Uniform Log-Sobolev inequality for Boltzmann measures with ...

Uniform Log-Sobolev inequality for
Boltzmann measures with exterior
magnetic field over spheres ∗
Zhengliang Zhang, † Bin Qian, ‡ Yutao Ma §
April 2009
Abstract
In this paper, We obtain the uniform Log-Sobolev inequality for the Boltzmann
measures by reducing multi-dimensional measures to one-dimensional
measures, and then applying the characterization on the constant of Log-
Sobolev inequality for a probability measure on the real line .
Keywords: Boltzmann measure, Log-Sobolev inequality, T 2 -transportation inequality,
Poincaré inequality.
AMS Classification Subjects 2000: 60E15 39B62 26Dxx
1 Introduction
Let S n−1 be the unit sphere in R n (n ≥ 3), and µ the normalized Lebesgue measure
on S n−1 , i.e. µ = σ n−1 /s n−1 , where σ n−1 and s n−1 are the uniform surface measure
and total volume respectively on S n−1 . For any h > 0 and e 1 = (1, 0, · · · , 0) ∈ S n−1 ,
let µ h be the probability on S n−1 given by
dµ h (x) = eh
c n (h) dµ(x), x ∈ Sn−1 ,
where c n (h) is the normalizing constant. This is the so-called Boltzmann measure
with exterior magnetic field [9].
∗ Research supported by National Science Funds of China (10626041, 10871153) and China
Scholarship council 2007U13020.
† Department of Mathematics and Statistics, Wuhan University, 430072-Hubei, China. E-mail
address: zzl731205@yahoo.com.cn; zhlzhang.math@whu.edu.cn
‡ Department of Mathematics and Statistics, Wuhan University, 430072-Hubei, China. E-mail
address: binqiancn@yahoo.com.cn
§ Department of Mathematics, Beijing Normal University, 100875-Beijing, China. E-mail address:
mayt@bnu.edu.cn
1

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
∫
Ent µ (f 2 ) ≤ C |∇f| 2 dµ,
where the entropy is defined by
∫
(∫
Ent µ (f 2 ) = f 2 log f 2 dµ −
) (∫
f 2 dµ log
)
f 2 dµ .
To be short we write µ ∈ LSI(C) for this inequality. Its property was well introduced
by Gross [11]. Though here we just study the Log-Sobolev inequality, it’s
well-known that Poincaré, transportation and Log-Sobolev inequalities are essential
tools in the study of concentration of measures (see e.g. [12], [10]), and there exist
some relations among the three types of inequalities: log-Sobolev inequality is
strictly stronger than Talagrand’s transportation cost inequality ([12], [6] and [2]),
which is stronger than Poincaré inequality (see [4], section 4.1).
The main method we applied is to reduce the proof of the measures µ x on spheres
to that of a probability measures ν x on [0, π], and then give an estimate on the
characterization of a probability measure on R which satisfies a logarithmic Sobolev
inequality.
2 Main result
In this paper, we get the uniform Log-Sobolev inequality in h > 0 for the Boltzmann
measures. The main results are shown as follows:
Theorem 2.1. For any h > 0, the probability measures µ h satisfy the uniform Log-
Sobolev inequality, that is, for any smooth function f : S n−1 → R, the following
inequality holds:
Ent µh (f 2 ) ≤ C LS
∫S n−1 |∇ S n−1f| 2 dµ,
where C LS is independent of h > 0, but depends on the dimension n.
We know that Log-Sobolev inequality is stronger than T 2 -transportation inequality,
and T 2 -transportation inequality stronger than Poincaré inequality, so we have
the following corollaries.
Corollary 2.2. For any h > 0, the probability measures µ h ∈ P I(C P ), C P independent
of h, that is, for any smooth function f : S n−1 → R, if the following inequality
holds:
where C P is independent of h > 0.
Var µh (f) ≤ C P
∫S n−1 |∇ S n−1f| 2 dµ,
2

Corollary 2.3. For any h > 0, the probability measures µ h ∈ T 2 (C), C independent
of h.
Remark: In fact, in [7], F.Q. Gao and L.M. Wu have already given the uniform
Poincaré inequality for the Boltzmann measures. Moreover, they find a more precise
uniform spectral gap. By their result on Poincaré inequality and an additional
condition given by Miao and the first author in [15], we can also obtain Corollary
2.3 quickly without Theorem 2.1 .
3 Idea of Proof
In this paper, following an idea in [2], the method consists in reducing the proof
to some one dimensional measures for which precise estimates of the log-Sobolev
constant are known see [3].
Notice that the case h = 0, i.e. µ h = µ, is the simplest one, and for any h > 0
fixed, it’s well-known that Log-Sobolev inequality holds for the measures µ h , so we
just need to consider the cases that h is large enough. Through the whole paper,
we assume that h is large enough.
Let ν h be the image of µ h by the map x → d(e 1 , x). It is a probability on [0, π]
with density
ρ n,h := dν h
dθ = 1
C n (h) eh cos θ (sin θ) n−2 ,
where C n (h) is the normalizing constant, whose order of h is e h h − n−1
2 as h goes to
infinity (see Lemma 5.1).
In [2], F. Barthe and the first author give the following lemma, which makes it
possible for us to reduce to the discussion on one-dimensional measures.
Lemma 3.1. Let λ be a probability measure on S n−1 with
dλ(x) = φ(d(x, e 1 ))dσ n−1 (x), x ∈ S n−1 ,
where φ is measurable. Let ν be the image of λ by the map x → d(x, e 1 ). Assume that
ν satisfies a Log-Sobolev inequality with constant C LS , then λ satisfies a Log-Sobolev
inequality with constant max(C LS , c n−2 ), where c n−2 is the Log-Sobolev constant of
the uniform probability σ n−2 /s n−2 on S n−2 , that is, for every smooth function f :
S n−1 → R,
∫
Ent λ (f 2 ) ≤ max(C LS , c n−2 ) |∇ S n−1f| 2 dλ.
S n−1
As far as the measure µ h is concerned, by the Lemma 3.1 we just need to discuss
the measure ν h . The generator associated with ν h is
L h f(θ) = d2
dθ 2 f + (−h sin θ + (n − 2) cot θ) d dθ f
It’s well-known that Bakry-Emery criteria gives a curvature condition to get the
Log-Sobolev inequality. However, the curvature condition requires a non-negative
lower bound.
3

Theorem 3.2. (Bakry and Emery [1]) Let dν = e −Ψ dx be a probability measure
on a manifold M, such that Ψ ∈ C 2 (M) and D 2 Ψ + Ric ≥ ρI n , ρ > 0. Then
dν ∈ LSI(ρ).
For the measure dν h = e h cos θ (sin θ) n−2 dθ/C n (h) on [0, π], we have
Ψ = −(h cos θ + (n − 2) log sin θ) + log C n (h),
D 2 Ψ = h cos θ + (n − 2) csc 2 θ.
Since θ = π − ε the second derivative D 2 Ψ behaves like −h + (n − 2)ε −2 , the
curvature (or the second derivative if you prefer) is not bounded below by some
positive constant for all h, and then Bakry-Emery criterion does not apply.
4 Proofs of main result
Here we’ll make use of the refined characterization from Barthe and Roberto [3]. Of
course, we may also use that one from Bobkov, and Götze [5], after all, we don’t
want to give a sharp estimate on the logarithmic Sobolev constants. Now we state
the refined characterization as follows.
Theorem 4.1. ([3]) Let µ, ν be Borel measures on R with µ(R) = 1 and dν(x) =
n(x)dx. Let m be a median of µ. Let C be the optimal constant such that for every
smooth function f : R → R, one has
∫
Ent µ (f 2 ) ≤ C f ′2 dν.
Then max(b − , b + ) ≤ C ≤ 4 max(B − , B + ), where
(
) ∫
1
x
1
b + = sup µ([x, ∞)) log 1 +
x>m
2µ([x, ∞)) m n ,
(
)
e 2 ∫ x
1
B + = sup µ([x, ∞)) log 1 +
x>m
µ([x, ∞)) m n ,
(
) ∫
1
m
1
b − = sup µ((−∞, x]) log 1 +
x

where m h is the median of ν h , and it tends to 0 as h → +∞ (see Lemma 5.2 in
appendix part). Moreover, we also give the speed rate of m h : there exist positive
constants M 0 , M, s.t. M 0 ≤ h(1 − cos m h )) ≤ M (see Lemma 5.3).
By a change of variables θ = arccos x, for arbitrary α, β ∈ (0, π), we have
∫ β
α
ρ n,h (θ)dθ =
∫ β
α
∫ β
α
e h cos θ (sin θ) n−2
C n (h)
1
ρ n,h (θ) dθ =
∫ cos α
cos β
∫ cos α
dθ =
cos β
C n (h)
e hx (1 − x 2 ) n−1
2
e hx (1 − x 2 ) n−3
2
C n (h)
Since we just want to prove that C LS is bounded, c = 1/2 or c = e 2 has no direct
influence on our proof. For simplicity, we’ll just estimate S − (h, 1) and S + (h, 1),
denoted by S − (h), and S + (h) respectively for short. In order to avoid some trouble,
we’ll denote all the constants independent of h by the uniform constant C.
(I) First we study the supper bound on S − (h). Let r = h(1 − t),
S − (h) =
sup
α∈(0,m h )
= sup
t∈(cos m h ,1)
(∫ α
)
ρ n,h (θ)dθ log
0
∫ 1
·
t
(
1 +
e hx (1 − x 2 ) n−3
2 dx log
(1 +
( ∫ t
cos(m h )
dx
e hx (1 − x 2 ) n−1
2
1
∫ α
ρ 0 n,h(θ)dθ
Since 0 < cos m h < t < 1 (when h is large enough), we have
∫ 1
t
∫ 1
e hx (1 − x) n−3
2 dx ≤
∫ 1
t
t
)
e hx (1 − x 2 ) n−3
n−3
2 dx ≤ 2 2
e hx (1 − x) n−3
2 dx =
e h
h n−1
2
.
∫ h(1−t)
0
dx.
α
dx,
) (∫ mh
)
1
ρ n,h (θ) dθ
C n (h)
∫ 1
t
ehx (1 − x 2 ) n−3
2 dx
∫ 1
t
e −y y n−3
2 dy,
and (noting that 0 < M 0 ≤ h(1 − cos m h ) ≤ M (see Lemma 5.3))
)
e hx (1 − x) n−3
2 dx,
∫ t
cos(m h )
dx
e hx (1 − x 2 ) n−1
2
≤ h ∫ n−3
2 h(1−cos mh )
e h h(1−t)
e y dy
y n−1
2
≤ h n−3
2
e h
∫ M
h(1−t)
e y dy
,
y n−1
2
Thus with C n (h)h n−1
2 e −h ≤ C (see Lemma 5.1)), we get
S − (h) ≤
≤
sup
r∈(0,M)
sup
r∈(0,M)
2 n−3
2
h
2 n−3
2
h
∫ r
0
r n−1
2
n−1
2
2 n+1
2 e M
≤
(n − 1)(n − 3) · 1
h
e −y y n−3
2 dy log
(1 +
(
log
1 + n − 1
2
(
sup r log
r∈(0,M)
C
∫ r
0 e−y y n−3
Cr − n−1
2
) e
M
2 dy
n−3
r − n−3
2
2
1 + n − 1 Cr − n−1
2
2
) (∫ M
)
r
)
e y dy
y n−1
2
5

Clearly, S − (h) is uniformly bounded in h > 0.
(II) Second we study the supper bound on S + (h).
(∫ π
) (
) (∫
1
α
)
1
S + (h) = sup ρ n,h (θ)dθ log 1 + ∫ π
α∈(m h ,π) α
ρ α n,h(θ)dθ m h
ρ n,h (θ) dθ
∫ )
t
= sup e hx (1 − x 2 ) n−3
C n (h)
2 dx log
(1 + ∫ t
t∈(−1,cos m h ) −1
−1 ehx (1 − x 2 ) n−3
2 dx
( ∫ )
cos mh
dx
·
t
e hx (1 − x 2 ) n−1
2
(1) When −1 < t < 0,
First we work on the first term of S + (h) in the right hand side,
∫ t
−1
∫ t
e hx (1 − x 2 ) n−3
n−3
2 dx ≤ 2 2
then on the third term,
−1
e hx (1 + x) n−3 2 n−3
2
2 dx ≤
h n−1
0
2 e h ∫ h(1+t)
e y y n−3
2 dy,
∫ cos mh
t
dx
e hx (1 − x 2 ) n−1
2
≤
∫ 0
t
dx
e hx (1 + x) n−1
2
+
∫ cos mh
0
dx
e hx (1 − x) n−1
2
,
notice that
∫ cos mh
0
dx
e hx (1 − x) n−1
2
∫ cos mh
≤ 2 n−1
2 +
1/2
≤ 2 n−1
2 +
dx
e hx (1 − x) n−1
2
e −h/2 h n−1
2
(h(1 − cos m h )) n−1
2
≤ C (independent of h),
and
∫ 0
t
dx
e hx (1 + x) n−1
2
∫ h
= h n−3
2 e
h
h(1+t)
dy
e y y n−1
2
∫ +∞
≤ h n−3
2 e
h
h(1+t)
dy
e y y n−1
2
.
Let q = h(1 + t), by the monotonicity of x log(1 + C ) in x > 0,
x
S + (h) ≤ sup C2 n−3
2
q∈(0,h)
+ sup
q∈(0,h)
2 n−3
2
h
h n−1
∫ q
0
∫
1 q
2 e h
0
e y y n−3
2 dy log
(
=: sup I 1 (h) + sup I 2 (h).
q∈(0,h)
q∈(0,h)
i). When lim h→+∞ q = 0,
e y y n−3
2 dy log
(
6
1 + h n−1
2 e h C n (h)
1 + h n−1
2 e h C n (h)
∫
2 n−3 q
2
0 ey y n−3
∫
2 n−3 q
2
0 ey y n−3
2 dy
2 dy
) ∫ +∞
q
)
dy
e y y n−1
2

I 2 (h) ≤ 2 n−3
2
h
= 2 n−3
2
n−1
2
∫ q
0
n−3
2
y n−3
2 dy
∫ +∞
q n−1
2
h
1
q n−3
2
≤C q (
)
h log 1 + Ce 2h q − n−1
2 → 0.
I 1 (h) → 0,
)
dy
log
(1 + h n−1
2 e h C n (h)
∫
q y n−1
2 2 n−3 q
2 y n−3
2 dy
0
(
n−1
2
log 1 +
h )
n−1
2 e h C n (h)
, (C
2 n−3
2 q n−1
n (h) ∼ eh
2
ii). When lim h→+∞ q = +∞ : note that 0 < q < h,
(
)
I 1 (h) ≤ C2 n−3 q n−3
2 e q
2
h n−1
2 e log 1 + h n−1
2 e h C n (h)
h 2 n−3
2 q n−3
2 e q
≤ C log(1 + Ce2h )
h
I 2 (h) ≤ 2 n−3
2 q n−3
2 e q
log(1 + Ce 2h )e −q 2 n−3
q− 2 = C log(1 + Ce2h )
h
n − 3 h
h n−1
2
Clearly I 1 (h), I 2 (h) are bounded in h in the case −1 < t < 0, so is S + (h).
(2) When 0 ≤ t < cos m h :
Let p = h(1 − t) ≥ h(1 − cos m h ) ≥ M 0 > 0,
∫ t
−1
∫ cos mh
t
∫ t
e hx (1 − x 2 ) n−3
n−3
2 dx ≤ 2 2
dx
e hx (1 − x 2 ) n−1
2
≤ h n−3
2
e h
−1
e hx (1 − x) n−3 2 n−3
2 e h
2 dx ≤
∫ h(1−t)
h(1−cos m h )
By the monotonicity of x log(1 + C ) in x > 0,
x
S + (h) ≤
≤
sup
p∈(M 0 ,2h)
sup
p∈(M 0 ,2h)
·
2 n−3
2
h
(∫ p
·
2 n−1
2
p
( ∫ p/2
M 0
∫ 2h
p
e −y y n−3
2 dy log
(
e y y − n−1
h(1−cos m h )
∫ +∞
p
2 dy
)
e −y y n−3
2 dy log
(1 +
e y y − n−1
2 dy +
∫ p
p/2
h n−1
2
e y y − n−1 h n−3
2
2 dy ≤
e h
∫ 2h
p
∫ p
1 + e−h h n−1
2 C n (h)
2 n−3
2
∫ +∞
∫ 2h
p
e −y y n−3
2 dy
C
)
e −y y n−3
2 dy,
M 0
e y y − n−1
2 dy.
)
)
e
p
−y y n−3
2 dy
)
e y y − n−1
2 dy , (C n (h) ∼ eh
h n−1
2
).
7

Notice that
∫ +∞
p
e −y y n−3
n−1
2 dy ≤ p 2
∫ +∞
1
= e −p p n−1
2
≤ Ce −p p n−1
2
≤ Ce −p p n−3
2 .
e −pz z n−3
2 dz
∫ +∞
0
∫ +∞
Then by the monotonicity of x log(1 + C ), we have
x
)
S + (h) ≤
≤
sup
p∈(M 0 ,2h)
sup
p∈(M 0 ,+∞)
2 n−1
2 Cp n−3
2
pe p
C
(
e − p 2 p
n−3
(
log
0
e −pz (z + 1) n−3
2 dz
e −pz e M 0z/2 dz
(
1 + Ce p p − n−3 2
2 ·
n − 3
(
) log
2
0 + 2 n−3
2
p
2 M
− n−3
e p 2
M n−3
2
0
1 + Ce p p n−3
2
)
+ ep 2 n−3
2
p n−3
2
From the quality above, we see that S + (h) is still bounded as p → +∞. Therefore,
we get that S + (h) is bounded in h in the case 0 < t < cos m h .
Summarizing the cases (1) and (2), we obtain S + (h) is uniformly bounded in
h > 0.
By the boundness of S − (h) and S + (h) and Theorem 4.1, we finish our proof.
.
)
5 Appendix
Lemma 5.1. There exist positive constants C 1 (n) and C 2 (n) such that for say h ≥ 1,
C 1 (n)e h h − n−1
2 ≤ C n (h) ≤ C 2 (n)e h h − n−1
2 .
Lemma 5.2. The median m h of the measure ν h tends to 0 as h → +∞.
Proof. By the definition of median, we have
∫ π
e h cos θ (sin θ) n−2 dθ = C n(h)
m h
2
However
∫ π
m h
e h cos θ (sin θ) n−2 dθ =
∫ cos mh
−1
≤ 2 n−3
2 + 2 n−3
2
= 2 n−3
2 + 2 n−3
2
e hx (1 − x 2 ) n−3
2 dx
∫ cos mh
0
e h
h n−1
2
∫ h
∼ e h h − n−1
2 .
e hx (1 − x) n−3
2 dx
h(1−cos m h )
e −y y n−3
2 dy
∼ e h cos mh · {(1 − cos m h ) n−3
2 h
−1 ∨ h − n−1
2 } (as h large).
8

By the comparison of the order of h, cos m h must tend to 1 as h → +∞. Thus
we reach the conclusion required.
Lemma 5.3. {h(1 − cos m h ); h > 0} is uniformly bounded in h. Moreover, there
exist positive constants M 0 , M, s.t. 0 < M 0 ≤ h(1 − cos m h ) ≤ M when h is large
enough.
Proof. Since
and by lim h→+∞ cos m h = 1
∫ mh
0
e h cos θ (sin θ) n−2 dθ = C n(h)
,
2
We have
∫ mh
0
∫ 1
e h cos θ (sin θ) n−2 dθ = e hx (1 − x 2 ) n−3
2 dx
cos m h
≃2 n−3
2
∫ 1
=2 n−3
2 e h h − n−1
2
cos m h
e hx (1 − x) n−3
2 dx
∫ h(1−cos mh )
0
e −y y n−3
2 dy,
∫ h(1−cos mh )
0
e −y y n−3
2 dy ≃ 2
1−n
2 e −h h n−1
2 Cn (h)
{
≤ 2 1−n
2 e −h h n−1
2 2 n−3
2 + 2 n−3
2
≤ 1 2 h n−1
2 e −h + 1 2
→ 1 2
∫ +∞
0
∫ +∞
0
e h
h n−1
2
e −y y n−3
2 dy
∫ +∞
e −y y n−3
2 dy (as h → +∞).
0
e −y y n−3
2 dy
}
Clearly, by the inequality above we know that h(1−cos m h ) doesn’t tend to infinity
as h → +∞, that is, {h(1 − cos m h ); h > 0} uniformly bounded in h. Moreover,
from lim h→+∞ 2 1−n
2 e −h h n−1
2 C n (h) > 0, we have
lim h(1 − cos m h) > 0.
h→+∞
Example : When n = 3, dν h (θ) = eh cos θ sin θ
dθ, where C 3 (h) is the normalized
C 3 (h)
constant, given as follows:
C 3 (h) =
∫ π
0
e h cos θ sin θdθ = eh − e −h
.
h
9

m h is the median of ν h , by the definition of median, we have
Then
Moreover,
∫ mh
0
e h cos θ (sin θ) n−2 dθ = C 3(h)
,
2
( )
1
m h = arccos
h log eh + e −h
.
2
lim h(1 − cos m h) = log 2.
h→+∞
References
[1] D. Bakry and M. Emery, Diffusions hypercontractivies, In Sém. Proba. XIX,
LNM, 1123, Springer (1985), 177-206.
[2] F. Barthe and Z.L. Zhang, Poincare, transportation and logarithmic Sobolev
inequalities for harmonic measures on spheres, to submit.
[3] F. Barthe and C. Roberto, Sobolev inequalities for probabilty measures on the
real line, Studia Mathematica, 159 (3) (2003), 481-497.
[4] S. G. Bobkov, I. Gentil and M. Ledoux, Hypercontractivity of Hamilton-Jacobi
equations, J. Math. Pures Appl. 80 (2001) 669-696.
[5] S. G. Bobkov and F. Götze, Exponential integrability and transportation cost
related to logarithmic Sobolev inequalities, J. Funct. Anal. 163 (1999), 1-28.
[6] P. Cattiaux and A. Guillin, On quadratic transpotation cost inequalities, J.
Math. Pures Appl. 86 (2006) 342-361.
[7] F.Q. Gao and L.M. Wu, Transportatoin-Information inequalities for Gibbs
measures. (in preparation)
[8] H. Gjellout, A. Guillin and L. Wu, Transportation cost-information inequalities
and applications to random dynamical systems and diffusons, Ann. Proba., Vol
32, No. 3B (2004), 2702-2732.
[9] J. Glimm and A. Jaffe, Quantum Physics, A Functional Integral Point of View,
Springer-Verlag, New York Heidelberg Berlin.
[10] A. Guionnet and B. Zegarlinski, Lectures on logarithmic Sobolev inequalities,
in: Séminaire de Probabilité XXXVI, Lecture Notes in Math. 1801, Springer,
Berlin, 2003, 1-134.
[11] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061-
1083.
10

[12] M. Ledoux, The Concentration of Measures Phenomenon, Math. Surveys Monographs
89, Amer. Math. Soc., Providence, RI, 2001.
[13] K. Marton, Bounding ¯d-distance by informational divergence: a method to
prove measure concentration, Ann. Prob. 24 (1996) 857-866.
[14] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links
with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), no. 2, 361–
400.
[15] Z.L. Zhang and Y. Miao, An equivalent condition between Poincare inequality
and T 2 -transportation cost inequality, Acta Appl. Math.DOI 10.1007/s10440-
008-9379-z.
11