A link between the log-Sobolev inequality and Lyapunov condition

We give an alternative look at the log-Sobolev inequality (LSI in short) for log-concave measures by semigroup tools. The similar idea yields a heat flow proof of LSI under some quadratic Lyapunov condition for symmetric diffusions on Riemannian manifolds provided the Bakry-Emery's curvature is bounded from below. Let's mention that, the general $\phi$-Lyapunov conditions were introduced by Cattiaux-Guillin-Wang-Wu [8] to study functional inequalities, and the above result on LSI was first proved subject to $\phi(\cdot)=d^2(\cdot, x_0)$ by Cattiaux-Guillin-Wu [9] through a combination of detective $L^2$ transportation-information inequality $\mathrm{W_2I}$ and the HWI inequality of Otto-Villani. Next, we assert a converse implication that the Lyapunov condition can be derived from LSI, which means their equivalence in the above setting.Comment: 8 pages, modified according to the referee's review, more minor corrections in version

Corrections to local scale invariance in the non-equilibrium dynamics of critical systems: numerical evidences

Local scale invariance (LSI) has been recently proposed as a possible extension of the dynamical scaling in systems at the critical point and during phase ordering. LSI has been applied inter alia to provide predictions for the scaling properties of the response function of non-equilibrium critical systems in the aging regime following a quench from the high-temperature phase to the critical point. These predictions have been confirmed by Monte Carlo simulations and analytical results for some specific models, but they are in disagreement with field-theoretical predictions. By means of Monte Carlo simulations of the critical two- and three-dimensional Ising model with Glauber dynamics, we study the intermediate integrated response, finding deviations from the corresponding LSI predictions that are in qualitative agreement with the field-theoretical computations. This result casts some doubts on the general applicability of LSI to critical dynamics.Comment: 4 pages, 2 figures, minor changes, version to appear in Phys. Rev. B as a Rapid Communicatio

The Impact of LSI (Large Scale Integration) on System Packaging

System packaging of LSI circuit