3.3 Recurrence
One of the most important objections raised against the possibility of describing the approach to the statistical equilibrium in time-reversible dynamical system (as the Hamiltonian ones), is the famous one due to Zermelo. He made use of a a theorem due to Poincaré stating that almost any initial condition of a reversible dynamical system evolves in such a way that the representative point passes arbitrarily close to the initial one infinitely many times (invoking no hypothesis such as ergodicity or mixing). He thus contrasted the idea of Boltzmann who was trying to build up a (kinetic) theory of approach to equilibrium. In particular, the paradoxical case treated by Zermelo is that of a gas initially contained in the left half part of a vessel.
Due to the Poincaré recurrence theorem, the evolution of the gas is such that the gas will soon or later go back close to such a strange configuration, and it will do that infinitely many times. Actually there is no paradox, and such recurrences of “strange” states may actually take place in real systems. The point is that the probability of such states is usually extremely small, and quickly vanishes as the number of degrees of freedom of the system goes to infinity.
Moreover, by another theorem due to Kac, under the ergodicity hypothesis, the time needed to observe the recurrence of a small set A of initial data is proportional to the inverse of its probability, so that if the latter is extremely small, the former is extremely large, so large that the most pathological recurrences are practically unobservable. In the Zermelo example, the a priori probability that the gas occupies the left part of the cubic vessel, independently of the velocities of its N particles, is p = 1/2N. By the Kac theorem, if the gas evolution is ergodic, the recurrence time of such a configuration isτ/p = τ2N, whereτ is a physically relevant time-unit (the minimum time-scale on which something happens in the gas: e.g. the single binary collision time-scale). With the normal densities involved in gasses, one has about N = 1025 particles per cubic meter. Even with a time unitτ of the order of the picosecond (10−12seconds), the Kac recurrence time of such an event, in a cubic meter, turns out to be of the order of 101024 seconds, i.e. a time enormously larger than the present estimate of the age of the universe, which amounts to about 1018seconds only.
Poincaré theorem
Theorem 3.4 (Poincaré recurrence theorem). Let (Ω,Φt,µ) a dynamical system, and let A be any measurable subset of Ω such that µ(A) > 0. Then, almost every point x ∈ A is such that Φt(x) ∈ A for arbitrarily large values of the time t (i.e. ∀T > 0 ∃t > T :Φt(x) ∈ A).
C PROOF. We report the non simple proof due to Kac [21], which makes use of a construc-tion that can be later used to estimate the recurrence times.
First, let us discretize the dynamics: we observe the system at times multiple ofτ, the latter being an arbitrary time-unit. The discrete flow consists then in the iterates of the map
Ψτ:=Φτ, (3.25)
with the obvious relations Ψnτ =Φnτ, Ψ`τ=Ψ`τ=Φ`τ. Let us consider the sequence of sets {An}n≥1, where A1 consists of the points of A whose first iterate belongs to A, and
An(n ≥ 2) consists of the points of A which exit A at the first iterate and re-enter A for the first time at the n-th one, i.e.
A1:= {x ∈ A :Ψτ(x) ∈ A} = A ∩Ψ−1τ (A) ; (3.26)
An := {x ∈ A :Ψkτ(x) ∉ A, k = 1,..., n − 1; Ψnτ(x) ∈ A} =
= A ∩C[Ψ−1τ (A)] ∩ ··· ∩C[Ψ−n+1τ (A)] ∩Ψ−nτ (A) (3.27) for n ≥ 2. Of course, An∩ Am= ; if n 6= m, since the same point of A cannot recur to A for the first time at two different times. A less obvious fact is thatS
n≥1Andiffers from A by a set of measure zero, at most, i.e.µ(Sn≥1An) =P
n≥1µ(An) = µ(A). Let us prove the latter fact. Taking in mind thatχC(A)= 1 − χA, thatχA∩B= χAχBand thatχΨ−1τ (A)= χA◦Ψτ, we can write the characteristic functions of the sets (3.26) and (3.27) as follows
χA1= χA(χA◦Ψτ) ;
χAn= χA[1 − χA◦Ψτ] ···[1 − χA◦Ψn−1τ ](χA◦Ψnτ) (n ≥ 2) .
By adding −1 and +1 to the first and last factor, one can rewrite the above characteristic functions as follows:
χA1 = [χA− 1 + 1][χA◦Ψτ− 1 + 1] =
= [1 − χA][1 − χA◦Ψτ] + 1 − [1 − χA] − [1 − χA◦Ψτ] ; χAn = [χA− 1 + 1][1 − χA◦Ψτ] ···[1 − χA◦Ψτn−1][χA◦Ψnτ− 1 + 1] =
= [1 − χA][1 − χA◦Ψτ] ···[1 − χA◦Ψn−1τ ][1 − χA◦Ψnτ] + + [1 − χA◦Ψτ] ···[1 − χA◦Ψn−1τ ] +
− [1 − χA][1 − χA◦Ψτ] ···[1 − χA◦Ψn−1τ ] +
− [1 − χA◦Ψτ] ···[1 − χA◦Ψn−1τ ][1 − χA◦Ψτn] (n ≥ 2) . One is thus naturally led to introduce the sets
B0 := C(A) ; (3.28)
Bn := {x ∉ A :Ψτ(x) ∉ A,...,Ψτn(x) ∉ A} =
= C(A) ∩C[Ψ−1τ (A)] ∩ ··· ∩C[Ψ−nτ (A)], (3.29) for n ≥ 1. Bnis the set of points out of A whose first n iterates stay all out of A. In terms of the Bn’s the characteristic functions of the sets A1, A2, . . . simplify to
χA1= χB1+ 1 − χB0− (χB0◦Ψτ) ; (3.30) χAn= χBn+ (χBn−2◦Ψτ) − χBn−1− (χBn−1◦Ψτ) (n ≥ 2) . (3.31) Upon integrating over the whole spaceΩand exploiting the invariance of the measure, one obtains
µ(A1) = µ(B1) + 1 − 2µ(B0) ;
3.3. RECURRENCE 61 µ(An) = µ(Bn) + µ(Bn−2) − 2µ(Bn−1) (n ≥ 2) .
Just in order to simplify the notation, let us define the numerical sequence {wn}n≥−1, where
w−1:= 1 ; wn:= µ(Bn), n ≥ 0 . (3.32) One thus gets
µ(An) = wn+ wn−2− 2wn−1, (3.33) which is valid for all n ≥ 1. Notice that by the definition (3.29) Bn+1⊆ Bnfor all n ≥ 0, and 1 = w−1> w0, so that the sequence of the wn is non increasing and bounded from below (by 0). Thus limn→+∞wn= inf{wn}exists finite. One has
µ µ
[
n≥1
A
¶
= X
n≥1
µ(An) =X
n≥1
(wn+ wn−2− 2wn−1) =
= lim
N→+∞
N
X
n=1
[(wn− wn−1) − (wn−1− wn−2)] =
= lim
N→+∞(wN− wN−1− w0+ 1) = 1 − w0=
= 1 − µ(C(A)) = µ(A) . Thus A \S
n≥1An has measure zero. SinceS
n≥1Anis the set of points of A that recur to A at least once, we have proved up to now that almost every point of A recurs to A at least once, i.e. that for almost any x ∈ A, namely for any x ∈S
n≥1An, there exists n ≥ 1 such thatΨτn(x) ∈ A. By substitutingΨτwithΨ`τ=Ψ`τ for an arbitrary positive integer
` ≥ 1, and repeating the whole reasoning made up to now, one also finds that, for almost any x ∈ A, namely for any x ∈S
n≥1An, there exists n ≥ 1 such thatΨ`τn (x) =Ψnτ`(x) ∈ A, whatever be` ≥ 1. As a consequence, the set D` consisting of points of A whose iterates from the`-th one on are out of A, i.e. D`:= {x ∈ A :Ψτn(x) ∉ A ∀n ≥ `}, has zero measure, whatever be` ≥ 1. Then, the set of points of A that recur to A at most a finite number of times, which is the union of the sets D`, has measure zero: µ(S`≥1D`) ≤P
`≥1µ(D`) = 0. In other words, almost every point of A recurs to A infinitely many times, and the theorem is thus proved. B
Mean recurrence time: Kac theorem
Let A ⊂Ωsuch that µ(A) > 0. For any x ∈ Sn≥1Anthe integer valued function
n(x) := min{ j ≥ 1 :Ψτj(x) ∈ A} (3.34) yields the first recurrence time of the point x to A, i.e. τn(x). The mean recurrence time of the setA is defined as the normalized integral of the latter quantity over A, namely
Tτ(A) := τ µ(A)
Z
A
n(x) dµ . (3.35)
Theorem 3.5 (Kac theorem). If the dynamical system (Ω,Ψτ,µ) is ergodic, then RAn(x)dµ = 1 for any set A ⊆Ωof positive measure, and the mean recurrence time isTτ(A) = τ/µ(A).
C PROOF. With reference to the proof of the Poincaré theorem, sinceS
n≥1An differs from A by a set of measure zero, and recalling that An is the set of points of A that recur for the first time to A at the n-th iterate, so that n(x) = k if x ∈ Ak, one has Now, by means of (3.32) and (3.33) one can write
N
Now, from the latter identity it follows that since the sequence of the partial sums PN
k=1kµ(Ak) is not decreasing, the sequence of the N(wN−1− wN) + wN−1 is not increas-ing, and is lower bounded (by 0). Thus limN→+∞[N(wN−1− wN) + wN−1] exists, and since limN→+∞wN exists, then the limit
L = lim
N→+∞N(wN−1− wN)
exists and, since {wN} is not increasing, it is L ≥ 0. But one can only have L = 0, since L > 0 would imply the existence of a constant c > 0 and of an M > 0 such that wN−1−wN>
c/N for all N > M (use the definition of limit); in such a case the seriesP
N(wN−1− wN) would diverge, which is impossible. Finally, one gets
Z
for any n, the limit set
B := \
This formula is general and requires no particular assumption. We are going to show that the ergodicity hypothesis (not used up to now) impliesµ(B) = 0. Indeed, let us consider the setΨτ(B). If x ∈Ψτ(B) thenΨ−1τ (x) ∈ B, i.e. Ψn−1τ (x) ∉ A for any n ≥ 0, i.e. Ψ−1τ (x) ∉ A
3.3. RECURRENCE 63 andΨn−1τ (x) ∉ A for any n ≥ 1, i.e.Ψ−1τ (x) ∉ A and x ∈ B. In other wordsΨτ(B) ⊆ B so that Ψn+1τ (B) ⊆Ψnτ(B) for any n ≥ 0. As a consequence, the limit set
C := \
n≥0
Ψnτ(B) (3.38)
exists. Now, due to the fact thatΨτ is injective (it is a bijection ofΩ), the image under Ψτof the intersection of sets is the intersection of the images of those sets1, which yields
Ψτ(C) =Ψτ
µ
\
n≥0
Ψnτ(B)
¶
= \
n≥0
Ψτn+1(B) = C ,
i.e. the limit set C defined in (3.38) is invariant. If the system is ergodic, the invariant set C must have measure zero or one (by metric indecomposability). By the invariance of the measure,µ(Ψτn(B)) = µ(B) for any n ≥ 0, which in turn implies
µ(C) = µ µ
\
n≥0
Ψτn(B)
¶
= lim
n→+∞µ(Ψnτ(B)) = µ(B) .
Thus, eitherµ(B) = 1 or µ(B) = 0. The former possibility must be excluded, since other-wise A ⊆C(B) would have measure zero, whileµ(A) > 0 by hypothesis. B
The Kac theorem yields a simple formula for the mean recurrence time, which explains why the events with small probability recur rarely. However, the whole treatment, starting with formula (3.35) has a defect: for a continuous time dynamical system (Ω,Φt,µ), the time-step τ is completely arbitrary. As a consequence, the mean recurrence time Tτ has the obvious and useless limit T0= 0 as τ → 0, even if the system is not ergodic and the Kac theorem does not hold. Of course, for arbitrarily small values of τ, most part of the points x ∈ A are such that Φt(x) ∈ A for all 0 ≤ t ≤ τ, i.e. the set A1 is just a slight deformation of the set A, so that the fake recurrence is due to the fact that on the time-scaleτ nothing happens in the system.
Example 3.9. In a gas, the single binary collision time-scale, below which almost nothing happens, is roughly given by
tcoll=ρ−1/3− 2r0
pT/m ,
where ρ = N/V is the density of the gas, r0 is the effective interaction range of the molecular interaction (two molecules whose distance is larger than 2r0 do not see each other), T is the temperature of the gas andm is the mass of a single molecule.
Such a remark suggests a correction of the formula (3.35) for the mean recurrence time, namely
Tˆτ(A) := τ µ(A \ A1)
Z
A\A1
n(x) dµ , (3.39)
1In general, for a generic function f : X → Y , and any pair A, B ⊂ X , the inclusion f (A ∩ B) ⊆ f (A) ∩ f (B) holds, the equality being true iff f is injective.
due to Smoluchowski [21]. In this way, one is computing the mean recurrence time by making use only of those points x ∈ A that at time τ are out of A: A \ A1= {x ∈ A :Ψτ(x) ∉ A}. One has
Tˆτ(A) = τ
Pk≥2kµ(Ak)
µ(A) − µ(A1)= τ 1 − µ(A1) − µ(B) µ(A) − µ(A1) .
Now, if the system is ergodic,µ(B) = 0 and, noting that limτ→0µ(A1) = µ(A), one gets limτ→0
Tˆτ(A) = 1 − µ(A) limτ→0µ(A)−µ(Aτ 1) ,
which exists under the hypothesis that the limit at the denominator exists finite.
Chapter 4
Hamiltonian perturbation theory
An introduction to the so-called “canonical perturbation theory” in the Hamiltonian framework is here provided. The main idea of perturbation theory is that of trying to characterize the properties of a system close to an integrable one in terms of the properties of the latter. For example, one would like to know whether the first integrals of the integrable system survive the perturbation, and how long.
4.1 Quasi-integrable systems
Loosely speaking, the dynamical system ˙x = u(x) is said to be quasi-integrable, or close to integrable, in D ⊆Γif, for any x ∈ D, one has
u(x) = v(x) + δv(x) , (4.1)
where v(x) is a vector field such that the system ˙x = v(x) is integrable, i.e. solvable for generic initial conditions in some sense to be specified, whereasδv(x) is a small vector field perturba-tion, i.e.
kδvk ¿ kvk , (4.2)
where k · k is a suitable norm.
Remark 4.1. In general, the concept of closeness to integrability is a local one, i.e. D ⊂Γ. As a consequence, whether a system can be considered quasi-integrable or not depends on the initial conditions. Given a vector fieldu(x) onΓ, its splitting into integrable partv plus perturbation δv may be different in different regions ofΓ.
On the other hand, integrability is meaningful and useful if it is a global property of the system, i.e. if it holds all over the phase space of the system.
In restricting our attention to Hamiltonian systems, we recall here the following theorem, which completely characterizes integrability from a Hamiltonian point of view, and also con-stitutes the basement of most part of Hamiltonian perturbation theory.
Theorem 4.1 (Liouvulle-Arnol’d). Let the Hamiltonian system defined by H be integrable in Γ⊆ R2n in the sense of Liouville, and leta ∈ Rnsuch that the level set
Ma:= {(q, p) ∈Γ : F1(q, p) = a1, . . . , Fn(q, p) = an} 65
is non empty; let also M0a denote a (nonempty) connected component of Ma. Then, a func-tion S(q, a) exists, such that p · dq|M0a = dqS(q, a) and S is a complete integral of the time-independent Hamilton-Jacobi equation, i.e. the generating function of a canonical transforma-tionC : (q, p) 7→ (b, a) such that H(C−1(b, a)) = H(q,∂S(q, a)/∂q) = K(a).
Moreover, ifM0ais compact, then a small neighborhoodU of Ma0 is canonically diffeomorphic to Tn× B, where B ⊂ R+n, i.e. there exists a canonical transformation to angle-action variables F : U → Tn× B : (q, p) 7→ (ϕ, I) such that H(F−1(ϕ, I)) = h(I) and Fj(F−1(q, p)) = fj(I) for any
j = 1,..., n.
Thus, for Liouville-integrable Hamiltonian systems displaying compact families of level sets, canonical action-angle coordinates (ϕ, I) can be introduced, such that both the Hamilto-nian and all the first integrals depend on the action variables I only. In terms of the vari-ables (ϕ, I), the dynamics of the system becomes trivial: the Hamilton equations ˙ϕ = ∂h/∂I, I = −∂h/∂ϕ imply that I(t) = I(0) := I˙ 0andϕ(t) = ϕ(0) + ω0t, where
ω0:= ω(I0) :=∂h(I0)
∂I . (4.3)
The phase space of the system is thus locally “foliated” into invariant tori, on each of which the motion is a translation with a frequency vector (4.3) depending, in general, on the value of the action I0labeling the torusTn.
Remark 4.2. In the present lectures we mean T1= R/(2πZ), i.e. the (group of) real numbers modulo 2π. Of course,Tn= T1× · · · × T1
| {z }
n times
.
Example 4.1. Autonomous Hamiltonian systems with one degree of freedom (n = 1) are ob-viously integrable. Any connected and compact component of the level curve H(q, p) = E, not containing critical points of H, is a periodic orbit. In that case the action I = 2π1 H pdq. The latter quantity is obviously a function of the energy level E: I = f (E). In such a case one has H(F−1(ϕ, I)) := h(I) = f−1(I). In the simplest mechanical case where H = p2/2 + V (q), one has I =1πRq+
q− p2(E − V (q)) dq, where V (q±) = E and V0(q±) 6= 0. As an example, for the harmonic oscillator, withV (q) = ω2q2/2, one finds I = E/ω.
Definition 4.1. A Hamiltonian system is said to be quasi-integrable in D ⊆Γ if, for anyx ∈ D, its HamiltonianH can be written as
H(x) = h(x) + P(x) ,
where h is the Hamiltonian of an integrable system and P is a perturbation Hamiltonian such that kXPk ¿ kXhk with respect to a suitable norm k · k.
Quite often in the literature, as a definition of quasi-integrability on finds the condition supD|P| ¿ supD|h|; such a condition may turn out to be meaningless. As an example, consider the case h(x) ≡ C, C being a positive constant. If P(x) is bounded in D, by a suitable choice of C one can always satisfy the inequality |P(x)| ¿ |h(x)| = C ∀x ∈ D, but the Hamilton equations read ˙x = XP(x), so that the dynamics of the system is completely decided by the perturbation (on the other hand, any constant added to the Hamiltonian is is irrelevant to the dynamics of the system). Even ruling out the problem of the constants, the smallness (in the sense of the sup-norm) of the perturbation P by itself may be still meaningless.
4.1. QUASI-INTEGRABLE SYSTEMS 67 Example 4.2. Consider a perturbed pendulum, whose unperturbed, integrable Hamiltonian is h(q, p) = p2/2 − cos(q). Let P(q) = εsin(q/ε2), where ε is a small parameter. On any domain D = T1×[− ˆp, ˆp], with ε small enough one has supD|P| = ε ¿ supD|h|. However, the dynamics of the system, described by the equation ¨q = −sin(q) − cos(q/ε2)/ε, is clearly influenced by the per-turbation. More precisely, settingξ := q/ε2andτ := t/ε3/2, one gets d2ξ/dτ2= − cos(ξ) − ε sin(ε2ξ), so that, on a suitable time-scale, the dynamics is strongly influenced by the perturbation.
The smallness of the perturbing vector field XP is usually more restrictive than, and ac-tually implies, the smallness of the perturbing Hamiltonian P with respect to the integrable component h, up to constant terms.
Remark 4.3. Quasi-integrability does not mean, in general, closeness of the solution of the
“unperturbed” problem ˙x = v(x) to the solution of the perturbed one, equation (1.1): any kind of scenario is possible, depending on the class of problems treated, on the initial conditions, and so on.
Example 4.3. Let us consider a planetary system with a large mass star at rest in the origin, described by the Hamiltonian
with corresponding equations (in second order form)
¨ri= −Ms of a n one-body Kepler Hamiltonians. The perturbation P takes into account the interaction between pairs of planets. One can easily check that in the domain D := B × R3n, where B is a
the quasi-integrability of system (4.5) is guaranteed. Inside B the acceleration of each planet is due mainly to the interaction planet-star, with a small cumulative effect due to all the other planets. On the other hand, the condition supD|P| ¿ supD|h| might well contain the case of a pair of planets so close to each other that the effect of the interaction with the star is, for both of them, of minor importance; in such a case the system is still quasi-integrable, but the reference integrable system is no longer the one defined byh. A simple example of this kind is the system Sun-Moon: the integrable reference problem consists of the center of mass of the Earth-Moon system moving under the influence of the Sun, plus the relative motion of Earth and Earth-Moon around their center of mass (try to show this explicitly).
Hamiltonian perturbation theory was developed by Poincaré to solve the following funda-mental problem. Consider a quasi-integrable system with Hamiltonian of the form
H(ϕ, I) = h(I) + εP1(ϕ, I) + ε2P2(ϕ, I) + ε3. . . , (4.6) defined for (ϕ, I) ∈ Tn× B, B ⊆ Rn+; ε is the small parameter ordering the various terms of the perturbation P :=P
j≥0εjPj. One now looks for a canonical transformation Cε:Tn× B → Tn× B0: (ϕ, I) 7→ (θ, J) =Cε(ϕ, I)
such that the transformation isε-close to the identity, i.e. C0= idTn×B, and in the transformed Hamiltonian, the first order term of the perturbation, if not completely removed, is simplified as much as possible: for example, it is transformed into a term independent of the angles.
If one is able to find such a transformation, then one can iterate the procedure and try to remove/simplify the new second order term of the perturbation, and so on, with the aim of getting a Hamiltonian as close as possible to that of the unperturbed integrable system, H0. Notice that the removal of all the angle variables to order k ensures that every new, and thus every original action variable changes just a bit on a time-scale 1/εk. Indeed, suppose that the procedure described above works for the first k steps. As a result, one is left with a transformed Hamiltonian of the form
H(Cε−1(θ, J)) = h(J) + εZ1(J) + ··· + εkZk(J) + Rk(θ, J;ε) .
Suppose now that the remainder at the k-th step satisfies Rk= O(εk+1) and∂Rk/∂θj= O(εk+1)
∀ j = 1, . . . , n. Then, ˙Jj= {Jj, H ◦Cε−1} = {Jj, Rk}, i.e.
|Jj(t) − Jj(0)| = Z t
0
¯
¯
¯
¯
∂Rk
∂θj
(θ(s), J(s);ε)
¯
¯
¯
¯ds ≤ cjεk+1t = (cjε)(εkt) ,
which means |Jj(t) − Jj(0)| = O(ε) over times t ≤ 1/εk ( j = 1,..., n). On the other hand, since the canonical transformation (ϕ, I) 7→ (θ, J) is ε-close to the identity, one has
|Ij(t) − Ij(0)| = |Ij(t) − Jj(t) + Jj(t) − Jj(0) + Jj(0) − Ij(0)| ≤
≤ |Ij(t) − Jj(t)| + |Jj(t) − Jj(0)| + |Jj(0) − Ij(0)| =
= O(ε) + O(ε) + O(ε) = O(ε) ,
again over times t ≤ 1/εk. Thus the original action variables too undergo a slow change, so that, on a long time interval 0 ≤ t < 1/εk, the dynamics of the perturbed system resembles that of the unperturbed, integrable one.
Remark 4.4. The ε-closeness to the identity of the canonical transformation Cε prevents one from improving the estimate on the variation of the original action variables: this can be done with the new actions only, getting for example a reduced action variation |Jj(t) − Jj(0)| = O(ε2) over shorter times |t| < 1/εk−1. However, in going back to the original actions such a sharper control is lost and the best one can do is decided by the canonical transformation.