In order to motivate the concept of ergodicity, which will be given below, we show how the problem may arise of identifying the time-average of a function with its mean value with respect to some pre-assigned invariant probability measure. To this end, let us consider a Newtonian system consisting of N interacting particles, the j-th one having vector of coordinates rj and of momenta pj = mj˙rj, m1, . . . , mN being the masses of the particles. The equations of motion of the system are ˙pj = fj, j = 1, . . . , N , where fj is the total force acting on the j-th particle. For the sake of simplicity, in the sequel we make repeated use of the following abuse of notation. If Φt denotes the flow of the system, given a function g(q, p), we write g(t) := g(q(t), p(t)) := g(Φt(q0, p0)), where the dependence on the initial point (q0, p0) is left understood.
Let us consider the function
G(r, p) := 1 2
N
X
j=1
rj· pj . (1.1)
One easily checks that
G(t) = K(t) −˙ V (t) , (1.2)
where K =PN
j=1|pj|2/(2mj) is the kinetic energy of the system, whereas V := −1
2
N
X
j=1
rj · fj (1.3)
is the so-called virial (function) of the system, introduced by Clausius [15]1. Let us denote the
1We stick to the original definition given by Clausius, but notice that nowadays the virial is often defined without the pre-factor −1/2.
7
time average at a finite time t of a function f along the orbit s 7→ Φs(r0, p0), 0 ≤ s ≤ t, as ft(r0, p0) := 1
t Z t
0
f (Φs(r0, p0)) ds ; (1.4) the time average (over an infinite time-interval) of f is defined as the limit, if it exists, of (1.4) when t → +∞, namely
f (r0, p0) := lim
t→+∞ft(r0, p0) . (1.5)
Now, taking the time average of the equality (1.2) leads to the following
Theorem 1.1 (Clausius virial theorem). If the positions and the velocities of all the particles of the system are bounded along the whole orbit originating at (r0, p0), then K(r0, p0) = V (r0, p0), i.e. the time average of the kinetic energy equals the time average of the virial.
C PROOF. The time average at a finite time of equation (1.2) yields Kt−Vt = G(t) − G(0)
t ,
which tends to zero when t → +∞ under the hypotheses made, since G(t) is bounded, uniformly in t. B
Remark 1.1. The existence of the time-averages of the functions involved in (1.6) (i.e. K and V ) must be proved: the Birkhoff theorem provides precise conditions to such an aim.
Notice also that the virial theorem holds on single orbits, i.e. dependently on the choice of the initial point. In general, the theorem becomes really meaningful when one knows further conditions ensuring the boundedness of G(t), uniformly in t and in the initial point (r0, p0), at least when the latter varies in some open set. An interesting example is that of Hamiltonian systems defined by
H(r, p) =
N
X
j=1
|pj|2 2mj
+ U (r1, . . . , rN) = K(p) + U (r) ,
where K is the kinetic energy and U is the potential energy function of the system.
Theorem 1.2. Suppose that the surface of constant energy SE = {H = E} is compact; then the equality
1 2
N
X
j=1
rj · ∂U
∂rj = K (1.6)
holds, together with the obvious one E = K + U .
1.1. GENERAL FORMULATION 9 C PROOF. The compactness of SE ensures two things. First of all, the Hamiltonian flow ΦtH exists for all times t, so that one can compute, in principle, the time average of the right hand side of (1.2). Secondly, the function (1.1) is bounded, uniformly in time, i.e. there exists a constant c > 0 such that |G(ΦtH(r, p))| ≤ c for any t ∈ R; thus the time-average of dG/dt vanishes. B
The computation of the time-average f , as defined in (1.5), would require the knowledge of the solution of the equations of motion at any time and for any initial condition. Thus, time-averages are not computable, in general and in principle.
Remark 1.2. It is important to understand when a time-average can be replaced by (i.e. is equal to) a mean value with respect to some probability measure, i.e. by a computable object.
Such a question represents the starting point and the motivation of ergodic theory.
1.1.1 Homogeneous case
An interesting case [34] where the virial theorem (1.6) simplifies, if valid, is that of systems whose potential energy function U is homogeneous of degree s, i.e. U (λr) = λsU (r) for any positive λ. Then, by Euler’s theorem on homogeneous functions, PN
j=1rj · ∂U/∂rj = sU (r1, . . . , rN) (differentiate the relation U (λr) = λsU (r) with respect to λ and compute it at λ = 1). In such a particular case, (1.6) reads sU = 2K, which, together with E = K + U , yields
U = 2
s + 2 E , K = s
s + 2 E , (1.7)
for s 6= −2, whereas in the special case s = −2 the two equations (the virial one and the conservation of energy) collapse into one and the same, with the necessary condition E = 0.
Remark 1.3. Relations (1.7) imply that both U and K (when they exist) do not depend on the initial point, but only on s and E. Moreover, being K > 0, the second equations of (1.7) implies that if E > 0 then s < −2 or s > 0, whereas if E < 0 then −2 < s < 0.
Actually, such a treatment of the homogeneous case, is very often only formal, and may lead if misunderstood, to paradoxical or even wrong results. The most notable case is that of gravitational systems, for which
U = −1 2
X
i,j=1 i6=j
Gmimj
|ri− rj| (1.8)
is homogeneous of degree s = −1. In such a case (1.7) would yield U = 2E and K = −E (which imply E < 0). As an example of paradoxical result, if in the latter equation one imposes that K = 3N T /2, interpreting T as the temperature of the system, then one gets a negative specific heat: dE/dT = −3N/2, i.e. the energy decreases by increasing the temperature, contrary to what happens in ordinary gasses and contrary to the stability principles of thermodynamics.
Such a phenomenon is known in the literature as the gravo-thermal catastrophe. However, one has to take into account that the constant energy surface SE, in the case of gravity, is not compact, whatever be the value of the energy E, so that the virial theorem, as formulated above, does not hold in general, due to two serious problems. First of all, there is no general condition that ensures the lower boundedness of U , uniformly in time, with the possibility to have blow-up of the solution in a finite time (not necessarily due to collisions). Secondly, even if the motion is initiated in a bounded region of the physical space, there is no general condition that prevents the unbounded growth of the system and, even worse, the escape of particles to infinity. The latter phenomenon would yield a function G, defined in (1.1), growing linearly or even super-linearly with time, in which case the virial equation (1.6) is false. With reference to the above mentioned gravo-thermal catastrophe, we also stress that the apparently natural choice of interpreting the kinetic energy of the system as proportional to the temperature is just made by analogy with the theory of ordinary gasses, and there is no actual reason why such a choice should be correct.
The most striking application of the virial theorem in astrophysics concerns the so-called dark matter, i.e. matter that is supposed to interact only through gravity, thus being invisible within the whole range of the electromagnetic spectrum. Dark matter is supposed to exist in order to explain, for example, the dynamics of galaxies at the border of galaxy-clusters, which seems not to agree with the standard laws of gravitation (even taking into account relativistic corrections), when one supposes that such objects experience only the effect of the visible matter. The first hypothesis of existence of dark matter was made by Zwicky in 1937 [50], who made use of the virial theorem in order to give an estimate of the mass contained in a galaxy cluster as a function of the measured velocities and inter-distances of its components.
Exercise 1.1. Prove the following estimate of the mass of a cluster based on the virial theorem:
2rv2
G ≤ Mcluster ≤ 2RV2
G ,
where
r−1 := 1/rmin ; v2 := vmin2 ; R−1 := 1/rmax ; V2 := vmax2 ,
and rmin/max is the minimal/maximal distance between two stars or galaxies in the cluster and vmin/max is the minimal/maximal velocity a a star or galaxy of the cluster may have.
It turns out that when one makes use of the experimental data, one gets an upper bound 2RVmax2 /G which exceeds by a huge amount the estimate one gets on the basis of the observed electromagnetic emission. Of course, other evidences supporting the existence of dark matter exist, and do not make use of the virial theorem. On the other hand, investigating the validity of the virial theorem in the astrophysical context goes on to be a very interesting and nontrivial research topic.
1.2. THEORY OF GASSES 11