Uniform Logarithmic Sobolev Inequality for Boltzmann Measures with Exterior Magnetic Field over Spheres
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Uniform Logarithmic Sobolev Inequality for Boltzmann Measures with Exterior Magnetic Field over Spheres Zhengliang Zhang · Bin Qian · Yutao Ma
Received: 10 March 2010 / Accepted: 11 September 2011 / Published online: 21 September 2011 © Springer Science+Business Media B.V. 2011
Abstract In this paper, we obtain the uniform logarithmic Sobolev inequality for the Boltzmann measures by reducing multi-dimensional measures to one-dimensional measures, and then applying the characterization on the constant of logarithmic Sobolev inequality for a probability measure on the real line. Keywords Boltzmann measure · Logarithmic Sobolev inequality · Talagrand’s transportation inequality · Poincaré inequality Mathematics Subject Classification (2000) 60E15 · 39B62 · 26Dxx 1 Introduction Let S n−1 be the unit sphere in Rn (n ≥ 3), and μ the normalized Lebesgue measure on S n−1 , i.e. μ = σn−1 /sn−1 , where σn−1 and sn−1 are the uniform surface measure and total volume Research supported by National Science Funds of China (11101313, 10871153, 11001034) and China and Shanghai Postdoctoral Scientific Program (20110490667, 11R21412200). Z. Zhang () Department of Mathematics and Statistics, Wuhan University, 430072 Hubei, China e-mail: [email protected] Z. Zhang e-mail: [email protected] B. Qian Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, China e-mail: [email protected] B. Qian School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, China Y. Ma Department of Mathematics, Beijing Normal University, 100875 Beijing, China e-mail: [email protected]
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respectively on S n−1 . For any h > 0 and e1 = (1, 0, . . . , 0) ∈ S n−1 , let μh be the probability on S n−1 given by dμh (x) =
ehe1 ,x dμ(x), cn (h)
x ∈ S n−1 ,
where cn (h) is the normalizing constant. This is the so-called Boltzmann measure with exterior magnetic field [8]. We say that a probability measure μ on a manifold X satisfies a logarithmic Sobolev inequality if there exists a constant C > 0 such that for every smooth function f : X → R, one has Entμ (f 2 ) ≤ C |∇f |2 dμ, where the relative entropy is defined by
Entμ (f ) = 2
f log f dμ − 2
2
2
f dμ log
2
f dμ .
The properties on logarithmic Sobolev inequality are well introduced by Gross [9]. Though here we just study the logarithmic Sobolev inequality, it’s well-known that Poincaré inequality, transportation inequality and logarithmic Sobolev inequality are essential tools in the study of concentration of measures (see e.g. [10, 11]). Moreover, there exist some relations among the three types of inequalities: logarithmic Sobolev inequality is strictly stronger than Talagrand’s transportation cost inequality ([6, 11] and [3]), which is stronger than Poincaré inequality (see [5], Sect. 4.1). The main method we applied is to reduce the proof of the measures μh on spheres to that of a probability measures νh on [0, π], and then estimate the constant of logarithmic Sobolev inequality by the charac
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