2.5 The Clausius virial theorem
3.1.1 Characterization of ergodic systems
Theorem 3.2 (Characterization of ergodic systems). Let (Ω,Φt,µ) be a dynamical system. The following properties are mutually equivalent:
1. Equality of time-average and mean value - The time-average and the mean value of µ-integrable functions coincide, i.e. for any f ∈L1(Ω,µ)
f (x) = 〈f 〉µ , µ-a.e. inΩ. (3.7)
2. Equality of frequency of visit and measure - For any measurable set B ⊆Ω, the fre-quency of visit relative toB (i.e. the fraction of time the system spends in B) when starting atx ∈Ωcoincides with the measure ofB, i.e.
νB(x) := χB(x) = µ(B) , µ-a.e. . (3.8) 3. Metric indecomposability - There exists no partition of Ω into two proper invariant
subsets of nontrivial measure, i.e. for any measurable setB ⊂Ω
Φt(B) = B ∀t ⇒ either µ(B) = 0 or µ(B) = 1 . (3.9)
4. Absence of first integrals - The system does not admit nontrivial first integrals, i.e.
µ-integrable invariant functions that are not a.e. constant: if f ∈L1(Ω,µ)
f (Φt(x)) = f (x) ∀t, µ-a.e. ⇒ f (x) ≡ const. µ-a.e. . (3.10)
CPROOF. We prove 1 ⇒ 2 ⇒ 3 ⇒ 1, and 3 ⇔ 4.
• 1 ⇒ 2 is trivial: 2 corresponds to the choice f = χB in 1, andχB∈L1(Ω,µ) by definition.
• 2 ⇒ 3 since, given a measurable invariant set B, then µ(B) = νB(x) = χB(x) = χB(x), the latter being either 1 if x ∈ B, or 0 if x ∈Ω\ B.
• 3 ⇒ 1: we prove the equivalent statement ¬1 ⇒ ¬3, where ¬ is the logic negation symbol.
Suppose there exists f ∈L1(Ω,µ) such that f (x) 6= 〈f 〉µ in a set of positive measure. Let us consider, the sets B>= {x ∈Ω: f (x) > 〈f 〉µ} and B< = {x ∈Ω: f (x) < 〈f 〉µ}. At least one out of B> and B< has positive measure, otherwise the complement of their union has measure one, which is equivalent to say that f (x) = 〈f 〉µ a.e. inΩ, contrary to the
hypothesis. Suppose now that µ(B>) > 0. Then µ(B>) < 1, otherwise f (x) > 〈f 〉µ a.e.
in Ω, whose expectation yieldsD fE
µ> 〈 f 〉µ, contrary to conclusion (3.6) of the Birkhoff theorem. Finally, B>is an invariant set:
Φt(B>) = {x ∈Ω : x =Φt( y), y ∈ B>} =
= {x ∈Ω : y =Φ−t(x), f ( y) > 〈f 〉µ} =
= {x ∈Ω : f (Φ−t(x)) > 〈f 〉µ} =
= {x ∈Ω : f (x) > 〈f 〉µ} = B>,
where in the last step use has been made of conclusion (3.5) of the Birkhoff theorem. If µ(B>) = 0, the above reasoning applies certainly to B<. We thus have an invariant set of nontrivial measure.
• 3 ⇒ 4: we prove ¬4 ⇒ ¬3. Suppose f ∈L1(Ω,µ) invariant and f 6= const. a.e. in Ω. Consider the sets C>= {x ∈Ω: f (x) > 〈f 〉µ} and C<= {x ∈Ω: f (x) < 〈f 〉µ}. Both sets are invariant, since f is invariant, and at least one of them has positive measure, otherwise f (x) = 〈f 〉µ:= const. a.e. inΩ, contrary to the hypothesis. Suppose first that µ(C>) > 0;
thenµ(C>) < 1, otherwise f (x) > 〈f 〉µ a.e., whose expectation yields 〈f 〉µ> 〈 f 〉µ, absurd.
Ifµ(C>) = 0, the same reasoning applies to C<. Thus, a metric decomposition ofΩexists.
• 4 ⇒ 3: we prove ¬3 ⇒ ¬4. Suppose that an invariant measurable set B exists such that 0 < µ(B) < 1. Consider then the function χB(x), which is clearly integrable and obviously not a.e. constant; moreoverχB(Φt(x)) = χΦ−t(B)(x) = χB(x), i.e. the given function isΦt -invariant, so that a nontrivial first integral exists. B
Due to such a theorem, one has the following
Definition 3.2. A dynamical system (Ω,Φt,µ) is said to be ergodic if any out of the four equiv-alent properties 1 − 4 of Theorem 3.2 holds.
Remark 3.1. Notice that for ergodic systems the time-average f of an integrable function is a.e.
independent ofx, i.e. constant, the value of the constant being the mean value 〈f 〉µ.
Example 3.4. Hamiltonian systems (SE,ΦtH, dµmc) with one degree of freedom are ergodic.
Indeed, the non singular, connected, compact components of the constant energy surface are closed curves, so that the motion is periodic and the system is clearly metrically indecomposable.
To be specific, let us consider the system defined by H = p2/2 + V (q), with a lower bounded potential energyV . On the constant energy surface one has p = ±p2(E − V (q)), defined on the interval I = [q−, q+], where E − V (q) is positive inside I and vanishes at q∓. If the latter zeros are simple the closed curve just defined is non singular, i.e. does not contain equilibrium points.
One has
dµmc = W−1 dΣ
|∇H|= W−1p ˙q2+ ˙p2 dt p p2+ (V0)2 =dt
W =
= W−1dq
|p|= W−1 dq
p2(E − V (q)) ,
3.1. EQUILIBRIUM: ERGODICITY 51 where W(E) =H dt = 2Rqq−+dq/p2(E − V (q)), namely the period of the motion. Observe that dµmc= dt/W just means that the measure coincides with the frequency of visit.
Example 3.5. The non resonant translations on the torus (3.2), i.e. the non resonant quasi-periodic motions of integrable Hamiltonian systems, are ergodic. We prove such a statement by showing that the time average and the mean value of any function f ∈L1(Tn,µT). We start by
Notice that the vanishing of the latter limit is ensured by the existence of a positive constantcN such that |k·ω0| ≥ cN> 0 for any k such that 0 < |k| ≤ N; if |k| were unbounded the denominator k ·ω0could be arbitrarily small, out of control. Now, the trigonometric polynomials are dense in C0(Tn), which in turn is dense inL1(Tn,µT) [34]. Thus, for any f ∈L1(Tn,µT) and anyε > 0, there exists anN and a trigonometric polynomial PN such that
k f − PNk1= Z
Tn| f − PN| dµT< ε . As a consequence, one has also
| 〈 f 〉 − 〈PN〉 | =
Finally, we observe that the existence of the time-average g of a function g means that for any ε > 0 there exists τε> 0 such that for any t > τε one has |gt− g| < ε (recall that gt:=1t Rt
0 g ◦Φsds is the time-average up to time t). We now have
| 〈 f 〉 − f | ≤ | 〈 f 〉 − 〈PN〉 | + |〈PN〉 − f | =
Integrating the latter inequality on the torus and taking into account that the measure µT is invariant, one gets
k 〈 f 〉 − f k1≤ 3ε + k f − PNk1< 4ε .
Sinceε is arbitrary, it follows that k〈f 〉− f k1= 0, i.e. f (ϕ) = 〈 f 〉 a.e., so that the system is ergodic.
Example 3.6. Consider the Hamiltonian dynamical system (3.1). Suppose that H admits an independent first integral F, i.e. a differentiable, not identically constant function such that {F, H} = 0. Then the system cannot be ergodic. Indeed the conclusion follows from property 4 of the ergodic characterization theorem.
Remark 3.2. In absence of additional first integrals in a given regularity class that is strictly contained inL1(SE,µmc), nothing can be said concerning the ergodicity of system (3.1). For example, excluding the presence of differentiable first integrals - which is a generally unsolvable non trivial problem - does not imply at all that the system is ergodic.
The ergodicity notion just given is obviously the concept on which equilibrium statistical mechanics is founded. For example, the time-averages appearing in the statement of the virial theorem can be replaced by mean values for ergodic systems. Moreover, for the latter systems, any dynamical measure performed through a time-average over a sufficiently long time and for a certain set of initial conditions is likely to give a predictable result.
Though the problem of establishing whether a given Hamiltonian dynamical system is er-godic or not is still open, in general, the notion of erer-godicity by itself is partially unsatisfactory.
On the one hand, the request to have ergodicity in the space of integrable functions might be too strong. In fact, to build up thermodynamics one would like the coincidence of time-average and mean value for a few physically relevant smooth functions. On the other hand, the pos-sibility of exchanging averages is useful if one starts at equilibrium, but does not imply at all that the system will approach equilibrium if had started out of it.