interested reader is referred e.g. to [42, 43, 44]. A monograph devoted to the (formal) virial theorem and its applications in astrophysics is found in [16]. Dark matter and many other topics are treated in the textbooks [8, 11].
An example of homogeneous system to which the virial theorem can be applied is the following.
Exercise 1.2. Consider the system of N identical particles interacting through harmonic forces, defined by the Hamiltonian
H =
N
X
j=1
|pj|2 2m +
N
X
i,j=1
k|ri− rj|2
4 ,
where k > 0. Show that i) the virial theorem holds; ii) the equations of motion of each particle become trivial in the center-of-mass frame. Solve such equations for any initial condition and compute explicitly the time-averages of U and K.
1.2 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 density ρ 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 fj acting 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 potential φ(r), so that one can write
fj = −
N
X
k=1 k6=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 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 1.4. If φ(r), and so Uint, is lower bounded, the virial theorem holds for the gas system.
Indeed, |rj| ≤ √
3L/2 (half-diagonal length) for any j = 1, . . . , N , whereas the kinetic energy K = E − Uint of 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
nnfj,nw (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 Pn exerted 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 1.3. Generalize the derivation of the Clausius law (1.10) to a vessel of general form, having piecewise smooth boundary and volume V . Hint: i) take a partition of the boundary of
1.2. THEORY OF GASSES 13 the vessel into small portions labelled by the outward normal n, with area ∆σn, and define the local pressure on each small portion of the boundary as the total normal force per unit surface:
Pn := ∆σ1
n
PN
j=1fj,nw ; ii) assume that such a local pressure is the same all over the surface:
Pn = P for any n; iii) take the limit of the partition to infinitesimal portions and use the divergence theorem (notice that ∇ · r = 3).
Concerning the virial of the internal forces one gets
Vint:= −1
The virial theorem, i.e. K =V = Vint+Vwall, yields the Clausius equation K =Vint+ 3
2P V . (1.12)
Upon defining the temperature T of the gas by K := 3
2N T , (1.13)
and dividing by 3V /2, equation (1.12) becomes
P = ρT − 2Vint
3V , (1.14)
where ρ := N/V is the number density of the gas. The Clausius equation, in the form (1.14), 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. In equation (1.14) there are three unknown quantities, namely the pressure P , the time averaged virial of the internal forces Vint, 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 (1.9) 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 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 (1.15) of the internal virial is now simplified by first replacing the local density (1.16) 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 requires that φ(r) ∼ 1/r3+η as r → +∞, for some η > 0; moreover, φ(r) ∼ 1/r3−µ as r → 0+, for some µ > 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 explanation 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/r3. A very rough way to avoid the divergence of the integral in (1.17), due to the repulsive part of the potential φ, is to artificially displace the lower extreme of integration in the radial integral to a suitable positive 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 [40]. Here we follow still another way, modeling the potential φ(r) as follows:
φ(r) := φ0[1 − θ(r − σ)] − θ(r − σ)g(r) , (1.18)
1.2. THEORY OF GASSES 15 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 (1.18) 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 (1.18) is
φ0(r) = −φ0δ(r − σ) − θ(r − σ)g0(r) . (1.19) Either first integrating by part and then making use of (1.18), or making use of (1.19), the integral in (1.17) is easily computed to yield
Vint= −πρN
By inserting the virial expression (1.20) into (1.14), and taking into account that within the mean field approximation Vint=Vint, one obtains
P = ρT + αρ2 , (1.21)
Relation (1.21) is the simplest model equation of state for real gasses. Notice that if α < 0, i.e. if the attractive component of the potential prevails on the repulsive one, then, according to (1.21) 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.
Exercise 1.4. Show that the condition expressing closeness to the ideal case, namely Vwall
Vint
, reads T |α|ρ, which means low density and/or high temperature.
We finally notice that the temperature entering the equation of state (1.21) is defined by the time average (1.13), so that it is apparently unknown, unless one is really able to perform such a computation. However, the law (1.9) 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)r2 dr = αρN (1.23)
where α is defined in (1.22). By inserting (1.23) into (1.9), taking the time average, and taking into account that K = 3N T /2, one gets
T = 2
3(ε − αρ) , (1.24)
where ε := E/N is the specific energy (energy per particle) of the system. Equation (1.24) expresses the temperature of the gas as a function of the specific energy and of the density,
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 (1.24), one can rewrite the equation of state (1.21) as
P = 2
3ρε + 1
3αρ2 , (1.25)
which provides the pressure of the gas as a function of its density and of its specific energy.
Chapter 2
Probabilistic approach
In many cases, instead of trying to control the details of the dynamics of a given system, it is convenient to approach the problem from the point of view of probability theory, trying to characterize the statistical aspects of the dynamics itself. To such a purpose, the phase space of the system has to be endowed with a probability measure that does not evolve along the flow, so that mean values of observables are independent of time. One of the most important results of such an approach is the deduction of the macroscopic laws of thermodynamics for mechanical systems with many degrees of freedom.
2.1 Probability measures and integration
Given a set Ω (think e.g. to a differentiable manifold) let us denote by 2Ω the power set of Ω, i.e. the set of all, proper and improper, subsets of Ω (recall that the notation is due to the fact that for a finite set of s elements the dimension of its power set is 2s).
Definition 2.1. A set σΩ ⊆ 2Ω is called a σ-algebra on Ω if 1. it contains Ω;
2. it is closed with respect to complementation, i.e. A ∈ σΩ⇒ Ac ∈ σΩ; 3. it is closed with respect to countable union, i.e. {Aj}j∈N∈ σΩ ⇒S
j∈NAj ∈ σΩ.
Notice that the complement of a countable union of sets is the countable intersection of the complements of those sets, which means that closure with respect to complementation and countable union implies closure with respect to countable intersection.
Due to the fact that 2Ω is a σ-algebra and the intersection of σ-algebras is still a σ-algebra, if F ⊂ 2Ω denotes a set of subsets of Ω, the smallest σ-algebra containing F always exists and is usually denoted by σΩ(F ), which is also refereed to as the σ-algebra generated by F . In this respect, if Ω is endowed with a topology, a σ-algebra particularly relevant to applications is the one generated by the open sets of Ω, which is called the Borel σ-algebra of Ω.
17
Definition 2.2. Given a set Ω and a sigma-algebra σΩ on it, a probability measure on Ω is a nonnegative function µ : σΩ→ [0, 1] which is
• normalized, i.e. µ(Ω) = 1;
• countably additive, i.e. additive with respect to countable unions of pairwise disjoint sets:
{Aj}j∈N ∈ σΩ and Ai∩ Aj = ∅ ∀i 6= j ⇒ µ(S
j∈NAj) =P
j∈Nµ(Aj).
The triple (Ω, σΩ, µ) is called a probability space. A set A ⊂ Ω is said to be µ-measurable if A ∈ σΩ. Moreover, if A is measurable and µ(A) = 0, then any set B ⊂ A is assumed to have measure zero. The general additivity law is readily proven by observing that A \ B, B \ A and A ∩ B are pairwise disjoint sets whose union yields A ∪ B. Moreover, A \ B and A ∩ B are disjoint sets, their union being A, so that µ(A \ B) = µ(A) − µ(A ∩ B). Thus, one gets
µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B) ≤ µ(A) + µ(B) , (2.1) the equality sign holding iff A ∩ B has measure zero, which is true, in particular, when the intersection is empty.
Definition 2.3. A property relative to the elements ω ∈ Ω is said to hold µ-almost everywhere (in short µ-a.e.) in the measurable set A ⊆ Ω if it holds ∀ω ∈ A \ B and B has measure zero.
If Ω is finite or countably infinite, one can always build up a probability measure on the largest σ-algebra 2Ω, in the natural way, by assigning a function p : Ω → [0, 1] : ω 7→ pω such that P
ω∈Ωpω = 1. Indeed, Given A ∈ 2Ω, since A = S
ω∈A{ω}, then, by the countable additivity of the measure µ one has
µ(A) = µ [
ω∈A
{ω}
!
=X
ω∈A
µ({ω}) .
Thus, the measure µ(A) of any measurable set A is completely determined by the value of the measure of all its singletons (i.e. subsets consisting of a single element), and one has to assign pω := µ({ω}). The normalization of the sum of the pω’s follows taking A = Ω in the above displayed equation and using µ(Ω) = 1. If Ω is uncountable, the latter procedure does not work, in general.
Example 2.1. Consider the case Ω := [0, 1]. A natural (probability) measure µ on Ω should be such that if 0 ≤ a ≤ b ≤ 1, then µ([a, b]) = b − a. Observe that the singletons are the set consisting of a single point ω of Ω, and that by shrinking any interval to a single point one gets µ({ω}) = 0 ∀ω ∈ Ω. Thus, one cannot define such a natural measure on singletons. Moreover, if one tries to define the candidate measure at hand on the uncountable power set 2Ω, it can be proven that no such measure exists: the power set is too large.
2.1. PROBABILITY MEASURES AND INTEGRATION 19 With a (probability) measure µ on Ω, one defines an integration over Ω as follows. First of all, if χA denotes the characteristic (or indicator) function of the measurable set A (χA(x) = 1 if x ∈ A and zero otherwise), one defines
Z
In this way one can define the integration of the so-called simple functions, namely functions that are (finite) linear combinations of characteristic functions of given sets. Thus, if S = P
More general functions are then approximated through sequences of simple functions. More precisely, if F ≥ 0, one sets
For a function F with non constant sign one then introduces the positive part F+ := max{0, F } and negative part F− = max{0, −F } = − min{0, F } of F (notice that both F+ and F− are nonnegative by definition).
Definition 2.4. A function F is said to be (absolutely) integrable over B ⊆ Ω with respect to the measure µ if
Notice that the latter definition of integrability is equivalent to require that both R
BF± dµ
Definition 2.5. The space of integrable functions over Ω with respect to µ is denoted by L1(Ω, µ). In general, for any p ≥ 1 one defines
Of particular interest in the sequel will be the spaces L1 and L2 of integrable and square integrable functions, respectively.
Definition 2.6. Given two probability measures µ and ν on Ω (i.e. defined on the same σΩ), µ is said to be absolutely continuous with respect to ν if for any set A such that ν(A) = 0 it follows µ(A) = 0.
If µ is absolutely continuous with respect to ν, then it turns out (Radon-Nikodym theorem) that µ has a density, namely there exists a nonnegative ν-integrable function % : Ω → R+ such that
µ(A) = Z
A
dµ = Z
A
% dν
for any measurable set A. One writes the above condition in short as dµ = %dν, or % = dµ/dν, referring to the latter as the Radon-Nikodym derivative of µ with respect to ν.
The most relevant case in applications is that of measures absolutely continuous with respect to the Lebesgue measure (the unique countably additive measure defined on the Borel σ-algebra of Rnand such that the measure of a multi-rectangle is the product of the lengths of the sides), in which case one writes dµ = % dV , where dV denotes the Lebesgue volume element in Rn.
In probability theory, the integral of F with respect to a probability measure µ over Ω is referred to as the expectation or mean value of the random “variable” F , and is denoted by
hF iµ= Eµ(F ) :=
Z
Ω
F dµ . The above formula implies for example that hχAiµ = µ(A).
Exercise 2.1. Let A = [0, 1] ∩ Q be the set of rationals in [0, 1]; then the Lebesgue measure V (A) of A is zero. Moreover, the Dirichlet function D(x) - defined on [0, 1] as D(x) = 1 if x is irrational and D(x) = 0 otherwise - is not Riemann integrable but is integrable with respect to the Lebesgue measure over [0, 1], the value of the integral being exactly one. Indeed, since A is countable, it can be covered by a sequence of intervals {Ij}j∈N such that Ij is centered at xj ∈ A and V (Ij) = ε/2j+1, where ε is arbitrarily small. Then, since A ⊂ S
jIj, it follows V (A) ≤ V (S
jIj) ≤ P
j≥0V (Ij) = ε, and the arbitrariness of ε implies V (A) = 0. For what concerns the Dirichlet function, observe that D(x) = χAc, so that R D(x)dV = R χAcdV = V (Ac) = 1 − V (A) = 1.
A good reference for probability theory is [25]. Abstract measure and integration theory is extensively treated in the analysis monograph [45].