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Condensation of Eigen Microstate in Statistical Ensemble and Phase Transition
In a statistical ensemble with microstates, we introduce an
correlation matrix with the correlations between microstates as its elements.
Using eigenvectors of the correlation matrix, we can define eigen microstates
of the ensemble. The normalized eigenvalue by represents the weight factor
in the ensemble of the corresponding eigen microstate. In the limit , weight factors go to zero in the ensemble without localization of
microstate. The finite limit of weight factor when indicates a
condensation of the corresponding eigen microstate. This indicates a phase
transition with new phase characterized by the condensed eigen microstate. We
propose a finite-size scaling relation of weight factors near critical point,
which can be used to identify the phase transition and its universality class
of general complex systems. The condensation of eigen microstate and the
finite-size scaling relation of weight factors have been confirmed by the Monte
Carlo data of one-dimensional and two-dimensional Ising models.Comment: 9 pages, 16 figures, accepted for publication in Sci. China-Phys.
Mech. Astro
gravity theories in the Palatini Formalism constrained from strong lensing
gravity, capable of driving the late-time acceleration of the
universe, is emerging as a promising alternative to dark energy. Various
gravity models have been intensively tested against probes of the expansion
history, including type Ia supernovae (SNIa), the cosmic microwave background
(CMB) and baryon acoustic oscillations (BAO). In this paper we propose to use
the statistical lens sample from Sloan Digital Sky Survey Quasar Lens Search
Data Release 3 (SQLS DR3) to constrain gravity models. This sample can
probe the expansion history up to , higher than what probed by
current SNIa and BAO data. We adopt a typical parameterization of the form
with and
constants. For (CDM), we obtain the best-fit value of the
parameter , for which the 95% confidence interval that is
[-4.633, -3.754]. This best-fit value of corresponds to the matter
density parameter , consistent with constraints from other
probes. Allowing to be free, the best-fit parameters are . Consequently, we give and the
deceleration parameter . At the 95% confidence level, and
are constrained to [-4.67, -2.89] and [-0.078, 0.202] respectively.
Clearly, given the currently limited sample size, we can only constrain
within the accuracy of and thus can not distinguish
between CDM and gravity with high significance, and actually,
the former lies in the 68% confidence contour. We expect that the extension of
the SQLS DR3 lens sample to the SDSS DR5 and SDSS-II will make constraints on
the model more stringent.Comment: 10 pages, 7 figures. Accepted for publication in MNRA
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