Chapter 8 Microcanonical Ensemble

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Chapter 8

Microcanonical ensemble

8.1 Deﬁnition

We consider an isolated system withN particles and energyE in a volumeV . By definition, such a system exchanges neither particles nor energy with the surroundings. Uniform distribution of microstates. The assumption, that thermal equilibrium implies that the distribution functionρ(q, p) of the system is a function of its energy, d ∂ρ ρ(q, p) =ρ(H(q, p)), ρ(q, p) = E˙ 0, dt ∂H ≡ leads to to a constantρ(q, p), which is manifestly consistent with the ergodic hypothesis and the postulate of a priori equal probabilities discussed in Sect. 7.5. Energy shell. We consider a small butfinite shell [E,E+Δ] close to the energy surface. The microcanonical ensemble is then defined by

1 E < H(q, p)

Γ(E,V,N)= d3N q d3N p (8.2) � �Espace enclosed by the energy surfaceE. We then have that Γ(E) =Φ(E+Δ) Φ(E). − 85 86 CHAPTER 8. MICROCANONICAL ENSEMBLE

Taking withΔ 0 the limit of an inﬁnitesimal shell thickness we obtain → ∂Φ(E) Γ(E) = Δ Ω(E)Δ,Δ E. (8.3) ∂E ≡ �

Density of states. The quantityΩ(E),

∂Φ(E) Ω(E) = = d3N q d3N pδ(E H(q, p)) , (8.4) ∂E − � � is the density of states at energyE. The distribution of microstatesρ is therefore given in terms ofΩ(E), as

1 E < H(q, p)

8.2 Entropy

The expectation value of a classical observableO(q, p) can be obtained by averaging over the probability densityρ(q, p) of the microcanonical ensemble,

1 O = d3N q d3N pρ(q, p)O(q, p) = d3N q d3N p O(q, p) � � Γ(E,V,N) � � �Eobservable. It is instead a function of the overall number of available states.

POSTULATE The entropy is, according to Boltzmann, proportional to the logarithm of the number of available states, as measured by the phase space volumeΓ:

Γ(E,V,N) S=k ln (8.5) B Γ (N) � 0 �

We will now discuss the ramiﬁcation of this deﬁnition.

Normalization factor. The normalization constantΓ 0(N) introduced in (8.5) has two functionalities.

–Γ 0(N) cancels the dimensions ofΓ(E,V,N). The argument of the logarithm is consequently dimensionless. 8.2. ENTROPY 87

– The number of particlesN is one of the fundamental thermodynamic variables. The functional dependence ofΓ 0(N) onN is hence important.

Incompleteness of classical statistics. It is not possible to determineΓ 0(N) correctly within classical statistics. We will derive lateron thatΓ 0(N) takes the value

3N Γ0(N)=h N! , (8.6)

34 2 in quantum statistics, whereh=6.63 10 − m kg/s is the Planck constant. We consider this value also for classical statistics.· – The factorh 3N deﬁnes the reference measure in phase space which has the dimensions [q]3N [p]3N . Note that [h] = m (kg m/s) = [q][p]. –N! is the counting factor for states obtained by permuting particles.

The factorN! arises from the fact that particles are indistinguishable. Even though one may be in a temperature and density range where the motion of molecules can be treated to a very good approximation by classical mechanics, one cannot go so far as to disregard the essential indistinguishability of the molecules; one cannot observe and label individual atomic particles as though they were macroscopic billiard balls.

We will discuss in Sect. 8.2.1 the Gibbs paradox, which arises when on regards the con- stituent particles as distinguishable. In this case there would be no factorN! inΓ 0(N). The entropy as an expectation value. We rewrite the deﬁnition (8.5) of the entropy as

Γ (N) S= k ln 0 = k d3N q d3N pρ(q, p) ln Γ (N)ρ(q, p) , (8.7) − B Γ(E,V,N) − B 0 � � � � � � where we have used thatρ(q, p) = 1/Γ(E,V,N) within the energy shell and that

d3N q d3N pρ(q, p) = d3N q d3N p/Γ(E,V,N)=1. � � �E

(1) additivity;

(2) consistency with the deﬁnition of the temperature;

(3) consistency with the second law of thermodynamics;

(4) adiabatic invariance.

8.2.1 Additivity; Gibbs paradox

The classical HamiltonianH(q, p) =H kin(p) +H int(q) is the sum of the kinetic energy Hkin(q) and of the particle-particle interactionH int(q). The condition

E < H(q, p)

limiting the available phase space volumeΓ(E,V,N) on the energy shell, as defined by (8.2) could then be fulfilled by a range of combinations ofH kin(p) andH int(q). The law of large numbers, which we will discuss in Sect. 8.6, implies however that both the kinetic and the interaction energies take well defined values for large particle numbers N. Scaling of the available phase space. The interaction between particles involves only pairs of particles, with the remainingN 2 N particles moving freely within the available volumeV . This consideration suggest− ≈ together with an equivalent argument for the kinetic energy that the volume of the energy shell scales like

d3N q d3N p= Γ(E,V,N) V N γN (E/N, V/N) . (8.8) ∼ � �Egas. Extensive entropy. Using scaling relation (8.8) for the volume of the energy shell and 3N the assumption thatΓ 0(N)=h N! we thenfind that the entropy defined by ( 8.5) is extensive: Γ(E,V,N) V N γN (E/N, V/N) S=k ln =k ln B Γ (N) B h3N N! � 0 � � � V γ =k N ln + 1 k N s(E/N, V/N) (8.9) B N h3 ≡ B � � � � where we have used the Stirling formulaN! √2πN(N/e) N , namely that ln(N!) N(ln(N) 1). ≈ ≈ − Gibbs paradox. The extensivity of the entropy result in (8.9) from the fact that N N V /N! (V/N) . Without the factorN! inΓ 0(N), which is however not justifiable within≈ classical statistics, the entropy would not be extensive. This is the Gibbs paradox. Additivity in the case of identical thermodynamic states. Two systems have identical thermodynamic properties if they intensive variables are the same, viz temperature 8.2. ENTROPY 89

T , pressureP , particle densityN/V and energy density E/N. It then follows directly from (8.9) that

S(E,V,N)=k B (N1 +N 2)s(E/N, V/N)=S(E 1,V1,N1) +S(E 2,V2,N2),

where we have used thatN=N 1 +N 2.

Additivity for two systems in thermal contact. Two systems deﬁned byE 1,V 1, N1 and respectivelyE 2,V 2,N 2 in thermal contact may allow energy such that the total energyE=E 1 +E 2 is constant. For the argument of the entropy we have then:

Γ(E,V,N) Γ(E ,V ,N ) Γ(E E ,V ,N ) = 1 1 1 − 1 2 2 (8.10) Γ0(N) Γ0(N1) Γ0(N2) E �1 The law of large numbers tells us that the right-hand-side is sharply peaked at its maxi- mum valueE 1 =E max and that the width of the peak has a width scaling with √Emax. We hence have

S(Emax)

Γ(E ,V ,N )Γ(E E ,V ,N ) S(E ) =k ln max 1 1 − max 2 2 , max B Γ (N )Γ (N ) � 0 1 0 2 �

from which follows thatS(E,V,N) =S(E 1,V1,N1) +S(E 2,V2,N2). Note that the entropyS(E max) is extensive and that the term ln(E max) in (8.11) is hence negligible in the thermodynamic limitN . ∼ →∞

8.2.2 Consistency with the deﬁnition of the temperature

Two systems with entropiesS 1 =S(E 1,V1,N1) andS 2 =S(E 2,V2,N2) in thermal contact may exchange energy in the form of heat, with the total entropy,

∂S1 ∂S2 0 = dS= dE1 + dE1, dE1 = dE 2 , (8.12) ∂E1 ∂E2 − 90 CHAPTER 8. MICROCANONICAL ENSEMBLE

becoming stationary at equilibrium. Note that the total energyE 1 +E 2 is constant. The equilibrium condition (8.12) implies that there exists a quantityT , denoted temperature, such that ∂S 1 ∂S 1 = = 2 . (8.13) ∂E1 T ∂E2

The possibility to deﬁne the temperature, as above, is hence a direct consequence of the conservation of the total energy. From the microcanonical deﬁnition of the entropy one only needs that the entropy is a function only of the internal energy, via the volume Γ(E,V,N) of the energy shell, and not of the underlying microscopic equation of motion.

8.2.3 Consistency with the second law of thermodynamics We would like to confirm here that the statistical entropy, defined by (8.5) fulfills the second law of thermodynamics

“If an isolated system undergoes a process between two states at equilibrium, the entropy of theﬁnal state cannot be smaller than that of the initial state.”

Free expansion. Both the energyE and and the number of particle stays constant during a free expansion, de- ﬁned by the absence of external heat transfer, ΔQ=0. The volumeΓ(E,V,N) of the energy shell increase with increasing volumeV . We hence have Γ(E,V,N) S(E,V ,N)>S(E,V ,N) ,S=k ln ,V > V , 2 1 B Γ (N) 2 1 � 0 � where have made use of the fact that the normalization factorΓ 0(N) remains constant. Dynamical constraints. Dynamical constraints (viz bouncing from the wall) are miti- gated when the volume is increased.

EQUIVALENCE The second law is equivalent to saying that the entropy rises when dynamical constraints are eliminated,

8.2.4 How thick is the energy shell? The definition Sect. 8.5 of the entropy involves the volumeΓ(E,V,N) of a shell of state space of widthΔ centered around the energyE. It seems therefore that the entropy S=S Δ(E,V,N) depends on an unspecified parameterΔ. Is the entropy then not uniquely specified? 8.3. CALCULATING WITH THE MICROCANONICAL ENSEMBLE 91

Reference energy. For smallΔ we may use the approximation ∂Φ(E) Γ(E,V,N) Ω(E)Δ,Ω(E) = , (8.14) ≈ ∂E whereΩ(E) is the density of states, as deﬁned previously in ( 8.3). In order to decide whether a givenΔ is small or large we need a reference energyΔ 0. One may take f.i. Δ0 k BT , which corresponds, as shown in Sect. 3.4.1, in order of magnitude to the thermal∼ energy of an individual particle. Thermodynamic limit. The entropy involves the logarithm ofΓ(E,V,N), Γ(E,V,N) Ω(E)ΔΔ Ω(E)Δ ln = ln 0 = ln 0 + ln(Δ/Δ ), Γ Γ Δ Γ 0 � 0 � � 0 0 � � 0 � N ∝ where we have taken care the arguments of the logarithms� �� are� dimensionless. The key insight resulting from this representation is that the exact value of bothΔ andΔ 0 is irrelevant in the thermodynamic limitN as long as →∞ ln(Δ/Δ ) N. 0 � Energy quantisation will ensure this� condition� in quantum statistics. � � Shell vs. sphere. We may also considering the limit of largeΔ to the extend that we may substitute the volumeΦ(E) for the the volumeΓ(E,V,N) of the energy shell, ln Γ(E,V,N) ln Φ(E) , (8.15) ≈ where we have disregarded the� normalization� factor� Γ 0.� The reason that (8.15) holds stems from the fact that the volume and surface of a sphere with radiusR of dimension 3N 3N 1 3N, compare Sect. 8.3.1, scale respectively likeR andR − . This scaling leads to ln Φ(E) ln R3N = 3N ln(R) ∼ 3N 1 ln Γ(E,V,N) ln R − Δ = (3N 1) ln(R) + ln(Δ) , � � ∼ � � − 3N ln(R) � � � � ∼ which corroborates (8.15). We did not perform here� an analysis�� of the� units involved, neglected in particular the reference energyΔ 0.

8.3 Calculating with the microcanonical ensemble

In order to perform calculations in statistical physics one proceeds through the following steps. 1) Formulation of the Hamilton function

H(q, p) =H(q 1, . . . , q3N , p1, . . . , p3N , z), wherez is some external parameter, e.g., volumeV.H(q, p) speciﬁes the microscopic interactions. 92 CHAPTER 8. MICROCANONICAL ENSEMBLE

2) Determination of the phase spaceΓ(E,V,N) and calculation of the density of states Ω(E,V,N): Ω(E,V,N)= d3N q d3N pδ(E H(q, p)). − � � 3) Calculation of the entropy from the volumeΦ(E) of the energy sphere:

Φ(E) S(E,V,N)=k ln . B Γ � 0 � 4) Calculation of P, T,µ:

1 ∂S µ ∂S P ∂S = , = , = . T ∂E − T ∂N T ∂V � �V,N � �E,V � �E,N

5) Calculation of the internal energy:

U= H =E(S,V,N). � � 6) Calculation of other thermodynamic potentials and their derivatives by application of the Legendre transformation:

F(T,V,N) =U TS, − H(S,P,N) =U+PV, G(T,P,N) =U+PV TS. −

7) One can calculate other quantities than the thermodynamic potentials, for instance, probability distribution functions of certain properties of the system, e.g., momenta/velocity distribution functions. If the phase space density of a system of N particles is given by

ρ(q, p) =ρ(�q1,...,�qN ,�p1,...,�pN ),

then the probability ofﬁnding particlei with momentum�p is

ρ (�p) = δ(�p �p ) i � − i � = d3q . . . d3q d3p . . . d3p ρ(�q,...,�q ,�p, ..,�p, ..,�p )δ(�p �p). 1 N 1 N 1 N 1 i N − i � � 8.3.1 Hyperspheres Let us calculate for later purposes the volume

n n Ωn(R) = d x=R Ωn(1) (8.16) n x2

of a hypersphere ofn dimensions and radiusP.

Spherical coordinates. We notice that the volumeΩ n(1) of the sphere with unity radius enters the determinant of the Jacobian when transforming via

n n 1 d x= dx 1 . . . dxn =Ω n(1) nR − dR (8.17)

euclidean to spherical coordinates. This transformation is valid if the integrand depends exclusively on the radiusR. Gaussian integrals. In order to evaluate (8.16) we make use of the fact that we can rewrite the Gaussian integral

+ + + ∞ x2 ∞ ∞ (x2+...+x2 ) n/2 dx e− = √π, dx1 ... dxn e− 1 N =π . �−∞ �−∞ �−∞ as

n/2 ∞ R2 n 1 ∞ y (n 1)/2 dy π = e− Ωn(1) nR − dR=nΩ n(1) e− y − 2√y �0 �0 n ∞ y n 1 = Ω (1) e− y 2 − dy , (8.18) 2 n �0 where we have used (8.17), x2 =R 2 y and that 2RdR= dy. i i ≡ Gamma function. With the� deﬁnition

∞ z 1 x Γ(z) = dx x − e− , �0 of theΓ-function we then obtain from ( 8.18) that

n πn/2 πn/2 = Ω (1)Γ(n/2),Ω (1) = . (8.19) 2 n n (n/2)Γ(n/2)

Note that we did evaluate the volume of a hypersphere for formally dimensionless variables xi. For later purposes we will need that

Γ(N)=(N 1)Γ(N 1),Γ(N)=(N 1)! (8.20) − − − whenever the argument of theΓ-function is an integer.

8.4 The classical ideal gas

We consider now the steps given in Sect. 8.3 in order to analyze an ideal gas ofN particles in a volumeV , deﬁned by the Hamilton function

N �p2 H(q, p) = i , 2m i=1 � 94 CHAPTER 8. MICROCANONICAL ENSEMBLE

wherem is the mass of the particles.

Phase space volume. We will make use of (8.15), namely that the volumeΓ(E,V,N) of the energy sphere E < H < E+Δ.

can be replaced by the volume of the energy shell,

Φ(E) = d3N q d3N p=V N d3N p , (8.21) 3N p2 2mE 3N p2 2mE � � i=1 i ≤ � i=1 i ≤ � � Ω3N (√2mE) � �� when it comes to calculating the entropy in the thermodynamic limit. We have identiﬁed the last integral in (8.21) as the volume of a 3N-dimensional sphere with radius √2mE.

Using (8.16) and (8.19),

π3N/2 Ω (√2mE) = (2mE) 3N/2 Ω (1),Ω (1) = , 3N 3N 3N (3N/2)Γ(3N/2)

we obtain

3N N Φ(E) =V Ω3N (1) √2mE . (8.22) � �

8.4.1 Entropy

Using (8.22) weﬁnd

3N N √ Φ(E) V 2mE Ω3N (1) S(E,V,N)=k ln = k ln (8.23) B h3N N! B  � h3N �N!  � �     for the entropy of a classical gas. It is easy to check, that the argument of the logarithm is dimensionless as it should be.

LargeN expansion. For N >> 1, one may use the Stirling formula,N! √2πN(N/e) N , to expand theΓ-function for integer argument as ≈

ln Γ(N) = ln (N 1)! − (N 1) ln(N 1) (N 1) � � ≈ −� −� − − N ln(N) N, ≈ − 8.4. THE CLASSICAL IDEAL GAS 95

in order to simplify the expression forS(E,V,N). Using ( 8.19) we perform the following algebraic transformations toΩ 3N (1):

3N π 2 lnΩ 3N (1) = ln 3N 3N � 2 Γ 2 � 3N 3N 3N 3N = lnπ � � ln 2 − 2 2 − 2 � � � � 3N 2π 3N = ln + 2 3N 2 2π 3/2 3 lnN =N ln + + . 3N 2 O N � � � � �� We insert this expression in Eq. (8.23) and obtain:

V (2mE)3/2 2π 3/2 3 S=k N ln + ln + (lnN 1) . (8.24) B  h3 3N 2 − −   � � � � lnN!   N   � ��  Sackur-Tetrode Equation. Rewriting (8.24) as

4πmE 3/2 V 5 S=k N ln + (8.25) B 3h2N N 2 � ��

we obtain the Sackur-Tetrode equation. Equations of state. Now we can diﬀerentiate the Sackur-Tetrode equation (8.25) and thus obtain

- the caloric equation of state (3.5) for the ideal gas:

1 ∂S 3 1 3 = =Nk , E= Nk T=U T ∂E B 2 E 2 B � �V,N

- the thermal equation of state for the ideal gas:

P ∂S k N = = B , PV=Nk T . T ∂V V B � �E,T

Gibbs Paradox. If we hadn’t considered the factorN! when working out the entropy, then one would obtain that

4πmE 3/2 3 S =k N ln V + . classical B 3h2N 2 � �� 96 CHAPTER 8. MICROCANONICAL ENSEMBLE

With this deﬁnition, the entropy is non-additive, i.e., E V S(E,V,N)=Ns , N N � � is not fulﬁlled, as mentioned already in Sect. 8.2.1. This was realized by Gibbs, who introduced the factorN! and attributed it to the fact that the particles are indistinguishable.

8.5 Fluctuations and correlation functions

We use, as deﬁned in Sect. 7.2.3, the probability distribution functionρ M (q, p) in phase space, which could be evaluated as a time average

1 M ρ(q, p) = lim δ(3N) q q (l) δ(3N) p p (l) M M − − →∞ l=1 � � � � � ofM measurements of the states (q (l), p(l)) visited along a given trajectory. We have pointed out in (8.1) thatρ(q, p) is uniformly distributed on the energy shell for a microcanonical ensemble. Observables. A classical observableB(q, p) corresponds to a function

B(q, p) =B(q 1, . . . , q3N , p1, . . . , p3N ), phase state. We start with the basic properties. Expectation value. The average value ofB(q, p) is

1 M B(q, p) = lim B q(l), p(l) � � M M →∞ l=1 � � � = d3N q d3N pρ(q, p)B(q, p). �Γ Variance. The Variance of a function on phase space is (ΔB)2 = (B B ) 2 = B2 B 2 =σ 2 . −� � − � � B The variance provides information� about the� fluctuation� � effects around the average value. Higher momenta.A momentum of ordern is defined generically as µ = B n . n � � Probability distribution of an observable. We may define with

ρ(b) = d3N q d3N pρ(q, p)δ(b B(q, p)) = δ(b B(q, p) (8.26) − � − � � also the probability distributionρ(b) of the observableB. The expectation value ofB is then given as B = db bρ(b). � � � 8.5. FLUCTUATIONS AND CORRELATION FUNCTIONS 97

8.5.1 Characteristic function and cumulants The characteristic function of an observable

izB 3N 3N izB(q,p) ϕB(z) = e = d q d pρ(q, p)e . (8.27) � � � is both a function of the dummy variable∗ z and a functional † of the observableB= B(q, p). Generating functional. The characteristic function of a random variableB serves as a generating functional, as it allows via suitable diﬀerentions (with respect to the dummy variablez),

1 dn ∞ (iz)n µn = ϕB(z) , ϕB(z) = µn , (8.28) in dzn z=0 n! � n=0 � � to recover the momentsµ n of underlying� probability distribution.

Cummulants. In (8.27) we did expandϕ B(z) in powers ofz. Alternatively one may expand the logarithm ofϕ B(z) in powers ofz. One obtains (here without proof)

∞ (iz)n z2 lnϕ (z) = ζ =µ iz σ 2 +.... (8.29) B n n! 1 − 2 n=1 � The coeﬃcients ζ =µ = B 1 1 � � ζ =σ 2 = B 2 B 2 2 � � − � � ζ3 =... are called cumulants. Extensivity. Cummulant expansions like (8.29) are found in many settings, they form, e.g., the basis of high-temperature expansions. The reason is that cumulants are (in contrast to the moments) consistent with extensivity requirements, f.i. with respect to the volumeV:

B V, B 2 V 2, ζ = B 2 B 2 N . � � ∼ � � ∼ 2 � � − � � ∼ All cumulants are size-extensive.

8.5.2 Correlations between observables We deﬁne in analogy to (8.26) the joint distribution functionρ(a, b), ρ(a, b) = δ(a A)δ(b B) (8.30) − − = � d3N q d3N pρ(q, p)�δ a A(q, p) δ b B(q, p) − − � � � � � ∗ A dummy variable doesn’t have a physical meaning. It solely serves for computational purposes. † A functional is a function of another function. 98 CHAPTER 8. MICROCANONICAL ENSEMBLE

of two observablesA=A(q, p) andB=B(q, p). Cross correlation function. The joint distribution functionρ(a, b) allows to investigate the cross correlations C = (A )(B ) (8.31) AB − − = � da dbρ(a, b) a A � b B −� � −� � � � � �� betweenA andB. Uncorrelated variables. Two observable are uncorrelated if they are statistically independent. The joint distribution factorizes is this case,

ρ(a, b) =ρ A(a)ρB(b). The cross correlations vanish consequently: C = A A B B AB −� � −� � = A A B B �� −� � �� −� � �� =� 0. � � �

8.6 Central limit theorem

ConsiderM statistically independent observablesB m(q, p). The central limit theorem states then that the probability density of the observable

B(q, p) = Bm(q, p) m=1 � is normal distributed with an extensive mean B M and with the width proportional to √M. � �∼

8.6.1 Normal distribution An observableY is said to be normal distributed if the probability distribution function ρ(y)= δ(y Y) has the form of a Gauss curve, � − �

2 2 1 (y y0) /(2σ ) ρ(y) = e− − (8.32) √2πσ2 The mean and the variance are + ∞ Y = dyρ(y)y=y , � � 0 �−∞ and + ∞ (Y ) 2 = dyρ(y)(y y )2 =σ 2 − − 0 �−∞ � � 8.6. CENTRAL LIMIT THEOREM 99

respectively. The even moments of the normal distribution arefinite, (2n)! Y 2n = σ2n , 2nn! with the odd moments Y 2n+1 = 0� vanishing� due to the reflection symmetry with respect � � to the meany 0. Characteristic function. The characteristic function of a normal distribution is also a Gauss function: + ∞ 1 2 2 izY izy izy0 z σ f(z) = e = dy e ρ(y) =e e− 2 . (8.33) �−∞ � � For the proof consider the overall exponent (y y )2 y2 2y y 2izyσ 2 +y 2 izy − 0 = − 0 − 0 − 2σ2 − 2σ2 (y y izσ2)2 1 = − 0 − + izy z2σ2 . − 2σ2 0 − 2 8.6.2 Derivation of the central limit theorem For the derivation of the central limit we define a variableY as 1 Y X1 +X 2 +...+X M , ≡ √M � � whereX 1,X 2,...,X M areM mutually independent variables such that X i = 0 for convenience. � � Characteristic function of independent variables. We make now use of the fact, that the characteristic function of the sum independent variables, such as forY , factorizes:

izY iz(X1+X2+...+X )/√M e− = e− M � � � M � izX /√M = e− j �j=1 � � M izX /√M = e− j . j=1 � � � Using the representation izX /√M A (z/√M) ln e− j (8.34) j ≡ � � for the logarithm of the characteristic function exp( izX j/√M) of the individual random variables we thenﬁnd � − �

M izY e− = exp Aj(z/√M) . (8.35) � j=1 � � � � 100 CHAPTER 8. MICROCANONICAL ENSEMBLE

LargeM expansion. IfM is a large number,A j(z/√M) can be expanded as

2 3 z z 1/2 A (z/√M) A (0) + A��(0) + (M − ) . (8.36) j � j 2M j M O