The Foundations ofStatistical Mechanics

Nino ZanghìUniversità di Genova

Contents

0 Title slide . . . . . . . . . . . . . . . . . . . 1

1 The Foundamental Probability Space in Clas-sical Mechanics 41.1 Classical mechanics . . . . . . . . . . . . . 61.2 Liuoville’s Theorem . . . . . . . . . . . . . 81.3 Micro-canonical Measure . . . . . . . . . . 111.4 Measure preserving dynamical systems . 141.5 Poincaré recurrence theorem . . . . . . . . 161.6 Newtonian systems . . . . . . . . . . . . . 171.7 Identical particles . . . . . . . . . . . . . . 201.8 Confined systems . . . . . . . . . . . . . . 271.9 Systems with a variable number of particles 28

2 Macroscopic Variables 302.1 Local quantities . . . . . . . . . . . . . . . 312.2 Hydrodynamics fields and locally conserved

quantities . . . . . . . . . . . . . . . . . . . 332.3 The one-particle distribution function . . 352.4 Coarse-graining . . . . . . . . . . . . . . . 362.5 Two systems in thermal contact . . . . . . 402.6 Subsystem of a larger system . . . . . . . 412.7 Hydrodynamic description . . . . . . . . . 432.8 Kinetic description . . . . . . . . . . . . . . 442.9 Macroscopic description and autonomous

evolution of the macro-state . . . . . . . . 45

1 The Foundamental ProbabilitySpace in Classical Mechanics

In statistical mechanics an “ensemble” is anidealization consisting of a large (possibly infinite)number of virtual copies of a system, considered all atonce, each of which represents a possible state thatthe real system might be in. Josiah Willard Gibbs inthe preface to his celebrated book on the ElementaryPrinciples in Statistical Mechanics published in1902, a year before his death, introduced this notion.In in the language of modern probability theory, anensemble is nothing but a probability space(Ω,F, IP), with the sample space Ω representing the setof possible microscopic states of a physical system and

IP providing their probability distribution; F is asuitable family of sets of micro-states (its choice ismainly dictated by mathematical considerations).The precise mathematical expression for theprobability space has a distinct form, as the samplespace Ω depends on the type of mechanics underconsideration (quantum or classical). In each case themeasure IP is a probability distribution overmicrostates, but the notion of a microstate isconsiderably different. In quantum mechanics, themicrostates are vectors in a Hilbert space while inclassical mechanics are points in the phase space ofthe system.

1.1 Classical mechanics

A classical system with r degrees of freedom isdescribed by r generalized coordinates q = (q1, . . . qr) andr generalized momenta p = (p1, . . . , pr). The totality ofsuch points forms the the phase space Γ = x = (q, p),which is a smooth manifold of dimension 2r, locallyhomeomorphic to R2r. The dynamics of the system isgiven by Hamilton’s equations

dX

dt= vH(X) , (1)

where vH is the vector field on Γ

vH(x) = vH(q, p) =(∂H(q, p)∂p

,−∂H(q, p)∂q

). (2)

with H(x) = H(q, p) the Hamiltonian of the system. Thesolution map of Hamilton’s equations Tt : Γ→ Γ, t ∈ R,that is the map that transform an initial state X0 intoXt = Tt(X0), the state at time t, is a flow on Γ, called theHamiltonian flow.

Flow A flow on a space Ω is a continuos family oftransformations Tt : Ω→ Ω parametrized by t ∈ R (the time)obeying the rules:

• T0 is the identity function on Γ;

• Ts Tt = Tt+s,

• T−1s = T−s, whenever T−1

s is well-defined.

For an Hamiltonian flow T−1s is well defined and in addition to the

aforesaid properties we have

∂Tt(x)∂t

= vH(Tt(x)) . (3)

1.2 Liuoville’s Theorem

The volume element in phase space,

dx =r∏k=1

dqkdpk (4)

does not depend of the choice of coordinates and isinvariant under canonical transformations. Thus, itdefines a notion of volume in phase space,

|A| =∫Adx , (5)

for A a Borel subset of Γ, which depends only on theintrinsic (symplectic) geometry of phase space. |A| iscalled the Lebesgue volume, or measure, of A. In

particular, the volume is invariant under the canonicaltransformations realizing the time evolution of thesystem. This is the content of a celebrated theorem.

Proposition 1 (Liouville s Theorem)

The Lebesgue volume of a measurable subset A ofphase space Γ does not change in the course of time.That is, if At is the set of phase points obtained byevolving the points in A for a time t,

At = X ∈ Γ : T−t(X) ∈ A , (6)

then|A| = |At| (7)

Equivalently, the theorem says that the Jacobiandeterminants

det(∂Tt(x)∂x

)are always unity.

Proof This is most easily seen (in the case oftransformations continuously connected with theidentity at least) by showing that Hamiltonian flowsare incompressible. To do this we calculate thedivergence of eq. (2),

div vH = ∂

∂q

∂H(q, p)∂p

− ∂

∂p

∂H(q, p)∂q

= 0 .

Then the conservation of the phase-space volumefollows from the application of Gauss’ theorem since

the surface integral of vH is the rate of change of thevolume within the surface.

This result obviously applies to any flows built up frominfinitesimal canonical transformations, not only toHamiltonian time evolution. (In fact, the conservationof phase-space volume holds for all canonicaltransformations.) Liouville s Theorem has profoundimplications both in statistical mechanics and indynamical systems theory. In particular, it impliesthat there are no attractors in Hamiltonian dynamics.

1.3 Micro-canonical Measure

Hamiltonian dynamics describes the motion of anisolated system. For example, the complete

microscopic state of an isolated classical system of Npoint particles is specified at any time t by a point X inits phase space Γ, with X = (Q1,P 1 . . . ,QN ,PN)representing the positions and the momenta of all theparticles. Given an X at some time t0, the micro-stateat any other time t ∈ R, Xt, is given (as long as thesystem stays isolated) by the evolution generated bythe Hamiltonian H(x) of the system. Such an evolutiontakes place on an energy surface, ΓE, specified byH(x) = E. It is useful—especially when dealing withmacroscopic systems—to actually think of ΓE as anenergy shell of thickness ∆E E, i.e., as the thesubset of space phase

ΓE,∆E = x ∈ Γ : E ≤ H(x) ≤ E + ∆E . (8)

Suppose that the particles are confined in a box Λ ⊂ R3

of finite volume V = |Λ|. Then |ΓE,∆E| is finite (if theparticles were not confined the volume would beinfinite). Then one can define the probability measure

IPE,∆E A = |A ∩ ΓE,∆E||ΓE,∆E|

, A ∈ B(Γ) , (9)

with B(Γ) the Borel algebra Γ, which is called themicrocanonical measure. It is simply the Lebesguemeasure restricted to the shell of constant energy andnormalized to the volume of the shell. Eventually, onemay pass to the limit ∆E → 0 of both sides of eq. (9)and obtain in this way the probability measure IPE onthe surface ΓE.

N.B. When no confusion will arise, we shall denoteΓE,∆E also by ΓE, assuming, whenever needed, that a

certain tolerance ∆E has been specified from theoutset.

1.4 Measure preserving dynamical systems

In view of Liouville theorem,

IPE

T−1t (A)

= IPE A , (10)

for all A ∈ B(Γ) and t ∈ R. This is condition of what iscalled a measure preserving dynamical system. Thelatter is defined as a probability space (Ω,F, IP) and acontinuous family of measure-preservingtransformation Tt on it fulfilling eq. (10) for IPE = IP andall A ∈ F.In the theory of dynamical systems, it is alsoconsidered the case of a discrete time dynamics

obtained by iterating a given measurabletransformation T : Ω→ Ω (the initial state ω0 evolvesinto the state ω1 = T (ω0), which evolves into ω2 = T (ω1)and so on . . . ) satisfying the condition

IP(T−1(A)

)= IP (A) , ∀A ∈ F. (11)

Isomorphisms of dynamical systems Twomeasure-preserving dynamical systems (Ω,F, IP, T ) and(Ω′,F′, IP′, T ′) are isomorphic if there a mappingφ : Ω→ Ω′ that preserve the structure. This means that(Ω,F, IP) and (Ω′,F′, IP′) are isomorphic as measurespaces and that φ T = T ′ φ.

1.5 Poincaré recurrence theorem

Poincare’s recurrence theorem can be stated for ageneral measure preserving dynamical system(Ω,F, IP, Tt).

Proposition 2 (Poincaré recurrence theorem)

Let A be a subset of Ω such that IP A > 0. Then foralmost every ω ∈ Ω (i.e., except for a set of omega’s ofIP-measure 0) there exist arbitrarily large t such thatTt(ω) ∈ A.

Proof First of all, rather than to consider continuoustime let us discretize it and consider T = Tτ , where τ isa fixed unit of time. A point ω ∈ A eventually returns to

A if there is k ≥ 1 for which T k(ω) ∈ A. Let B the set ofall those points of A which will never return in A. Notethat if ω ∈ B, then T n(ω) 6∈ B for each n ≥ 1. ThusB ∩ T−n(B) = ∅ for n ≥ 1 and henceT−k(B) ∩ T−(n+k)(B) = ∅ for each n ≥ 1 and each k > 0.Then the sets B, T−1(B), T−2(B), . . . are pairwise disjointand each has measure IP B. Since IP Ω = 1, byadditivity of the measure of disjoint sets, IP B mustbe zero.

1.6 Newtonian systems

We shall provide now more details on a system of Npoint particles moving in physical space E3 ' R3. Thephase space Γ of such a system is the Cartesian

product of the the one-particle phase spaces Γ1p,

Γ = Γ1p × · · · × Γ1p︸ ︷︷ ︸N times

= Γ1pN . (12)

whereΓ1p = Q1p ×P1p , (13)

with Q1p and P1p the one-particle configuration spaceand configuration space, respectively. Thus a genericphase point has coordinates x = (x1, . . . , xN), wherexi = (qi,pi) ∈ Γ1p. If the particles are not confined,clearly Q1p ' R3, while Q1p = Λ for particles confined ina bounded region Λ. If relativistic effect are neglected(=no maximal speed), P1p ' R3.Let m1, . . . ,mN be the masses of the particles. If (as it isusually assumed) the particles i and j interact trough

pair potentials V ij(q), then the Hamiltonian is of thestandard form eq. (12) with

K(p) =N∑i=1

pi2

2miand U(q) = 1

2∑i 6=j

V ij(qi − qj) , (14)

i.e.,

H =N∑i=1

pi2

2mi+ 1

2∑i 6=j

V ij(qi − qj) (15)

(If there there an external field with potential energy φi,the Hamiltonian contain the additional term∑N

i=1 φi(qi) .)The Hamiltonian above defines a Newtonian systemN point particles. As it is easily seen, for such anHamiltonian, Hamilton’s equations become Newton’s

equations

mid2Qi

dt2= −∇i

∑j 6=i

V ij(Qi −Qj) , i = 1, . . . , N (16)

1.7 Identical particles

If the particles are identical, they have the same massm and interact trough the same pair potential V . Thustheir motion is governed by the Hamiltonian

H = 12m

N∑i=1

pi2 + 1

2∑i 6=j

V (qi − qj) (17)

Note that H is a function on Γ which is symmetricunder permutations of the particles. Indeed, since the

particles are identical, no physical distinction can bemade between points in Γ1p

N that differ only in theordering of the particle phase points. Thus, two pointsin Γ1p

N x = (x1, . . . , xN) , xi ∈ Γ1p,

x′ = π(x) = (xπ−1(1), . . . , xπ−1(N))(18)

where π is a permutation of the particles indices, bothdescribe the same state of the system.Therefore the true phase space of the N-particlesystem is not the the Cartesian product Γ1p

N, but thespace obtained identifying points in Γ1p

N representingthe same state. We shall denote such a space by NΓ1p

and we shall call it the natural phase space of Nidentical particles. It can be characterizedmathematically in two equivalent ways:

(i) Subtract from Γ1pN all the coincedence points and

form the space

Γ1pN6= =

(x1, . . . , xN) ∈ Γ1p

N : xi 6= xj ,∀i 6= j. (19)

Then identify all points that differ by apermutation; in other words, “divide” the space bythe action of the permutation group SN, obtainingin this way the space Γ1p

N6=/SN. This is the natural

phase space NΓ1p. Note that the first step of theconstruction guarantees that the phase pointsrepresent exactly N particles (if the coincidencepoints were not removed, there would be phasepoints with a number of particles less than N ).

(ii) A point x in NΓ1p can be considered as a set of N

points in the one-particle phase space Γ1p, i.e.,

NΓ1p 3 x = x1, . . . xN xi ∈ Γ1p , xi 6= xj if i 6= j .(20)

In can be shown that NΓ1p is a smooth manifold.

Misconceptions about identical particles

The fact that the phase space of N identical particles isNΓ1p and not Γ1p

N has usually been overlooked and israrely mentioned in textbooks of classical mechanicsand statistical mechanics. The textbooks tend tounderline that the proper description of identicalparticles can be achieved only within the framework ofquantum mechanics. This is probably due to two

reasons. One is the following (wrong) argument:“Particles are identical if they cannot be distinguishedby means of measurements. So, if particles have thesame mass, charge, etc., they could be distinguishedonly by their location in space, as it is the case inclassical mechanics. However, in quantum mechanics,particles do not have trajectories. Therefore theycannot be be distinguished if they have the same mass,charge, etc.. Thus the notion of identical particles ispurely a quantum one without any classicalequivalent.” This argument is completely wrong.The notion of identical particles is older than quantummechanics. It was already recognized by Gibbs. Inorder to correctly calculate the entropy change in aprocess of mixing to identical fluids or gases (at the

same temperature, etc.), Gibbs postulated that statesdiffering only by permutations of identical particlesshould not be counted as distinct. Gibbs postulation isexplained by what we have said above and summarizehere: for non identical particles, the propermathematical description of the state is in terms of thevector x = (x1, . . . , xN); the vector is an orderedsequence of numbers, and so it keeps track of thelabels of the particles; if the particles are identical, it isthe set x = x1, . . . , xN the right mathematicalrepresentation, since a set of objects does non involveany order of the objects themselves.Another reason for overlooking the classicaldescription of identical particles in terms of NΓ1p couldbe the following. The dynamics of a classical system

involves only local properties in phase space. To bemore precise, the time evolution of a classicalN-particle system is a continuos curve in NΓ1p. Theone curve in NΓ1p corresponds to N ! curves in Γ1p

N.These curves do not intersect and are locally identical.Then one can just pick up one of them to describe thedynamics. In other words, the dynamics do not mixthe N ! factorial copies of NΓ1p whose union is just thespace Γ1p

N. So, the description of identical particles bymeans of the “wrong” space Γ1p

N is only redundant (asthe description of the electromagnetic field in terms ofthe vector potential), but has not dynamicalimplications (this redundancy manifests itself onlywhen we count the states, as in the case considered byGibbs). The situation in quantum mechanics is

different, as the global properties of NQ1p, theconfiguration space of N identical particles, thenbecome essential, since the the wave function of thesystem ψ(q) “sees” the global properties ofconfiguration space.

1.8 Confined systems

Usually, the particles are confined in a box (vessel)Λ ⊂ R3 of finite volume V = |Λ|. In this case, Newton’sequations eq. (16) for Qi ∈ Λ must be supplemented byconditions on the boundary ∂Λ. The simplest one is areflecting boundary condition. As the name says,when a particle reaches a reflecting boundary it isreflected back into the box. In molecular dynamics, in

order to mimic a liquid or solid without increasing thenumber of particles to be simulated, one uses periodicboundary conditions: a particle, going through aboundaries returns to the box from the opposite side.

1.9 Systems with a variable number ofparticles

A central problem in statistical mechanics is the studyof the properties of the subsystems of a large system.If the particles are identical, a natural way to specify asubsystem is by considering the particles in a certainregion ∆ ⊂ Λ. A subsystem so defined will be called alocal-subsystem associated with ∆. Since in thecourse of time the number of particles in ∆ may

change, the microscopic state of the subsystem isspecified by the number n of particles in ∆ and by theirstate x = (x1, . . . xn). Thus the phase space of the localsubsystem associated with ∆ is

Γ∆ =N⋃n

nΓ1p ∆ (21)

where Γ1p ∆ is the one-particle phase space restricted to∆ and N is the number of particles in Λ. If we wish tolet unspecified the number of particles in Λ, we mayset N =∞ in eq. (21).

2 Macroscopic Variables

In classical mechanics a physical quantity isrepresented by a real valued (measurable) function Yon the phase space Γ. If the micro-state of the systemis X, the value of the quantity is

Y = Y (X) . (22)

If the quantity Y is measured and it is found in theinterval Y ±∆Y we gather information about theactual micro-state: we know that is in the region ofphase space of those micro-states X such thatY −∆ ≤ Y (X) ≤ Y + ∆Y .Not any old function Y on phase space represents amacroscopic quantity. To be a candidate for that, the

function has to vary slowly on the macroscopic scale.This is so for locally conserved quantites.

2.1 Local quantities

A local quantity is one which is associated with any(measurable) subset ∆ ⊂ Λ and can be expressed asthe integral of a (possibly singular) density. This is sofor the following functions on phase phase describingthe number of particles N, the momentum P and theenergy E of the particles in any (measurable) subset∆ ⊂ Λ:

NX(∆) =∫

∆nX(r) d3r , where nX(r) =

N∑i=1

δ(r −Qi) , (23)

PX(∆) =∫

∆pX(r) d3r , where pX(r) =

N∑i=1

P iδ(r −Qi) , (24)

EX(∆) =∫

∆e(r) d3r , where eX(r) =

N∑i=1

12m

P i2 + 1

2∑i 6=j

V (Qi −Qj)

δ(r −Qi) .

(25)

We have used the subscript X to express thedependence on the actual microscopic stateX = Qi,P iNi=1 (In the following, when no confusionwill arise, for easiness of notation, we shall drop thesubscript X.) N, P and E are known as themicroscopic hydrodynamics fields and n, p, and e astheir densities.

2.2 Hydrodynamics fields and locallyconserved quantities

The hydrodynamics fields are locally conserved, that is,their densities satisfy the continuity equations

∂n

∂t+∇ · JN = 0 , (26)

∂p

∂t+∇ · JP = 0 (27)

∂e

∂t+∇ · JE = 0 , (28)

where JN, JP , and JE are the associated flux densities(we leave as an exercise to compute their explicit form).We have used the symbol JP for denoting themomentum flux because it is a tensor )on the

macroscopic scale, related to the stress tensor). Thelocal conservation of the hydrodynamics fields is thereason why they are macroscopically relevant (asopposed to any old function on phase space). In fact, alocal excess of a locally conserved quantity cannotdisappear locally (which otherwise could happen veryrapidly), but can only relax by spreading slowly overthe entire system. Then the processes of transport ofsuch quantities are extremely long on the microscopicscale and thus are macroscopically relevant—they are,so to speak, the macroscopic manifestation of themicroscopic dynamics.

2.3 The one-particle distribution function

The macro-state X can be considered as a set of Npoints in the six dimensional one-particle space Γ1p

(section 1.7), somewhat analogous to the positionsQiNi=1 in Λ. We may then consider the density in thesix-dimensional one-particle phase space Γ1p

fX(q,p) =N∑i=1

δ(q −Qi)δ(p− P i) , (29)

which, when integrated on any (measurable) region∆ ⊂ Γ1p, ∫

∆fX(q,p) d3q d3p , (30)

gives, for any micro-state X = Qi,P iNi=1, the numberFX(∆) of particles that are in ∆, that is, for ∆ = ∆x ×∆p,

the number of particles whose positions are in ∆x andtheir momenta in ∆p. (In measure-theoretic language,it is the uniform point measure on the one-particlephase space.) It is the most known example ofone-particle distribution function. It was introduced byBoltzmann in order to analyze the macroscopicproperties of a dilute gas in order to provide amicroscopic explanation of the second law ofthermodynamics (indeed the first one).

2.4 Coarse-graining

Consider for example the number of particles of a gasin standard conditions contained in a cube of side 1mm, differences in this number of order 10 or even 1010,

are differences which do not make any difference. So,the appropriate macroscopic description is not givenby Y itself, but by the function

v

Y obtained by suitablecoarse-graining.Mathematically, the coarse graining of a function canbe done in the following way. Let the values Y of Y bereal (or integer) numbers. Divide these values intodisjoint intervals ∆Yν, ν = 1, 2, 3, . . . and let Yν a point in∆Yν, for example the middle point (of or a suitableaverage) of F in ∆Yν. Then the coarse-graining of Y isthe function

v

Y which assumes the constant value Yνfor all X such that Y (X) ∈ ∆Yν, that is,

v

Y (x) =∑ν

Yν 1∆Yν (Y (x)) , (31)

where 1A(y) is the characteristic function of the set A,

i.e., it is equal to 1 if y ∈ A and equal to zero otherwise.The box functions are discontinuous, so, if formathematical reasons one wishes to work with smoothfunctions, one may allow the 1∆Yν to be, with anydegree of precision, smooth approximations of thecharacteristic functions. Note that we have alreadyused a coarse-graining procedure at the verybeginning of Sec. ??, when we have said that themacroscopically appropriate phase space of an isolatedsystem is the energy shell of thickness ∆E rather thanthe energy surface.A macro-state M is then specified by one of the valuesof

v

Y ,M = Yν , (32)

andΓM = X :

v

Y (X) = Yν (33)

is the region in ΓE associated with macro-state M .Clearly, the ΓM are disjoint and the decomposition ??holds. See Fig. ??.The coarse-graining procedure extends trivially to afamily of n macroscopic quantities represented by thefunctions Y1, . . . Yn. All we have to do is to read eq. (22)and eq. (31) as vector equations, with Y = (Y1, . . . , Yn) ineq. (22) representing now the values of the family offunctions Y = (Y1, . . . , Yn), and ν and ∆Yν in eq. (31),being now, respectively, the multi-index ν = (ν1, . . . , νn)and the n dimensional box whose edges are theintervals ∆Yν1, . . . , ∆Yνn.

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M = Y and ΓM = X : Y (X) = Y . (34)

2.5 Two systems in thermal contact

To illustrate more concretely the decomposition ??, weconsider a macroscopic system of fixed total energy Econsisting of two parts, a and b, of energies EA and EB,between which energy can flow, but which areotherwise independent. See Fig. ??. The total energy

shell is ΓE = Γa × Γb and the Hamiltonian is

H(x) = Ha(xa) + Hb(xb) + λV (xa, xb) , (35)

where Ha and Hb are the Hamiltonians of the twosystems, and λV is a small interaction. A macroscopicdescription of the system is then given by the coarsegrained energy of one of the two systems, say Ha, sincewhen its value is EA, the energy of the other system isE2 ≈ E − E1 (having assumed that the total system isisolated and the interaction between the two systemsis small). Then

M = EA , and ΓM = X : Ha(X) = Ea , (36)

whence the partition ??.

2.6 Subsystem of a larger system

We now consider the case in which the a-system isdefined as the collection of the particles in ∆a of asystem contained in a larger vessel Λ, as depicted Fig.fig. 1. A macro-state M of a can now be specified bythe values Na, and Ea of the (coarse grained) number ofparticles NX(∆a) and energy EX(∆a). Then

M = NA, EA , and ΓM = X : NX(∆a) = Na , EX(∆a) = Ea .(37)

For a macroscopic system, the total interaction energyof the particles inside ∆a with those outside is(typically) negligible with respect to the energy EA.Then (typically) E ≈ EX(∆a) + EX(Λ \∆a) and and thusthe sets ΓM provide a partition ?? of the total phase

space ΓE.

2.7 Hydrodynamic description

When N is very large, we can give a coarse-graineddescription of a system of total constant energy E bycharacterizing its macro-state M in terms of the coarsegrained hydrodynamic fields as described in ??. Wedivide Λ into cells (e.g., cubes) ∆α, α = 1, . . . J, centeredaround the points rα,

Λ =J⋃α=1

∆α .

Assume that J is large, but still J N , so that so thateach cell contains a very large number of particles.

Then a macro-state M is specified by the(coarse-grained) values of the energy Eα, momentumP α and number of particles Nα in each cell ∆α

(analogously to what we have done in section 2.6 forthe region ∆a),

M = Nα,P α, EαJα=1 , ΓM =X : NX(∆α) = Nα ,PX(∆α) = P α , E(∆α) = EαJα=1

,

(38)whence ??.

2.8 Kinetic description

The kinetic description of a fluid consists in acoarse-grained specification of the one-particledistribution function. This is achieved in a manneranalogous to that for the hydrodynamic fields:

consider a partition of one particle phase-space Γ1p

into cells ∆α, α = 1, . . . K, centered around thepoints (rα,pα). As before, assume that 1 K N andspecify a macro-state M in terms of the(coarse-grained) values of the number of particles Nα

in each cell ∆α, M = NαKα=1 . Then

ΓM =X : FX(∆α) = NαKα=1

, (39)

and we have again ??.

2.9 Macroscopic description andautonomous evolution of the macro-state

To decide whether a certain macroscopic description isappropriate, e.g., whether the hydrodynamic or the

kinetic description is appropriate for the system athand, one should consider the time evolution of themacrostates

M0 = M(X0) −→Mt = M(Xt) , (40)

where Xt = T t(X0) is the Hamiltonian evolution of theinitial micro-state X0. The macroscopic descriptionprovided by M will be satisfactory if the evolutioneq. (40) of the macro-state is autonomous, i.e., is givenby a flow in space M of all macrostates. This importantfact has been proven for the hydrodynamics fields in asuitable regime (local equilibrium) and only for specialmodels. The discovery that the one-particledistribution function of a dilute gas evolvesautonomously according to an evolution equation has

been the everlasting contribution of Boltzmann tostatistical mechanics—the evolution equation carrieshis name. A rigorous proof of Boltzmann’sachievement has been obtained only recently, around1970, by Lanford.

∆a

Λ

Figure 1: The system a is the collection of all particles of asystem contained in a large vessel Λ that are in the region ∆a.