Exact analytic solution for the correlation time of a Brownian particle in a double‐well potential from the Langevin equation

1996 ◽  
Vol 105 (5) ◽  
pp. 2112-2118 ◽  
Author(s):  
Yu. P. Kalmykov ◽  
W. T. Coffey ◽  
J. T. Waldron
Keyword(s):  
Langevin Equation ◽  
Correlation Time ◽  
Analytic Solution ◽  
Brownian Particle ◽  
Exact Analytic Solution ◽  
Double Well Potential

scholarly journals An Old Universe in K-Essence Cosmology

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Rong-Jia Yang ◽  
Shuang Nan Zhang ◽  
Bohai Chen ◽  
Jing-Zhao Qi
Keyword(s):  
Analytic Solution ◽  
Exact Analytic Solution ◽  
Simple Hypothesis ◽  
Kinetic Factor ◽  
Λcdm Model ◽  
The Future ◽  
Constant Potential ◽  
The Universe

Models in which the Lagrangian contains only a kinetic factor and does not depend explicitly on the field itself in k-essence cosmology are considered. In the case of a constant potential, we obtain an exact analytic solution of k-essence, wk=−(3α−2)/3α, under a simple hypothesis, a=a0(t/t0)α, but without any assumption about the form of F(X). In purely kinetic k-essence model, the acceleration can only be induced after the matter-dominated epoch; the Universe is about 33.5−3.4+4.2 Gyr old now and about 3.1 Gyr old at z=3.9 which is consistent with the fact of quasar observation, while this observation contradicts with the prediction of ΛCDM model, and the year where a transition from deceleration to acceleration expansion is about from 18.43 Gyr to 25.2 Gyr after the beginning of the Universe. These results may offer clues to test observationally the k-essence scenario in the future.

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