Reversible Anosov diffeomorphisms and large deviations

9 Pages·0.15 MB·

by  Giovanni Gallavotti

Reversible Anosov diffeomorphisms and large deviations.
Giovanni Gallavotti∗
Abstract: the volume contraction obeys a large deviation rule.1
§1. – Anosov maps and thermodynamic formalism.
5 This section reviews the basic results of Sinai’s theory of transitive Anosov diffeomor-
9 phisms. It is meant for the reader with some familiarity with the subject and it should
9
be used for notations only. Expert readers will probably find the rest of the paper self
1
contained, without this section.
n
Let C be a d–dimensional, C∞, compact manifold and let S be a C∞, transitive Anosov
a
J diffeomorphism, [AA], [S]. If Wu,Ws denote the unstable or stable manifold at x ∈ C, we
x x

1 call Wu,δ,Ws,δ the connected parts of Wu,Ws containing x and contained in the sphere
x x x x
2 with center x and radius δ. Let d ,d be the dimensions of Wu,Ws: d = d +d . It is
u s x x u s
convenienttotakeδ alwayssmallerthanthesmallestcurvatureradiusofWu,Ws forx∈C.
1 x x
v Transitivity means that Wxu,Wxs are dense in C for all x∈C.
1 The map S can be regarded, locally near x, either as a map of C to C or of Wu to Wu ,
x Sx
1 or of Ws to Ws . The jacobian matrices of the ”three” maps will be d×d, d ×d and
0 x Sx u u
d ×d matrices denoted respectively ∂S(x), ∂S(x) , ∂S(x) . The absolute values of the
1 s s u s
0 respective determinants will be denoted, respectively, Λ(x), Λu(x), Λs(x) and are Ho¨lder
5 continuous functions, strictly positive (in fact Λ(x) is C∞), [S], [AA]. Likewise one can
9
define the jacobians of the n–th iterate of S which are denoted by appending a label n to
/
n Λ,Λ ,Λ and are related to the latter by:
u s
y
d
-
o n−1 n−1 n−1
a Λn(x)= Λ(Sjx), Λu,n(x)= Λu(Sjx), Λs,n(Sjx)= Λs,n(Sjx)
h (1.1)
j=0 j=0 j=0
c Y Y Y
: Λn(x)=Λu,n(x)Λs,n(x)χn(x)
v
i
X and χ (x)= sinα(Snx) is the ratio of the sines of the angles α(Snx) and α(x) between Wu
n sinα(x)
r and Ws at the points Snx and x. Hence χ (x) is bounded above and below in terms of a
a n
constant B >0: B−1 ≤χ (x)≤B, for all x (by the transversality of Wu and Ws).
n
A set E is a rectangle with center x and axes ∆u,∆s if:
1) ∆u,∆s are two connected surface elements of Wu,Ws containing x.
x x
2) for any choice of ξ ∈ ∆u and η ∈∆s the local manfolds Ws,δ and Wu,δ intersect in one
ξ η
∗
Fisica,Universita`diRomaLaSapienza, P.leMoro2,00185,RomaItalia.
1 Archivedinmp [email protected],#95-19
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and only one point x(ξ,η)=Ws,δ∩Wu,δ. The point x(ξ,η) will also be denoted ξ×η.
ξ η
3) the boundaries ∂∆u and ∂∆s (regarding the latter sets as subsets of Wu and Ws) have
x x
zero surface area on Wu and Ws.
x x
4) E is the set of points ∆u×∆s.
Note that any x′ ∈E can be regarded as the center of E because there are ∆′u,∆′s both
containing x′ and such that ∆u ×∆s ≡ ∆′u ×∆′s. Hence each E can be regarded as a
rectangle centered at any x′ ∈E (with suitable axes). See figure.
ξ×η
E
∆s η
x ∆u
ξ
Thecircleisasmallneighborhoodofx;thefirstpictureshowstheaxes;theintermediatepictureshowsthe
× operation and Wηu,δ,Wξs,δ; the third picture shows the rectangle E with the axes and the four marked
points are the boundaries ∂∆u and ∂∆s. The picture refers to a two dimensional case and the stable and
unstablemanifoldsaredrawnasflat,i.e. the∆’sareverysmallcomparedtothecurvatureofthemanifolds.
The center x is drawn in a central position, but it can be any other point of E provided ∆u and ∆s are
correspondinglyredefined. Oneshouldmeditateonthesymbolicnatureofthedrawinginthecasesofhigher
dimension.
The unstable boundary of a rectangle E will be the set ∂ E = ∆u × ∂∆s; the stable
u
boundary will be ∂ E = ∂∆u×∆s. The boundary ∂E is therefore ∂E = ∂ E∪∂ E. The
s s u
set of the interior points of E will be denoted E0. A pavement of C will be a covering
E = (E ,...,E ) of C by N rectangles with pairwise disjoint interiors. The stable (or
1 N
unstable) boundary ∂ E (or ∂ E) of E is the union of the stable (or unstable) boundaries of
s u
the rectangles E : ∂ E =∪ ∂ E and ∂ E =∪ ∂ E .
j u j u j s j s j
A pavement E is called markovian if its stable boundary ∂ E retracts on itself under the
s
action of S and its unstable boundary retracts on itself under the action of S−1, [S], [Bo];
this means:
S∂ E ⊆∂ E, S−1∂ E ⊆∂ E (1.2)
s s u u
Setting Mj,j′ =0,j,j′ ∈{1,...,N},ifSEj0∩Ej0′ =∅andMj,j′ =1otherwisewecallC the
set of sequences j =(j )∞ , j ∈{1,...,N} such that M ≡1. The transitivity of
k k=−∞ k jk,jk+1
thesystem(C,S)impliesthatM istransitive: i.e. thereisapowerofthematrixM withall
entries positive. The space C will be called the space of the compatible symbolic sequences.
If E is a markovianpavement and δ is small enough the map:
∞
X :j ∈C → x= S−kE ∈C (1.3)
jk
k=−∞
\
is continuous and 1−1 between the complement C ⊂ C of the set N = ∪∞ Sk∂E and
0 k=−∞
the complement C ⊂ C of X−1(N). This map is called the symbolic code of the points of
0
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C: it is a code that associateswith eachx6∈N a sequence of symbols j which arethe labels
of the rectangles of the pavement that are successively visited by the motion Sjx.
ThesymboliccodeX transformstheactionofS intotheleftshiftϑonC: SX(j)=X(ϑj).
Akeyresult,[S],is thatittransformsthevolume measureµ onC into aGibbs distribution,
0
[LR], [R], µ on C with formal hamiltonian:
0
−1 ∞
H(j)= h (ϑkj)+h (j)+ h (ϑkj) (1.4)
− 0 +
k=−∞ k=0
X X
where, see (1.1):
h (j)=−logΛ (X(j)), h (j)=logΛ (X(j)), h (j)=logsinα(X(j)) (1.5)
− s + u 0
If F is smooth on C the function F(j)=F(X(j)) can be represented in terms of suitable
functions Φ (j ,...,j ) as:
k −k k
∞
F(j)= Φ (j ,...,j ), |Φ (j ,...,j )|≤ϕe−λk (1.6)
k −k k k −k k
k=1
X
where ϕ>0,λ>0. In particular h (and h ) enjoy the property (1.6) (short range).
± 0
If µ ,µ are the Gibbs states with formal hamiltonians:
+ −
∞ ∞
h (ϑkj), h (ϑkj) (1.7)
+ −
k=−∞ k=−∞
X X
the distributions µ on C, images of µ via the code X in (1.3), will be the forward and
± ±
backward statistics of the volume distribution µ (corresponding to µ via the code X), [S].
0 0
This means that:
T−1
1
lim F(S±kx)= µ (dy)F(y)≡µ (F) (1.8)
± ±
T→∞T k=0 ZC
X
for all smooth F and for µ –almost all x ∈ C. The distributions µ are often called the
0 ±
SRB distributions, [ER]; the above statements and (1.7),(1.8) constitute the content of a
well known theorem by Sinai, [S].
An approximation theorem for µ can be given in terms of the coarse graining of C
+
generated by the markovian pavement E = T S−kE.2 If E ≡ ∩T S−kE
T k=−T j−T,...,jT k=−T jk
and if x is a point, arbitrarily chosen, in the coarse grain set E we define
j−T,...,jT W j−T,...,jT
the distribution µ by setting:
T,τ
2 Where ∨ denotes the operation which, given two pavements E,E′ generates a new pavement E∨E′: the
rectangles ofE∨E′ simplyconsistofalltheintersections E∩E′ ofpairsofrectangles E∈E andE′∈E′.
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−1
Λ (x )F(x )
µ (F)≡ µ (dx)F(x)= j−T,...,jT u,τ j−T,...,jT j−T,...,jT
T,τ T,τ −1
ZC P j−T,...,jT Λu,τ(xj−T,...,jT) (1.9)
τ/2−1 P
Λ (x)d=ef Λ (Skx)
u u
k=Y−τ/2
Then for all smooth F it is: lim µ (F) = µ (F). Note that equation (1.9)
T≥τ/2,τ→∞ T,τ +
can also be written:
e− τk=/2−−τ1/2h+(ϑkj0)F(X(j0)
µ (F)= j−T,...,jT (1.10)
T,τ P Pe− τk=/2−−τ1/2h+(ϑkj0)
j−T,...,jT
P
P
where j0 ∈ C is a compatible sequence, arbitrarily chosen, which agrees with j ,...,j
−T T
between −T and T. (i.e. X(j0)=x ∈E ).
j−T,...,jT j−T,...,jT
Notation: to simplify the notations we shall write q for the elements q = (j ,...,j ) of
−T T
{1,...,N}2T+1; and E will denote E and x a point of E .
q j−T,...,jT q q
Remark: Notethattheweightsin(1.10)dependonthespecialchoicesofthecentersx (i.e.
q
of j0); but if x varies in E the weight of x changes by at most a factor, bounded above
q q q
by some B <∞ and below by B−1, for all T ≥ 0: this is a consequence of the short range
properties of h expressed by (1.6)).
+
The last formula shows that the forwardstatistics of µ can be regardedas a Gibbs state
0
for a short range one dimensional spin chain with a hard core interaction. The spin at x is
the value of j ∈ {1,...,N}; the short range refers to the fact that, Λ (x) being smooth,
x u
the function h (j)≡logΛ (X(j)) can be representedas in (1.6) where the Φ play he role
+ u k
of”manyspins”interactionpotentialsandthehardcorereferstothefactthattheonlyspin
configurations j allowed are those with M ≡1 for all integers k.
jk,jk+1
§2. – Reversible dissipative systems and results.
Let (C,S) be a transitive, smooth Anosov system, see §1, and let Λ(x) = |det∂S(x)|;
let µ be the forward and backward statistics of the volume measure µ (i.e. the SRB
± 0
distributions for S and S−1).
Definition (A): The system (C,S) is dissipative if:
− µ (dx)logΛ(x)±1 =σ >0 (2.1)
± ±
ZC
Definition (B): The system (C,S) is reversible if there is an isometric involution i:C←→C,
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(i2 =1), such that: iS =S−1i.
We consider from now on only dynamical systems (C,S) verifying (A),(B).
Remarks:
1)(A),(B)implyσ =σ andΛ(x)=Λ(ix)−1;furthermoreiWu =Ws andthedimensions
+ − x ix
of the stable and unstable manifolds d ,d are equal: d =d and d=d +d is even.
s u u s u s
2) if Λ (x),Λ (x) denote the absolute values of the jacobian determinants of S as a map of
u s
Wu to Wu and of Ws to Ws , then Λ (x)=Λ (ix)−1.
x Sx x Sx u s
Definition: The dimensionless entropy production rate or the phase space contraction rate
at x∈C and over a time τ is the function ε (x):
τ
τ/2−1
x→ε (x)= 1 logΛ−1(Sjx)= 1 logΛ−1(x) (2.2)
τ
σ τ σ τ
+ +
j=X−τ/2
with Λ(x)d=ef τ/2−1Λ(Sjx) (see (1.9)).
−τ/2
Remarks: Q
i) by definition (see (2.1)):
T−1
1
hε i = lim ε (Sjx)≡ µ (dy)ε (y)=1 (2.3)
τ + T→+∞T τ + τ
j=0 Z
X
with µ –probability 1.
0
ii) Note that, on the other hand, lim ε (x) = 0 with µ –probability 1, by the re-
τ→∞ τ 0
versibility (B).
Here we prove the following fluctuation theorem:
Fluctuation theorem: There exists p∗ >0 such that the SRB distribution µ verifies:
+
1 µ ({ε (x)∈[p−δ,p+δ]})
+ τ
p−δ ≤ lim log ≤p+δ (2.4)
τ→∞σ+τ µ+({ετ(x)∈−[p−δ,p+δ]})
for all p, |p|<p∗.
The above theorem was first informally proved in [GC1],[GC2] where its interest for
nonequilibrium statistical mechanics was pointed out. Although I think that the physi-
cal interest of the theorem far outweighs its mathematical aspects, see also [G1], it appears
that it might be useful to write the explicit and formal proof (more detailed than [G2]) de-
scribed below. The theorem can be regarded as a large deviation result for the probability
distribution µ .
+
ThestrategythatIshallfollowtotakeadvantageoftheexistingliteratureisthefollowing.
First the function (2.2) is convertedto a function on the spin configurationsj ∈C (see §1):
τ/2−1
1
ε˜ (j)=ε (X(j))= L(ϑkj) (2.5)
τ τ
τ
k=X−τ/2
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where L(j)≡ 1 logΛ(X(j))−1 has a short range representation of the type (1.6).
σ+
The SRB distribution µ can be regardedas a Gibbs state µ with shortrange potential
+ +
onthespaceC ofthe compatiblesymbolicsequences,associatedwithaMarkovpartitionE.
Therefore there is a function ζ(s) real analytic in s for s ∈ (−p∗,p∗) for a suitable p∗ > 0,
strictly convex and such that if p<p∗ and [p−δ,p+δ]⊂(−p∗,p∗) it is:
1
logµ ({ε˜ (j)∈[p−δ,p+δ]})−−−→ max −ζ(s) (2.6)
τ + τ τ→∞ s∈[p−δ,p+δ]
and the difference between the r.h.s. and the l.h.s. tends to 0 bounded by Dτ−1 for a
suitable constant D. The function ζ(s) is the Legendre transform of the function λ(β)
defined as:
1
λ(β)= lim log eτβε˜τ(j)µ (dj) (2.7)
τ→∞τ +
Z
i.e. λ(β) = maxs∈(−p∗,p∗)(βs−ζ(s)), where p∗ can be taken p∗ = limβ→+∞β−1λ(β) and
the function λ(β) is a real analytic, [CO], strictly convex function of β ∈ (−∞,∞) and
asymptotically linear: β−1λ(β)−−−−−→ ±p∗.
β→±∞
Theabove(2.6)isa”largedeviationstheorem”foronedimensionalspinchainswithshort
range interactions, [L].
Hence it will be sufficient to prove the following; if I =[p−δ,p+δ]:
p,δ
1 log µ+({ε˜τ(x)∈Ip,δ±η(τ)}) <p+δ+η′(τ) (2.8)
σ τ µ ({ε˜ (x)∈I )} >p−δ−η′(τ)
+ + τ −p,δ∓η(τ) (cid:26)
with η(τ),η′(τ)−−−→0.
τ→∞
§3. – Thermodynamic formalism (proof the fluctuation theorem).
Let, for n odd, j = (j ,j ,...,j ) if X = (x,x+1,...,x+n−1), and call
X x x+1 x+n−1
X =x+(n−1)/2 the center of X. If j ∈C is an infinite spin configurationwe also denote
j the set of the spins with labels x ∈ X. The left shift of the interval X will be denoted
X
by ϑ; i.e. by the same symbol of the left shift of a (infinite) spin configuration j.
Let l (j ) = l(n)(j ,j ,...,j ), h+(j ) = h(n)(j ,j ,...,j ) be transla-
X X x x+1 x+n−1 X X + x x+1 x+n−1
tion invariant functions, i.e. functions such that l (j) ≡ l (j) and h+ = h+(j), and
ϑX X ϑX(j) X
such that the functions h (j), see (1.6), and L(j), see (2.5), can be written:
+
L(j)= l (j ), h (j)= h+(j )
X X + X X
XX=0 XX=0 (3.1)
|l (j )|≤b e−b2n, |h+(j )|≤be−b′n
X X 1 X X
for suitable constants b ,b ,b,b′. Then τε˜ (j) can be written as l (j ).
1 2 τ X∈[−τ/2,τ/2−1] X X
Henceτε˜τ(j)canbeapproximatedbyτε˜Mτ (j)= (M)lX(jX)wPhere (M) meanssumma-
tion overthe sets X ⊆[−1τ−M,1τ+M],while X is in [−1τ,1τ−1]. The approximation
2 2 P 2 2 P
is described by:
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|τε˜M(j)−τε˜ (j)|≤b e−b4M ≡η (M) (3.2)
τ τ 3 0
for suitable3 b ,b and for all M ≥0. Therefore if I =[p−δ,p+δ] and M =0 we have:
3 4 p,δ
≤µ ({ε˜0 ∈I })
µ ({ε (x)∈I }) + τ p,δ+b3/τ (3.3)
+ τ p,δ ≥µ ({ε˜0 ∈I })
(cid:26) + τ p,δ−b3/τ
It follows from the general theory of 1–dimensional Gibbs distributions, [R2], that the
µ –probability of a spin configurationcoinciding with j in the interval[−1τ,1τ],4
+ [−τ/2,τ/2] 2 2
is:
e− ∗h+X(jX)
P(j ) (3.4)
h P i [−τ/2,τ/2]
·
j′
[−τ/2,τ/2]
h i
∗ P
where denotes summation over all the X ⊂ [−τ/2,τ/2−1]; the denominator is just
the sum of terms like the numerator,evaluatedat a generic (compatible) spin configuration
P
j′ ; finally P verifies the bound, [R2]:
[−τ/2,τ/2]
B−1 <P(j )<B (3.5)
1 [−τ/2,τ/2] 1
with B a suitable constant independent of j and of τ (B can be explicitly esti-
1 [−τ/2,τ/2] 1
mated in terms of b,b′). Therefore from (3.3) and (3.4) we deduce for any T ≥τ/2:
µ (ε (x)∈I )≤µ (ε˜0 ∈I )≤
+ τ p,δ + τ p,δ+b3/τ (3.6)
≤B µ (ε˜0 ∈I )≤B µ (ε˜ ∈I )
2 T,τ τ p,δ+b3/τ 2 T,τ τ p,δ+2b3/τ
for some constant B > 0; and likewise a lower bound is obtained by replacing B by B−1
2 2 2
and b by −b .
3 3
Then if p < p∗ and I ⊂ (−p∗,p∗) the set of the rectangles E ∈ T S−kE with center
p,δ −T
x such that ε (x)∈I is not empty.
τ p,δ
W
We immediately deduce the lemma:
Lemma 1: the distributions µ and µ , T ≥ 1τ, verify:
+ T,τ 2
1 µ (ε (x)∈I ) < logB22 + 1 log µT,τ(ε˜τ∈Ip,δ)
log + τ p,δ∓2b3/τ τσ+ τσ+ µT,τ(ε˜τ∈−Ip,δ) (3.7)
τσ+ µ+(ετ(x)∈−Ip,δ±2b3/τ) >−loτgσB+22 + τσ1+ logµµTT,τ,τ(ε(˜ε˜ττ∈∈−IIpp,δ,δ))
for Ip,δ ⊂[−p∗,p∗] and for τ so large that p+δ+2b3/τ <p∗.
Hence (2.8) will follow if we can prove:
3 Onecancheckfrom(3.1),thattheconstants b3,b4 canbeexpressedassimplefunctionsofb1,b2.
4 i.e. thespinconfigurations j′ suchthatjx′ =jx,x∈[−21τ,21τ].
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Lemma 2: there is a constant b such that the approximate SRB distribution µ verifies:
T,τ
1 µT,τ(ε˜∈Ip,δ) ≤p+δ+b/τ
log (3.8)
σ τ µ (ε˜∈−I ) ≥p−δ−b/τ
+ T,τ p,δ (cid:26)
for τ large enough (so that δ+b/τ <p∗−p) and for all T ≥τ/2.
The latter lemma will be proved in §4 and it is the only statement that does not follow
from the already existing literature.
§4. – Time reversal symmetry and large deviations.
The relation (3.7) holds for any choice of the Markov partition E. Note that if E is a
Markov pavement also iE is such (because iS = S−1i and iWu = Ws); furthermore if E
x ix 1
and E are Markov pavements also E =E ∨E are markovian. Therefore:
2 1 2
Lemma 3: there exists a time reversal Markov pavement E, i.e. a Markov pavement such
that E =iE.
ThiscanbeseenbytakinganyMarkovpavementE andsettingE =E ∨iE . Alternatively
0 0 0
one could construct the Markov pavement in such a way that it verifies automatically the
symmetry [G2]. Since the center of a rectangle E ∈ E can be taken to be any point x
q T q
in the rectangle E we can and shall suppose that the centers of the rectangles in E have
q T
been so chosen that the center of iE is ix , i.e. the time reversalof the center x of E .
q q q q
Forτ largeenoughthesetofconfigurationsq =j suchthatε (x)∈I forallx∈E
[−T,T] τ p,δ q
is not empty5 and the ratio in (3.8) can be written, if x is the center of E ∈ T E, as:
q q −T
−1 −1 W
Λ (x ) Λ (x )
ετ(xq)∈Ip,δ u,τ q ετ(xq)∈Ip,δ u,τ q
= (4.1)
−1 −1
P Λ (x ) P Λ (ix )
ετ(xq)∈−Ip,δ u,τ q ετ(xq)∈Ip,δ u,τ q
P P −1
But the time reversal symmetry implies that Λ (x) =Λ (ix), see remark 2) following
u,τ s,τ
definition (B), §2.6 This permits us to change (4.1) into:
ετ(xq)∈Ip,δΛu−,τ1(xq) <maxqΛu−,τ1(xq)Λs−,τ1(xq)
(4.2)
Pετ(xq)∈Ip,δΛs,τ(ixq) (>minqΛu−,τ1(xq)Λs−,τ1(xq)
where the maxima aPre evaluated as q varies with ε (x )∈I .
τ q p,δ
By(1.1)wecanreplaceΛ−1(x)Λ−1(x)withΛ−1(x)B±1,see(1.9),(2.2);thusnotingthat
u,τ s,τ τ
5henNcoete>th−a∞t ,p∗fo=raslulp|sx|l<imps∗u.pτ→+∞ετ(Sτ/2x) and let p∈(−p∗+δ,p∗−δ); furthermore ζ(s) is smooth,
6 hereitisessentialthatΛu,τ(x)istheexpansionoftheunstablemanifoldbetweentheinitialpointS−τ/2x
andthefinalpointSτ/2xwhichisatrajectoryoftimelength τ,whichatitscentral timeisinx.
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bydefinitionofthesetofq’sinthemaximumoperationin(4.2)itis 1 logΛ−1(x )∈I ,
σ+τ τ q p,δ
we see that (3.8) follows with b= 1 logB.
σ+τ
Corollary: the above analysis gives us a concrete bound on the speed at which the limits in
(2.4) are approached. Namely the error has order O(τ−1).
This is so because the limit (2.6) is reached at speed O(τ−1); furthermore the regularity of
λ(s) in (2.6) and the size of η(τ),η′(τ) and the error term in (3.8) have all order O(τ−1).
Theaboveanalysisprovesalargedeviationresultfortheprobabilitydistributionµ : since
+
m is a Gibbs distribution, see (1.7), various other large deviations theorems hold fo them,
+
[DV], [E], [O], but unlike the above they are not related to the time reversal symmetry.
Acknowledgements: I am indebted to E.G.D. Cohen and J.L. Lebowitz for many discus-
sionsandfortheirencouragement. ThisworkispartoftheresearchprogramoftheEuropean
Network on: ”Stability and Universality in Classical Mechanics”, # ERBCHRXCT940460.
References.
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1981.
[E] Ellis, R.S.: Entropy, large deviations and statistical mechanics, New York, Springer
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