2.5 The Clausius virial theorem
2.5.2 Theory of gasses
Let us now apply the virial theorem to the case of gasses, which yields, in the ideal case of noninteracting particles, the Boyle law, i.e. the equation of state P = ρT, saying that the pressure P of the gas is equal to the number densityρ of the gas (number of particles per unit volume) times the absolute temperature T measured in energy units.
The gas model we consider consists of N particles of masses m1, . . . , mN enclosed in a vessel with ideally reflecting walls. The force fjacting on the j-th particle naturally splits up into two components, an internal one, due to the interaction with all the other particles of the system, and the impulsive force that the wall of the vessel exerts, inward and orthogonal to the wall itself, on the particle when hit by the latter. The pairwise interaction between the particles is supposed to be described by a short range central potentialφ(r), so that one can write
f j= −
N
X
k=1k6= j
φ0(rjk) ˆrjk+ fwj(t),
where rjk:= rj− rk, rjk:= |rjk|, ˆrjk:= rjk/rjk, and fwj(t) is the confining force due to the walls.
Notice that the latter force acts instantaneously reversing the normal component of particle’s momentum and preserving the parallel one. In particular, the kinetic energy of each particle is preserved in every collision of the particle with a wall of of the vessel, and the same holds
2.5. THE CLAUSIUS VIRIAL THEOREM 41 for the potential energy of interaction between particles. Thus, the total energy of the system is preserved, i.e.
The volume V occupied by the gas is constant and equals that of the vessel. For the sake of simplicity we here consider a cubic vessel of side-length L, the origin of the coordinates being placed at the center of the cube.
Remark 2.6. Ifφ(r), and so Uint, is lower bounded, the virial theorem holds for the gas system.
Indeed, |rj| ≤p
3L/2 (half-diagonal length) for any j = 1,..., N, whereas the kinetic energy K = E −Uintof the system is upper bounded, and as a consequence each |pj| is bounded.
Let us now examine the virial of the wall forces. Taking in mind that fwj(t) is zero if at time t the j-th particle is not on the boundary of the cube, and that fwj (t) = −P
nn fwj,n(t) if the particle is, at time t, on one of the internal faces of the cube with outward normal unit vector n, so that rj(t) · n = L/2, one gets
n is the sum over the six outward normal unit vector labeling the faces of the cube:
n = ± ˆx,± ˆy,± ˆz. It is now quite natural to define the wall pressure Pnexerted by the gas on the wall with outward normal unit vector n as the time average of the total force per unit surface exerted on that wall, i.e.
Pn:= 1
The total pressure of the is defined as the arithmetic average of the six wall pressures, i.e.
P :=1
where V = L3. The latter law is due to Clausius and one can show that it does not depend on the form of the vessel.
Exercise 2.8. Generalize the derivation of the Clausius law (2.61) to a vessel of general form, having piecewise smooth boundary and volume V . Hint: i) take a partition of the boundary of the vessel into small portions and define the local pressure on each small portion of the boundary as the total normal force per unit surface; ii) assume that such a local pressure is the same all over the surface; iii) take the limit of the partition to infinitesimal portions and use the divergence theorem.
Concerning the virial of the internal forces one gets
Vint:= −1 2
N
X
j=1
rj·
−
N
X
k=1k6= j
φ0(rjk) ˆrjk
= 1 4
N
X
j,k=1 j6=k
φ0(rjk)rjk . (2.62)
The virial theorem, i.e. K =V =V int+Vwall, yields the Clausius equation K =V int+3
2PV . (2.63)
Upon defining the temperature T of the gas by K :=3
2N T , (2.64)
and dividing by 3V /2, equation (2.63) becomes
P = ρT −2Vint
3V , (2.65)
where ρ := N/V is the number density of the gas. The Clausius equation, in the form (2.65), reads as the equation of state of real gasses, the correction with respect to the Boyle law being due to the interaction between the particles.
Exercise 2.9. Show that the canonical measure with parameter T, relative to the Hamiltonian H = K(p) yields exactly 〈K〉c= 3NT/2. Show that the same result holds if H = K(p) +U(r).
Remark 2.7. In the framework of statistical mechanics the definition (2.64) is an identity, and the validity of both of them, with the sameT, implies K = 〈K〉c, i.e. the identification of the time average with a particular phase-space average, i.e. the mean value with respect to the canonical measure. Notice that time-averages are actually impossible to compute, whereas mean values can be at least estimated. As a consequence, it is very important to know when the time average of observables of physical interest can be replaced by a computable expectation.
In equation (2.65) there are three unknown quantities, namely the pressure P, the time averaged virial of the internal forces V int, and T = 2K/(3N). Notice that the equation itself determines the value of one of such quantities when the other two are known. In the limit case of the ideal gas, where (by definition)φ(r) ≡ 0, one gets a simpler equation of state, namely the Boyle law P = ρT. Moreover, due to the law (2.60) of conservation of energy, in the ideal case one has K = K = E, and the temperature T = 2E/(3N) is determined by the initial conditions, so that no time average must be actually computed.
One easily realizes that for the model of ideal gas used above, the virial theorem holds for each particle separately, and, for any given particle, in the special case of the cubic geometry, it holds separately along any of the three orthogonal directions (prove both these statements).
It is then clear that, for very special initial conditions, one can get “pathological” results. For example, one can in principle think to set all the initial velocities of the particles directed orthogonally to one of the faces of the cube, say along the x-direction; four out of the six wall
2.5. THE CLAUSIUS VIRIAL THEOREM 43 pressures exactly vanish in this case. Though from the point of view of dynamics such a choice is perfectly admissible, it can be excluded from any reasonable probabilistic point of view. One instead expects, on the same line of reasoning, that the good initial conditions are those leading to equal or almost-equal wall pressures and single-particle kinetic energies of comparable size.
Of course, the case of the ideal gas is an extreme one. The Boyle law works rather well for most “real” gasses with a nonzero interaction between particles, provided the density is not too high and the temperature is not too low. In order to show this, and to get a model equation of state for real gasses, let us start by rewriting
Vint=1
The latter quantity is a local density of particles surrounding the j-th one: when integrated over some portion of the vessel ˆρj yields the number of particles inside such a portion, the j-th one being excluded. The expression (2.66) of the internal virial is now simplified by first replacing the local density (2.67) with its average value over the whole volume occupied by the gas, namely (N − 1)/L3→ ρ, in the large N and/or L limit. This is the so-called mean field the second step (where after a translation the cube is replaced by the whole space) being valid up to an irrelevant small remainder. The convergence of the (last) radial integral above re-quires thatφ(r) ∼ 1/r3+η as r → +∞, for some η > 0; moreover, φ(r) cannot diverge faster than 1/r2 as r → 0. Real intermolecular potentials display an asymptotic attracting tail ∼ 1/r6 at large r, the so-called Van der Waals force, which is due to charge fluctuations and whose expla-nation requires the use of quantum mechanics. On the other hand, due to the Pauli exclusion principle (again a purely quantum mechanical effect), real potentials exhibit a steep repulsive
“wall” at short distances, the divergence being much faster than 1/r2. A very rough way to avoid the divergence of the integral in (2.68), due to the repulsive part of the potentialφ, is to artificially displace the lower extreme of integration in the radial integral to a suitable pos-itive cutoff value. Another possibility consists instead in defining φ(r) = +∞ if r < σ, which models the repulsive interaction of hard spheres. In this case the contribution of such a hard-core repulsive component of the potential can be computed apart through methods of transport theory; see [29]. Here we follow still another way, modeling the potentialφ(r) as follows:
φ(r) := φ0[1 − θ(r − σ)] − θ(r − σ)g(r) , (2.69) where φ0 is the constant positive value of the potential if r < σ, g(r) is a positive function describing the interaction well of the potential if r > σ, and θ(·) is the Heaviside step function;
it is assumed that g(σ) = 0. Observe that the potential model (2.69) displays an upper bounded repulsive part. Taking into account that δ(r − σ)g(r) = δ(r − σ)g(σ) = 0, the derivative of the potential (2.69) is
φ0(r) = −φ0δ(r − σ) − θ(r − σ)g0(r). (2.70) Either first integrating by part and then making use of (2.69), or making use of (2.70), the integral in (2.68) is easily computed to yield
Vint= −πρN
·
φ0σ3− 3 Z +∞
σ g(r)r2 dr
¸
. (2.71)
By inserting the virial expression (2.71) into (2.65), and taking into account that within the mean field approximationV int=Vint, one obtains
P = ρT + αρ2, (2.72)
where
α :=2π 3
·
φ0σ3− 3 Z +∞
σ g(r)r2dr
¸
. (2.73)
Relation (2.72) is the simplest model equation of state for real gasses. Notice that ifα < 0, i.e.
if the attractive component of the potential prevails on the repulsive one, then, according to (2.72) the pressure of a real gas at a given temperature and density is less than the pressure of and ideal gas with the same temperature and pressure.
Remark 2.8. The condition expressing closeness to the ideal case obviously reads
1 ¿ Vwall
¯
¯
¯V int
¯
¯
¯
= P
|α|ρ2= T
|α|ρ+ 1
which means low density and/or high temperature, as anticipated above.
We finally notice that the temperature entering the equation of state (2.72) is defined by the time average (2.64), so that it is apparently unknown, unless one is really able to perform such a computation. However, the law (2.60) conservation of energy determines the temperature as a function of the density and of the energy per particle. Indeed, within the same mean field approximation used for the computation of the internal virial, one can express the total potential energy as
Uint= 2πρN Z +∞
0 φ(r)r2dr = αρN (2.74)
whereα is defined in (2.73). By inserting (2.74) into (2.60), taking the time average, and taking into account that K = 3NT/2, one gets
T =2
3(ε − αρ) , (2.75)
where ε := E/N is the specific energy (energy per particle) of the system. Equation (2.75) expresses the temperature of the gas as a function of the specific energy and of the density,
2.5. THE CLAUSIUS VIRIAL THEOREM 45 quantities that must not be computed by solving the equations of motion. In particular, the energy of the gas is completely determined by the initial conditions, which can be assigned in some reasonable way. For example, one can think of uniformly distributed particles inside the vessel, with velocities extracted according to the Maxwell-Boltzmann measure (i.e. the canonical one, restricted to the momenta). Finally, taking into account equation (2.75), one can rewrite the equation of state (2.72) as
P =2 3ρε +1
3αρ2, (2.76)
which provides the pressure of the gas as a function of its density and of its specific energy.
Chapter 3
Ergodic theory
Dynamical problems require the computation of time-averages, through which interesting quantities with a macroscopic (e.g. thermodynamic) meaning are defined. On the other hand, time-averages are almost impossible to compute, since one should know the solution to the equations of motion. One would thus like to replace them with mean values with respect to suitable probability measures. The possibility to do this is a nontrivial property of the specific system considered and of the particular invariant measure chosen for it. Such a property is called ergodicity.
Definition 3.1. Given a set Ω, a group Φt:X ×Ω→Ω of transformations (X = R or Z), and a probability measure µ on Ω, invariant with respect to Φt, the triplet (Ω,Φt,µ) is called a dynamical system.
We recall thatµ brings with it the definition of a σ-algebra σΩ, the domain ofµ, and that the elements ofσΩ are referred to as measurable sets.
Example 3.1. Given a Hamiltonian system defined by the time-independent Hamiltonian H, and value of the energy E such that SE is nonempty, compact and does not contain critical points ofH, the triplet
¡SE,ΦHt ,µmc
¢ (3.1)
where dµmc= W−1 dΣ|∇H|, defines a dynamical system. HereσSE is the Borelσ-algebra ofSE. If other first integrals besides the Hamiltonian exist, the actual dynamics takes place on the level set of all of them, and a suitable invariant measure must be introduced on the latter.
Example 3.2. Given an integrable Hamiltonian system defined by h(I), the dynamics of the system is given by ϕ(t) = ϕ(0) + ω0t, I(t) = I(0) := I0, where ω0= ∂h(I0)∂I is the vector of fre-quencies corresponding to the constant vector of actions I0. Given the torusT0= {I0} × Tn, the translation on the torus
(T0,ϕ 7→ ϕ + ω0t,µT), (3.2)
where dµT= dnϕ/(2π)n= dϕ1. . . dϕn/(2π)nis the normalized Lebesgue measure onTn, defines a dynamical system.
47
In the latter example one has to study apart the case of resonant, or rationally dependent frequenciesω01, . . . ,ω0n, i.e. when there exists k ∈ Zn\{0} such that k ·ω0= k1ω01+· · ·+ knω0n= 0. Indeed, suppose that r linearly independent integer vectors k(1), . . . , k(r)∈ Znexist, such that k(i)·ω0= 0, i = 1, . . . , r. Such integer vectors generate the so-called resonance module associated to the frequency vectorω0, namely the set Mω0:= {k ∈ Zn: k ·ω0= 0}, which is a subgroup of Zn. The number r := dim Mω0 of independent resonance relations is called the dimension of Mω0, and one has 0 ≤ r ≤ n−1. In the two extreme cases r = 0 and r = n−1 the frequency vector ω0is said to be nonresonant and completely resonant, respectively. In the case r ≥ 1 one can prove that a matrixA∈ SL(n, Z) (i.e. a n×n matrixAwith integer elements and unitary determinant) exists, such that its first r rows belong to Mω0\ {0}. Thus, (Aω0)i= 0 for any i = 1, . . . , r, and the change of variablesϕ0=Aϕ is such that ϕ0i(t) = ϕ0i(0) if i = 1,..., r, and ϕ0i(t) = ϕ(0) + (Aω0)it if i = r + 1,..., n, i.e. the actual dynamics takes place on a n − r dimensional torus Tn−r.
Example 3.3. Consider the translation onT2 with frequencies ω01= ν and ω02= 2ν, where ν is any real number different from zero. Then Mω0 = {k ∈ Z2: k = q(2,−1), q ∈ Z} has dimension one. The matrixA=
µ 2 −1
−3 2
¶
is such thatAω0= (0, ν)T, and maps the dynamics onT2 onto the 2π/ν-periodic one: ϕ01(t) = ϕ01(0),ϕ02(t) = ϕ02(0) + νt.
3.1 Equilibrium: ergodicity
The first problem that poses in ergodic theory concerns the existence of time-averages. This is solved in large by the following important theorem, whose proof is not reported here; the interested reader is referred e.g. to [18], [22] and [28].
Theorem 3.1 (Birkhoff theorem). Let (Ω,Φt,µ) be a dynamical system and f any function in L1(Ω,µ). Then, the time-averages of f , i.e. the limits
f (x) := lim
t→+∞
1 t
Z t
0
f (Φs(x)) ds (t ∈ R) (3.3)
and/or
f (x) := lim
t→+∞
1 t
t−1X
s=0
f (Φs(x)) (t ∈ Z) (3.4)
existµ-a.e. inΩ. The time average f is invariant with respect toΦt, i.e.
f (Φt(x)) = f (x) (3.5)
for any t and for any x such that f (x) exists. The mean value of f and f with respect to µ coincide:
D fE
µ= Z
Ωf (x) dµ(x) = Z
Ωf (x) dµ(x) = 〈f 〉µ . (3.6) In the continuous case, both the averages (3.3) and (3.4) exist and coincide.
3.1. EQUILIBRIUM: ERGODICITY 49 Once the existence of time-averages is established, the notion of ergodicity can be formal-ized. As previously motivated, the useful notion concerns the coincidence of time and space-averages. In order to justify such an equality, other interesting ideas were developed histori-cally. At an intuitive level, the time-average and the mean coincide if the orbit densely covers Ω, or if there is no way of partitioning Ω into two invariant proper subsets, or if the system admits only trivial first integrals (i.e. constant functions).