d
dt%(t, x(t)) =∂%
∂t + ∇x% · ˙x = ¡∂t% + {%, H}¢(t, x(t)) = 0 . Then, setting x =ΦtH( y) on both sides of (2.7) yields (2.6).B
Formula (2.6) obviously excludes any possibility of a strong convergenceρ(t, x) → ρeq(x) as t → ∞. Thus, the problem of the approach to equilibrium must be re-formulated in a different or a weaker form.
2.3 Equilibrium statistical mechanics
Let us focus our attention, for the moment, on the equilibrium invariant measures, whose den-sity ρeq satisfies the stationary Liouville equation {ρeq, H} = 0. Such an equation means that ρeq is a smooth first integral of the Hamiltonian system defined by H, and such first integrals can exist or not, depending (strongly) on H. On the other hand, if one looks for equilibrium densities that satisfy some requirements given a priori and independently of H, then the only reasonable choice is to setρeq= f (H), where f (·) is some function to be determined.
Statistical mechanics is that part of theoretical physics which aims to explain the laws of equilibrium thermodynamics relative to a given Hamiltonian system (e.g. a gas in a vessel) in terms of the sole knowledge of its Hamiltonian H(x). This is done by means of two particular equilibrium densities f (H) on the phase spaceΓof the system, both introduced by Boltzmann and Maxwell first, and later, in the form here reported, by Gibbs [17, 19, 39, 40].
2.3.1 Micro-canonical measure
The first possible choice consists in the so-called micro-canonical measure
dµmc:= W−1δ(H(x) − E) dV (x), (2.8) where δ(s) denotes the Dirac delta “function”, characterized by the two properties: δ(s) = 0 for any s ∈ R \ {0}, and δ(s) = +∞ in such a way that R
Rδ(s)ds = 1. It is easily shown that such a property implies that if f (s) is any function defined in [a, b] and continuous in 0 ∈]a, b[, thenRb
a f (s)δ(s)ds = f (0). Moreover, it turns out that δ(s) = dθ(s)/ds, where the Heaviside step functionθ(s) is so defined: θ(s) = 1 if s > 0, θ(s) = 0 if s < 0 and θ(0) = 1/2. By imposing that µmc(Γ) = 1, one gets for the normalization factor W appearing in (2.8) the following formula
W(E) = d dE
Z
H(x)≤EdV (x) = d
dEV ({H(x) ≤ E}) . (2.9)
The measure (2.8) is concentrated on the surface of constant energy
SE:= {x ∈Γ : H(x) = E} , (2.10)
and is used to describe the thermodynamics of isolated systems. The constant W in (2.8) is chosen in such a way thatµmc(SE) = 1, and plays a fundamental role in the theory; we come back below on this point. The micro-canonical measure (2.8) is clearly invariant with respect
to the Hamiltonian flowΦHt of the system ˙x = XH(x), sinceρeq(x) = W−1δ(E − H) is a “function”
of H and the Lebesgue volume element dV , due to (2.4), is invariant.
Sometimes, the micro-canonical measure is improperly and erroneously referred to as the measure assigning equal a priori probabilities to all the points of SE. In fact, suppose that x ∈SE and x + dx ∈SE+dE, with dE > 0. Then
dE = H(x + dx) − H(x) = ∇H(x) · dx = |∇H(x)|dx⊥,
where dx⊥ is the projection of dx along ∇H(x), which is orthogonal toSE. Thus at any point x ∈SE the volume element is given by
dV (x) = dx⊥dΣ(x) = dEdΣ(x)
|∇H(x)| ,
where dΣ(x) is the Lebesgue surface element at x ∈SE. Then, if B ⊂Γis Lebesgue measurable in Γ(positive volume) and such that B ∩SE=A, withA ⊆SE Lebesgue measurable onSE
(positive area), then
µmc(A) := W−1 Z
Bδ(E − H(x)) dV (x) = W−1 Z
Bδ(E − H(x)) dΣ(x)dx⊥=
= W−1 Z
A
µZ
δ(E − H(x))dE
¶ dΣ(x)
|∇H(x)|= W−1 Z
A
dΣ(x)
|∇H(x)| . (2.11) It follows that the normalization factor W, also known as the Boltmann statistical weight, is given by
W(E) = Z
SE
dΣ(x)
|∇H(x)| . (2.12)
Notice that the compactness ofSE and the condition |∇H| ≥ c > 0 ensure the existence of W. It thus turns out that a point x ∈SE is weighted with the inverse of |∇H(x)| = |XH(x)|, which has also an intuitive meaning: the slower is the phase-flow at some point the longer is the time spent by the system in its neighborhood, and the more relevant the point is to the statistics.
Notice that the probability measure (2.8) is not absolutely continuous with respect to the Lebesgue measure inΓ. Indeed, ifA is Lebesgue measurable on SE, the Lebesgue measure of A in Γ is obviously zero (a surface has no volume), but µmc(A) > 0. On the other hand, the measure (2.8), due to (2.11), can be regarded as absolutely continuous with respect to the Lebesgue measure onSE, with density 1/|∇H|.
We finally notice that the thermodynamics of an isolated system for which the micro-canonical measure (2.8) exists, is built up through the Boltzmann formula
S(E) = logW(E) , (2.13)
which yields the entropy S(E) of the system once one knows the Boltmann statistical weight.
Once the entropy (2.13) is known, the thermodynamics of the system is completely determined by the first principle of thermodynamics, i.e. the general law of energy conservation, together with the second one for reversible processes, which together read
dE + PdV − φ dN = TdS . (2.14)
2.3. EQUILIBRIUM STATISTICAL MECHANICS 27 Here E is the energy, P the pressure, V the volume (occupied by the system in the physical space),φ the chemical potential, N the number of particles, T the absolute temperature and S the entropy of the system defined in (2.13). Notice that actually S = S(E,V , N) since the Boltz-mann weight W = W(E,V , N). By means of (2.14), one can compute any interesting quantity, such as the temperature T = (∂S/∂E)−1, the pressure P = T∂S/∂V , or the chemical potential φ = −T∂S/∂N.
The micro-canonical mean value of a function F is given by
〈F〉mc:= W−1 Z
ΓFδ(E − H) dV = W−1 Z
SE
F dΣ
|∇H| . (2.15)
2.3.2 Canonical measure
If the Hamiltonian system under consideration is not isolated, but in contact with a thermal bath so that, in place of the total energy, the temperature is known exactly, then the so-called canonical measure must be used, namely the invariant measure
dµc:= Z−1e−T1H(x) dV (x), (2.16) where T is the absolute temperature of the system. The normalization constant
Z(T) = Z
Γe−T1H(x)dV (x) (2.17)
is known as the partition function of the system, whose existence requires of course some con-ditions (H must be both suitably lower bounded and upper unbounded) to be satisfied. Notice that the canonical partition function (2.17) is the Laplace transform of the micro-canonical statistical weight (2.12) with respect to the energy:
Z(T) = Z
Γe−T1H dV = Z
Γ
µZ
δ(E − H) dE
¶
e−1TH dV = Z +∞
0
e−ETW(E) dE , (2.18) (under the hypotheses on H ensuring the existence of Z it is not restrictive to integrate in E starting from 0). Such a substitution of the energy with the temperature as the independent variable yields a function Z = Z(T,V , N).
The link with thermodynamics is given by the Gibbs formula
F(T) = −T log Z(T) , (2.19)
which yields the free energy F = F(T,V , N) of the system once one knows the partition function (2.17). Since the free energy is F = U − TS, where U denotes the internal energy (which is a dependent variable in the canonical framework), the first and second principle of thermody-namics combine together to give
dF = −SdT − PdV + φ dN , (2.20)
from which one can compute the relevant thermodynamic quantities, such as entropy, pressure and chemical potential.
The mean value of a function F with respect to the canonical measure is linked to the micro-canonical mean value (2.15). Indeed, one has
〈F〉c :=
Z
ΓF dµc= Z−1 Z
ΓF e−T1H dV =
= Z−1 Z
e−ET µZ
ΓFδ(E − H) dV
¶ dE =
= R e−ET〈F〉mcW(E) dE R e−ETW(E) dE
. (2.21)
As an example, consider the internal energy U of the system, defined as the mean value of the Hamiltonian, namely U(T) = 〈H〉c=R Hdµc. By means of (2.21), taking into account that, obviously, 〈H〉mc= E, one finds the well known formula U(T) = T2d log Z(T)/dT.
Exercise 2.3. Consider the (n−1)-dimensional spherical surface of radius R in the n-dimensional space, defined by the equationPn
k=1x2
k= R2 (n ≥ 2), and show that the volumeΩn(R) of the re-gion it encloses is given by
Ωn(R) := πn2Rn Γ(n2+ 1) , whereΓ(z) :=R
R+tz−1e−tdt is the Euler Gamma-function, satisfyingΓ(z + 1) = zΓ(z). Hint: first, show thatΩn(R) =Ωn(1)Rn, and observe thatR
Rnf (x21+· · ·+ x2n) dx1. . . dxn=R
R+f (R2)Ω0n(R) dR;
then, use the latter trick to compute the Gaussian integralR
Rne−Pnk=1x2k dx1. . . dxn and compare the result with the known one, i.e.πn/2.
Exercise 2.4. Consider the perfect gas model defined by H =PN
i=1
|pi|2
2m , together with the con-dition that any particle vector position xi belong to a given domainD ⊂ R3 whose volume isV ; ideal elastic reflection of any particle impinging on the wall is assumed. ComputeW(E, V , N) and, by means of (2.14), determine the pressure of the gas, which yields the Boyle law:PV = NT.
Compute alsoZ(T, V , N) and get the pressure from (2.20). Compare the two results.
Exercise 2.5. Consider a system of N non interacting harmonic oscillators, whose Hamiltonian isH =12PN
j=1(p2j+ ω2jq2
j). Compute W(E, V , N) and Z(T, V , N).