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DYNAMICAL FOUNDATIONS of STATISTICAL MECHANICS

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Scuola Galileiana, Padova, A.A. 2017/2018

A. Ponno

May 31, 2018

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Contents

Preface 5

1 The Clausius virial theorem 7

1.1 General formulation . . . 7

1.1.1 Homogeneous case . . . 9

1.2 Theory of gasses . . . 11

2 Probabilistic approach 17 2.1 Probability measures and integration . . . 17

2.2 Invariant measures . . . 20

2.3 Equilibrium statistical mechanics . . . 24

2.3.1 Micro-canonical measure . . . 24

2.3.2 Canonical measure . . . 26

2.4 Equivalence of the two measures . . . 28

2.4.1 The Lebowitz-Percus-Verlet theory of fluctuations . . . 30

3 Stochastically thermostated systems 33 3.1 The Langevin equation . . . 33

3.2 The Fokker-Planck equation . . . 35

3.3 Hamiltonian case . . . 37

3.3.1 Free energy minimum . . . 38

3.3.2 Free energy decay . . . 40

3.3.3 Convergence to equilibrium . . . 43

4 Kinetic theory of gasses and fluids 45

Bibliography 47

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Preface

The present notes are devoted to the study of those mathematical aspects relevant to the dynamical foundations of statistical mechanics of classical Hamiltonian systems.

Equilibrium statistical mechanics, whose present formulation is due to Gibbs [22], is the physical theory that explains the laws of equilibrium thermodynamics of a macroscopic body, described by a large size Hamiltonian system, in terms of a privileged probability measure, named Gibbs measure, on its phase space. In referring to “large” size systems one has in mind numbers of “particles” that can be as large as the Avogadro number NA, i.e. the inverse of the atomic mass unit mu expressed in grams: NA= 1/mu = 6.02 · · · × 1023.

The problem that naturally arises concerns the role played by the dynamics. Indeed, at a microscopic level, particles move subject to both their mutual interaction forces and to forces due to external sources (e.g. gravity, confining forces, electromagnetic fields and so on). Thus it has to be explained how and why such an unsolvable microscopic dynamics may give rise to, or be compatible with the stationary Gibbs measure describing the collective properties of the whole system. In short, one should be able to explain thermodynamics starting from the Newton law.

In real experiments one usually measures physical quantities by taking the arithmetic mean of their values detected at different times. Such a procedure is used both in measuring the value of a quantity in stationary conditions and in measuring the evolution of its value in the course of time. Thus, it would be desirable to have at one’s disposal a theory concerning time averages along single orbits. The Clausius virial theorem represents a very interesting example in this direction. In that context, a second fundamental question arises, namely whether the

“experimentally” computed time-averages coincide or not with the theoretically computable expectation values with respect to the Gibbs measure. Such a property, specific of the system at hand, of the chosen probability measure, and of the class of interesting physical quantities, or “observables”, is called ergodicity. It must be stressed that, given a Hamiltonian system of some interest to matter physics, nobody is presently able to decide whether it is ergodic or not, neither with possible restrictions of the observables.

However, even ergodicity would not be sufficient to build up thermodynamics. Indeed, not only the existence of an equilibrium, but also the approach to equilibrium must be justified.

This requires a property stronger than (i.e. implying) ergodicity to hold, namely mixing. Such a property, which consists in the asymptotic de-correlation of observables in time, ensures that even starting with a measure that is not the Gibbs one, the latter will be approached in the

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course of time, in a weak sense. Also establishing the validity of the mixing property for a given Hamiltonian system of physical interest is still out of order. The only existing theory of approach to equilibrium rests on the addition of mutually balanced noise and dissipation to the conservative forces acting on the particles of the system. Such a theory is both mathematically elegant and conceptually plausible from a physical point of view, but in this way stochasticity is artificially inserted from outside instead of being obtained as a “collective” feature of the conservative system.

The approach to equilibrium of a given system may well take place notwithstanding the time reversal symmetry of the Hamilton equations that rule its dynamics at a microscopic level. More than this, the microscopic dynamics is characterized by the recurrence of almost every initial condition. The solution of such paradoxes helps to understand how an effective arrow of time occurs, in a statistical sense, when passing from the microscopic to the macroscopic level of description.

While deciding whether the dynamics of a given system displays some precise degree of stochasticity (ergodicity, mixing or more) is extremely difficult, it is usually easier to show whether some integrable behavior has to be expected. Such a perspective is meaningful since many systems in matter physics are close to integrable: two fundamental examples are crystals at low temperatures and gasses at high temperatures. In practice, by means of the canonical perturbation theory, one can try to build up one or more approximate constants of motion, or quasi-integrals, of the system. This procedure leads to absolutely nontrivial results, such as the KAM theorem, which ensures the preservation of most invariant tori of an integrable system under small perturbations. The latter result, together with the classical Poincar´e theorem on the nonexistence of smooth first integrals (besides the Hamiltonian) for quasi-integrable systems, is used to state that generic Hamiltonian systems are neither integrable nor ergodic [37]. However, one must never forget the role played by the energy-size relation characterizing the system: in the thermodynamics of ordinary matter the latter two quantities are strictly proportional. As a consequence, the validity of the above mentioned conclusion is doubtful in matter physics, and ergodicity or mixing might well old in the so-called thermodynamic limit, when the number of particles is ideally pushed to infinity at a constant (and usually small) value of the energy per particle.

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Chapter 1

The Clausius virial theorem

1.1 General formulation

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.

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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 .

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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.

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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.

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1.2. THEORY OF GASSES 11 Sharp statements of the virial theorem in the case of gravitational systems exist. The 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

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system is preserved, i.e.

N

X

j=1

|pj|2 2mj

+1 2

N

X

k,j=1 k6=j

φ(rkj) := K + Uint= E . (1.9)

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

Vwall := −1 2

N

X

j=1

rj(t) · fwj(t) = 1 2

X

n N

X

j=1

rj(t) · nfj,nw (t) = L 4

X

n N

X

j=1

fj,nw (t) ,

where P

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 L2

N

X

j=1

fj,nw .

The total pressure of the is defined as the arithmetic average of the six wall pressures, i.e.

P := 1 6

X

n

Pn .

One thus gets

Vwall = L3 4

X

n

Pn = 3

2P V , (1.10)

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

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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 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 . (1.11)

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

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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 4

N

X

j,k=1 j6=k

φ0(rjk)rjk = 1 4

N

X

j=1

Z

cube

ˆ

ρj(r)φ0(|r − rj|)|r − rj| d3r , (1.15)

where

ˆ

ρj(r) :=

N

X

k=1k6=j

δ(r − rk) . (1.16)

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 approximation, leading to

Vint= ρ 4

N

X

j=1

Z

cube

φ0(|r − rj|)|r − rj| d3r = ρN 4

Z

R3

φ0(r)r d3r = πρN Z +∞

0

φ0(r)r3 dr , (1.17)

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)

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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



φ0σ3− 3 Z +∞

σ

g(r)r2 dr



. (1.20)

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)

where

α := 2π 3



φ0σ3− 3 Z +∞

σ

g(r)r2 dr



. (1.22)

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,

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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.

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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 2 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 σ ⊆ 2 is called a σ-algebra on Ω if 1. it contains Ω;

2. it is closed with respect to complementation, i.e. A ∈ σ⇒ Ac ∈ σ; 3. it is closed with respect to countable union, i.e. {Aj}j∈N∈ σ ⇒S

j∈NAj ∈ σ.

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 2 is a σ-algebra and the intersection of σ-algebras is still a σ-algebra, if F ⊂ 2 denotes a set of subsets of Ω, the smallest σ-algebra containing F always exists and is usually denoted by σ(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

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Definition 2.2. Given a set Ω and a sigma-algebra σ on it, a probability measure on Ω is a nonnegative function µ : σ→ [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 ∈ σ and Ai∩ Aj = ∅ ∀i 6= j ⇒ µ(S

j∈NAj) =P

j∈Nµ(Aj).

The triple (Ω, σ, µ) is called a probability space. A set A ⊂ Ω is said to be µ-measurable if A ∈ σ. 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 2, in the natural way, by assigning a function p : Ω → [0, 1] : ω 7→ pω such that P

ω∈Ωpω = 1. Indeed, Given A ∈ 2, 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 2, it can be proven that no such measure exists: the power set is too large.

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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

χA dµ = Z

A

dµ := µ(A) ; Z

B

χA dµ = Z

A∩B

dµ = Z

χA∩B dµ = µ(A ∩ B) .

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

jsjχAj, one has Z

B

S dµ :=X

j

sj Z

B

χAj dµ =X

j

sjµ(Aj ∩ B) .

More general functions are then approximated through sequences of simple functions. More precisely, if F ≥ 0, one sets

Z

B

F dµ := sup

simple S:

0≤S≤F

Z

B

S dµ .

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

Z

B

|F | dµ = Z

B

F+ dµ + Z

B

F dµ exists finite.

Notice that the latter definition of integrability is equivalent to require that both R

BF± dµ exist finite, so that

Z

B

F dµ :=

Z

B

F+ dµ − Z

B

F dµ exists finite.

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

Lp(Ω, µ) :=

(

F : kF kp :=

Z

|F |p

1p

< +∞

) .

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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 σ), 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

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].

2.2 Invariant measures

Given a set Ω, suppose that a one-parameter group {Φt}t of transformations of Ω into itself is given. In the typical case, Ω is at least a topological space and, for any fixed t, Φt is at

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2.2. INVARIANT MEASURES 21 least a homeomorphism, i.e. a continuous bijection with continuous inverse. The group may be continuous, i.e. t ∈ R, or discrete, i.e. t ∈ Z. We will always have in mind the physical case where Ω is a smooth manifold, Φt being the flow of a given vector field on it.

Definition 2.7. A measure µ on Ω is said to be invariant with respect to Φt if

µ(Φt(A)) = µ(A) (2.2)

for any measurable A ⊆ Ω and any t.

The reason why one is particularly interested in invariant measures is that the mean value of a function F is the same as the mean value of F ◦ Φt if the expectation is taken with respect to a measure µ invariant with respect to Φt. Indeed, (2.2) means that

µ(Φt(A)) = Z

χΦt(A)(x) dµ(x) = Z

χA−t(x)) dµ(x) = Z

χA(y) dµ(Φt(y))

must be equal to µ(A) =R

χA(y)dµ(y) for any measurable A and any t, which implies dµ = dµ ◦ Φt for any t. Thus, one gets

F ◦ Φt

= Z

F (Φt(x)) dµ(x) = Z

F (y) dµ(Φ−ty) =

= Z

F (y) dµ(y) = hF i ,

i.e. one can compute mean values of dynamical variables, i.e. of compositions of the type F ◦Φt, even if the dynamics defined by Φt is unknown.

The physically interesting case is that of measures that are absolutely continuous with respect to the Lebesgue measure on Ω = Rn, with a smooth density %, when the one parameter group of transformation is the flow Φtu of the vector field u defining the dynamical system

˙x = u(t, x). In general, the density % is assumed to be explicitly dependent on the time t. The following proposition characterizes such invariant measures in terms of the dynamics of their related densities.

Proposition 2.1. The smooth density % of an invariant measure of the system ˙x = u(t, x) in Rn satisfies the continuity equation

∂t%(t, x) + ∇x· [%(t, x)u(t, x)] = 0 . (2.3) C PROOF. In order to simplify the notation, let us set x(t) := Φtu(y). The invariance

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condition (2.2) reads

0 = d

dtµ(Φtu(A)) = d dt

Z

Φtu(A)

%(t, x(t)) dV (x(t)) =

= Z

A

d dt



%(t, x(t)) det ∂x(t)

∂y



dV (y) =

= Z

A



t+ u(t, x(t)) · ∇x+ d

dtdet ∂x(t)

∂y



%(t, x(t)) dV (y) =

= Z

Φtu(A)

[∂t%(t, x(t)) + ∇x· (%(t, x(t))u(t, x(t)))] dV (x(t)) ,

which, having to be valid for all A, implies that the integrand in the last row above must vanish, which yields equation (2.3) with x = x(t) = Φtu(y). Since the equation has to be valid for all t and all y one can remove the constraint x = x(t) and get (2.3). B

Remark 2.1. Notice that in the proof above we made use of the identity d

dt det ∂x(t)

∂y



= [∇x· u(t, x(t))] det ∂x(t)

∂y



. (2.4)

Moreover, this is the only place where u(x) enters the game, which also implies the validity of (2.3) when u depends explicitly on time.

Exercise 2.2. Prove the latter identity. Hint: prove that det(I + A) = 1 + tr(A) + o() as

 → 0.

In the case of a standard Hamiltonian system, defined by the Hamiltonian H(x), the con- tinuity equation (2.3) becomes the so-called Liouville equation

∂%

∂t + {%, H} = 0 . (2.5)

Indeed, in this case the vector field u = XH = J ∇H is divergence free, i.e. ∇ · XH = 0 (show this explicitly), and ∇ · (%XH) = ∇% · XH = ∇% · J ∇H = {%, H}.

Proposition 2.2. If ∇x· u = 0 then the Lebesgue measure is Φtu-invariant.

C PROOF. If ∇x· u(t, x) = 0, (2.4) implies det(∂x(t)/∂y) = det(∂x(0)/∂y) = 1, so that dV (Φtu(y)) = dV (x(t)) = det ∂x(t)

∂y



dV (y) = dV (y) . B Thus, this is always the case for standard Hamiltonian systems.

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2.2. INVARIANT MEASURES 23 Proposition 2.3. The formal solution of the continuity equation (2.3), with initial condition ρ(0, x) := ρ0(x), is given by

ρ(t, x) = Z

δ x − Φtu(y) ρ0(y)dV (y) = ρ0 Φ−tu (x) det ∂Φ−tu (x)

∂x



. (2.6)

C PROOF. First of all, we observe that the identity of the two forms of the solution (2.6) follows by passing to the variable ξ = Φtu(y), which implies

Z

δ x − Φtu(y) ρ0(y)dV (y) = Z

δ(x − ξ)ρ0 Φ−tu (ξ) det ∂Φ−tu (ξ)

∂ξ



dV (ξ) =

= ρ0 Φ−tu (x) det ∂Φ−tu (x)

∂x

 .

Then, using the first form (the integral one) and setting x(t) = Φtu(y), one gets

∂ρ

∂t = ∂

∂t Z

δ(x − x(t))ρ0dV = − Z

˙x(t) · ∇xδ(x − x(t))ρ0dV =

= −∇x· Z

˙x(t)δ(x − x(t))ρ0dV = −∇x· Z

u(t, x(t))δ(x − x(t))ρ0dV =

= −∇x·

 u(t, x)

Z

δ(x − x(t))ρ0dV



= −∇x· (ρu) . B

For divergence free vector fields, ∇x · u = 0, such as the Hamiltonian ones, the solution (2.6) simplifies to

ρ(t, x) = ρ0 Φ−tu (x)

. (2.7)

Thus, the solution of the Liouville equation (2.5) is

%(t, x) = %0−tH(x)) . (2.8)

Notice that the Liouville equation (2.5) expresses the constancy of % along the characteristic curves (t, x(t)), where x(t) = ΦtH(y). Indeed, the condition %(t, x(t)) = %(0, x(0)), i.e.

%(t, ΦtH(y)) = %(0, y) , (2.9)

implies

d

dt%(t, x(t)) = ∂%

∂t + ∇x% · ˙x = (∂t% + {%, H}) (t, x(t)) = 0 . Then, setting x = ΦtH(y) on both sides of (2.9) yields (2.8).

One might hope that the density ρ, which evolves according to the Liouville equation (2.5), approaches, as t → +∞, an equilibrium (i.e. time-independent) density ρeq satisfy- ing {ρeq, H} = 0, at least for a reasonable class of initial densities ρ0(x). In such a case the invariant equilibrium measure dµeq = ρeqdV would be the privileged measure with respect to which one has to compute expectation values of functions defined on the phase space. How- ever, formula (2.8) obviously excludes any possibility of a strong convergence ρ(t, x) → ρeq(x) as t → ∞. Moreover, the following result holds.

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Proposition 2.4. Consider the Liouville equation (2.5) and let S be a real function satisfying [S(ρ)XH · ˆn]|∂Γ = 0 (ˆn(x) is the outward unit vector orthogonal to ∂Γ at x). Then R

ΓS(ρ)dV is preserved by the flow of the Liouville equation.

C PROOF. One has d

dt Z

Γ

S(ρ) dV = − Z

Γ

S0(ρ){ρ, H} dV = − Z

Γ

S0(ρ)∇ρ · J ∇H dV =

= − Z

Γ

∇S(ρ) · J∇H = − Z

Γ

∇ · [S(ρ)J∇H] =

= − Z

∂Γ

S(ρ)XH · ˆn dσ = 0 B

Thus, the Liouville equation possesses infinitely many constants of motion, and 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 density 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 set ρeq = 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 space Γ of the system, both introduced by Boltzmann and Maxwell first, and later, in the form here reported, by Gibbs [22, 24, 48, 49].

2.3.1 Micro-canonical measure

The first possible choice consists in the so-called micro-canonical measure

mc := W−1δ(H(x) − E) dV (x), (2.10)

where δ(s) denotes the Dirac delta “function”, characterized by the two properties: δ(s) = 0 for any s ∈ R \ {0}, and δ(0) = +∞ 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[, then Rb

af (s)δ(s)ds = f (0). Moreover, it turns out that δ(s) = dθ(s)/ds, where the Heaviside step function θ(s) is so defined: θ(s) = 1 if s > 0, θ(s) = 0 if s < 0 and θ(0) = 1/2. By

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2.3. EQUILIBRIUM STATISTICAL MECHANICS 25 imposing that R

Γmc = 1, one gets for the normalization factor W appearing in (2.10) the following formula

W (E) = d dE

Z

H(x)≤E

dV (x) = d

dEV ({H(x) ≤ E}) . (2.11)

The measure (2.10) is concentrated on the surface of constant energy

SE := {x ∈ Γ : H(x) = E} , (2.12)

and is used to describe the thermodynamics of isolated systems. The normalization constant W in (2.10) plays a fundamental role in the theory; we come back in a moment on this point. The micro-canonical measure (2.10) is clearly invariant with respect to the Hamiltonian flow ΦtH of the system ˙x = XH(x), since ρeq(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) = dxdΣ(x) = dEdΣ(x)

|∇H(x)| ,

where dΣ(x) is the Lebesgue surface element at x ∈SE. Then, if B ⊂ Γ is Lebesgue measurable in Γ (positive volume) and such that B ∩SE =A , with A ⊆ SE Lebesgue measurable onSE

(positive area), then

µmc(A ) := W−1Z

B

δ(E − H(x)) dV (x) = W−1 Z

B

δ(E − H(x)) dΣ(x)dx =

= W−1 Z

A

Z

δ(E − H(x))dE

 dΣ(x)

|∇H(x)| = W−1 Z

A

dΣ(x)

|∇H(x)| . (2.13) It follows that the normalization factor W , also known as the Boltmann statistical weight (or simply Boltzmann “factor”), is given by

W (E) = Z

SE

dΣ(x)

|∇H(x)| . (2.14)

Notice that the compactness of SE 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)|,

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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 (2.10) is not absolutely continuous with respect to the Lebesgue measure in Γ. Indeed, if A 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 (2.10), due to (2.13), can be regarded as absolutely continuous with respect to the Lebesgue measure onSE, with density 1/|∇H|.

The thermodynamics of an isolated system for which the micro-canonical measure (2.10) exists, is built up through the Boltzmann formula

S(E) = log W (E) , (2.15)

which yields the entropy S(E) of the system once one knows the Boltmann statistical weight.

Once the entropy (2.15) 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 + P dV − φ dN = T dS . (2.16)

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 (2.15). Notice that actually S = S(E, V, N ) since the Boltzmann weight W = W (E, V, N ). By means of (2.16), 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 hF imc:= W−1

Z

Γ

F δ(E − H) dV = W−1 Z

SE

F dΣ

|∇H| . (2.17)

2.3.2 Canonical measure

If the Hamiltonian system under consideration is thought of as non isolated, but in contact with a thermal bath so that, in place of the total energy, the temperature is kept fixed, then the so-called canonical measure must be used, namely the invariant measure

c := Z−1eT1H(x) dV (x) , (2.18) where T is the absolute temperature of the system. The normalization constant

Z(T ) = Z

Γ

eT1H(x) dV (x) (2.19)

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2.3. EQUILIBRIUM STATISTICAL MECHANICS 27 is known as the partition function of the system, whose existence requires of course some conditions (H must be both suitably lower bounded and upper unbounded) to be satisfied.

Notice that the canonical partition function (2.19) is the Laplace transform of the micro- canonical statistical weight (2.14) with respect to the energy:

Z(T ) = Z

Γ

eT1H dV = Z

Γ

Z

δ(E − H) dE



eT1H dV = Z +∞

0

eETW (E) dE , (2.20) (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 ) , (2.21)

which yields the free energy F = F (T, V, N ) of the system once one knows the partition function (2.19). Since the free energy is F = U − T S, where U denotes the internal energy (which is a dependent variable in the canonical framework), the first and second principle of thermodynamics combine together to give

dF = −SdT − P dV + φ dN , (2.22)

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 (2.17). Indeed, one has

hF ic :=

Z

Γ

F dµc= Z−1 Z

Γ

F eT1H dV =

= Z−1 Z

eET

Z

Γ

F δ(E − H) dV

 dE =

= R eET hF imcW (E) dE

R eETW (E) dE . (2.23)

Exercise 2.3. Consider the internal energy U of the system, defined as the mean value of the Hamiltonian, namely U (T ) := hHic = R Hdµc. By means of (2.23), taking into account that, obviously, hHimc = E, get the well known formula U (T ) = T2d log Z(T )/dT .

Exercise 2.4. Consider the (n−1)-dimensional spherical surface of radius R in the n-dimensional space, defined by the equation Pn

k=1x2k = R2 (n ≥ 2), and show that the volume Ωn(R) of the region it encloses is given by

n(R) := πn2Rn Γ(n2 + 1) ,

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where Γ(z) :=R

R+tz−1e−tdt is the Euler Gamma-function, satisfying Γ(z + 1) = zΓ(z). Hint:

first, show that Ωn(R) = Ωn(1)Rn, and observe that Z

Rn

f (x21+ · · · + x2n) dx1. . . dxn = Z

R+

f (R2)Ω0n(R) dR ;

then, use the latter trick to compute the Gaussian integralR

RnePnk=1x2k dx1. . . dxn and compare the result with the known one, i.e. πn/2.

Exercise 2.5. Consider the perfect gas model defined by H = PN i=1

|pi|2

2m, together with the condition that any particle vector position xi belong to a given domain D ⊂ R3 whose vol- ume is V ; ideal elastic reflection of any particle impinging on the wall is assumed. Compute W (E, V, N ) and, by means of (2.16), determine the pressure of the gas, which yields the Boyle law: P V = N T . Compute also Z(T, V, N ) and get the pressure from (2.22). Compare the two results.

Exercise 2.6. Consider a system of N non interacting harmonic oscillators, whose Hamilto- nian is H = 12PN

j=1(p2j + ωj2qj2). Compute W (E, V, N ) and Z(T, V, N ).

Exercise 2.7. Consider a system described by the Hamiltonian H =Pn j=1

hp2 j

2m + V (q)i

, where q = (q1, . . . , qn). Show that, if E ∝ n and V (q) = O(1), as n → ∞ the Boltzmann factor turns out to be

W (E, n) ∼ CnEn−22 Z

en−22E V (q)dnq .

Compare this result with the expression of the canonical partition function Z(T ) for the same system.

2.4 Equivalence of the two measures

In what follows we consider a specific class of observables, namely that of extensive and local functions F (q, p) on the phase space. The two properties are so defined: hF i = O(N ) and h(δF )2i = O(N ) as N → ∞, respectively, with δF := F − hF i. Here the averages h·i are meant both in the canonical and in the micro-canonical sense. In practice, such observables are of the form

F (q, p) =

N

X

j=1

Fj(q, p) ,

with hFji = O(1) and h(Fj− hFji)(Fk− hFki)i = cjk, where cjk are the entries of a symmetric, positive definite matrix1 such that cjj = O(1) and P

j6=kcjk = O(N ); for example, the latter

1For an observable of the form Fξ(q, p) :=P

jξjFj(q, p) one has 0 ≤(δFξ)2 = Pj,kξjcjkξk, which holds for any choice of ξ1, . . . , ξN.

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