Functional Inequalities For Boltzmann Measures On Circles

This thesis is a study on the theory of Log-Sobolev inequality and Spectral gaps for Boltzmann measures C_P??_h?with parameters h.In this paper,we first introduce the origin of Boltzmann mearsure and gives the definition of Poincaréinequality and Log-Sobolev inequality,which are related to the concentration of measure phenomenon.In the second part,an important theorem is given,which can simplify our problem by reducing the high dimensional case to one dimensional case.In this paper,we mainly study the Poincaréconstant and the Log-Sobolev constant of the Boltzmann measure?n=2?,and get the fact that the optimal Poincaréconstant C_P??_h??0 with speed 1/h when h??,while optimal logarithmic Sobolev constant C_LS??_h?is bounded at h>0,moreover,the estimate for C_p??_h?is sharp since when h=0,the lower and upper bounds coincide with C_p??_h?=1.Besides,our results can enhance to some extent the claim that the Log-Sobolev inequality is stronger than the Poincaréinequality.