over paths of trial densities.
Statistical closure of the underresolved dynamics is obtained at the level of
the value function, which equals the optimal cost of reduction with respect to
the resolved variables, and the evolution of the estimated statistical state is
deduced from the Hamilton-Jacobi equation satisfied by the value function. In
the near-equilibrium regime, or under a local quadratic approximation in the
far-from-equilibrium regime, this best-fit closure is governed by a
differential equation for the estimated state vector coupled to a Riccati
differential equation for the Hessian matrix of the value function. Since
memory effects are not explicitly included in the trial densities, a single
adjustable parameter is introduced into the cost function to capture a
time-scale ratio between resolved and unresolved motions. Apart from this
parameter, the closed equations for the resolved variables are completely
determined by the underlying deterministic dynamics
An optimization principle for deriving nonequilibrium statistical models of Hamiltonian dynamics
A general method for deriving closed reduced models of Hamiltonian dynamical
systems is developed using techniques from optimization and statistical
estimation. As in standard projection operator methods, a set of resolved
variables is selected to capture the slow, macroscopic behavior of the system,
and the family of quasi-equilibrium probability densities on phase space
corresponding to these resolved variables is employed as a statistical model.
The macroscopic dynamics of the mean resolved variables is determined by
optimizing over paths of these probability densities. Specifically, a cost
function is introduced that quantifies the lack-of-fit of such paths to the
underlying microscopic dynamics; it is an ensemble-averaged, squared-norm of
the residual that results from submitting a path of trial densities to the
Liouville equation. The evolution of the macrostate is estimated by minimizing
the time integral of the cost function. The value function for this
optimization satisfies the associated Hamilton-Jacobi equation, and it
determines the optimal relation between the statistical parameters and the
irreversible fluxes of the resolved variables, thereby closing the reduced
dynamics. The resulting equations for the macroscopic variables have the
generic form of governing equations for nonequilibrium thermodynamics, and they
furnish a rational extension of the classical equations of linear irreversible
thermodynamics beyond the near-equilibrium regime. In particular, the value
function is a thermodynamic potential that extends the classical dissipation
function and supplies the nonlinear relation between thermodynamics forces and
fluxes
Repeating voting with complete information
A committee is choosing from two alternatives. If required supermajority is not reached, voting is repeated indefinitely, although there is a cost of delay. Under suitable assumptions the equilibrium analysis provides a sharp prediction. The result can be interpreted as a generalization of the seminal median voter theorem known from the simple majority case. If supermajority is required instead, then the power to select the outcome moves from the median voter to the more extreme voters. Normative analysis indicates that the simple majority is not constrained efficient because it does not reflect the strengths of voters' opinion. Even if unanimity is a bad voting rule, voting rules close to unanimity may be efficient. The more likely it is to have a very many almost indifferent voters and some very opinionated ones, the more stringent supermajority is required for efficienc
On the oscillations in Mercury's obliquity
One major objective of MESSENGER and BepiColombo spatial missions is to
accurately measure Mercury's rotation and its obliquity in order to obtain
constraints on internal structure of the planet. Which is the obliquity's
dynamical behavior deriving from a complete spin-orbit motion of Mercury
simultaneously integrated with planetary interactions? We have used our SONYR
model integrating the spin-orbit N-body problem applied to the solar System
(Sun and planets). For lack of current accurate observations or ephemerides of
Mercury's rotation, and therefore for lack of valid initial conditions for a
numerical integration, we have built an original method for finding the
libration center of the spin-orbit system and, as a consequence, for avoiding
arbitrary amplitudes in librations of the spin-orbit motion as well as in
Mercury's obliquity. The method has been carried out in two cases: (1) the
spin-orbit motion of Mercury in the 2-body problem case (Sun-Mercury) where an
uniform precession of the Keplerian orbital plane is kinematically added at a
fixed inclination (S2K case), (2) the spin-orbit motion of Mercury in the
N-body problem case (Sun and planets) (Sn case). We find that the remaining
amplitude of the oscillations in the Sn case is one order of magnitude larger
than in the S2K case, namely 4 versus 0.4 arcseconds (peak-to-peak). The mean
obliquity is also larger, namely 1.98 versus 1.80 arcminutes, for a difference
of 10.8 arcseconds. These theoretical results are in a good agreement with
recent radar observations but it is not excluded that it should be possible to
push farther the convergence process by drawing nearer still more precisely to
the libration center.Comment: 30 pages, 3 tables, 8 figures, accepted to Icarus (26 Jul 2007
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