Download 5 Statistical Fluid Dynamics

5
Statistical Fluid Dynamics
In the previous chapter, we considered fluid turbulence from a heuristic viewpoint,
combining ideas based upon fundamental conservation laws with more qualitative ideas
about irreversible behavior. However, there have been numerous attempts to study
turbulence in a more quantitative and deductive fashion, beginning with the governing
dynamical equations for the fluid. None of these attempts is, or could be, mathematically
rigorous; all involve approximations that are difficult to justify and hard to test. In fact,
these more deductive theories are perhaps best viewed as approximations in which the
exact dynamical equations are replaced by stochastic-model equations having some (but
not all) of the same physical properties as the exact equations, in much the same way as
the quasigeostrophic equations contain only a part of the physics in the more exact
primitive equations.
In this chapter, we examine fluid turbulence from the standpoint of equilibrium and
nonequilibrium statistical mechanics. This is a complicated and controversial subject,
and our discussion will be introductory and elementary. In fact, we shall be much more
concerned with the philosophy behind the statistical methods than with the detailed
structure of the theory or with its successes and failures at describing real turbulence.
This philosophy appears to be applicable to a much wider range of problems than so far
considered. Readers who want a more thorough and complete description of statistical
turbulence theory should consult more specialized sources.1
This chapter continues a theme, begun in Chapter 4, to be continued in Chapter 6, that
much of our understanding of turbulence is based upon two general principles: the
conservation principle, that quantities like energy and potential vorticity are conserved
(apart from the effects of dissipation), and the irreversibility principle, that a turbulent
system tends toward ever greater complexity. In Chapter 4, we met the irreversibility
principle in the assumptions that a narrow spectral peak spreads out, and that nearby fluid
particles tend to move apart. In this chapter, we encounter the irreversibility principle
again, as a kind of macroscopic form of the Second Law.
1. The closure problem of turbulence
First we re-state the fundamental closure problem of turbulence by means of an
example. Consider a laboratory experiment in which water flows down a long, smooth,
cylindrical pipe. An experimenter controls the average pressure gradient along the pipe.
When this pressure gradient is sufficiently large, the flow is turbulent. As explained in
Chapter 4, the details of this turbulent flow are erratic and unpredictable, but the average
cross-sectional velocity is well described by a smooth, reproducible curve that can easily
be fit with simple analytic functions. The fundamental question is this: Can this simple,
smooth curve, obtained by averaging an enormous number of laboratory (or now perhaps
numerical) data, instead be calculated directly from the governing Navier-Stokes
equations, with no a posteriori averaging at all?
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The precise answer to this question is probably no. However, it might well be
possible to get a very good approximation to the average velocity curve with much less
work than that required to take and average all the measurements. Even more
importantly, a successful approximation method would almost certainly illuminate the
physics of turbulent flow. These are the hopes on which turbulence closure theory rests.
An obvious way to begin is by forming the equations for the averages themselves.
Readers of the previous chapter will readily accept that the governing equations of fluid
dynamics can always be written in the abstract form
dyi
= ∑ Aijk y j yk − ν i yi ,
dt
j ,k
(1.1)
where {yi} is a set of real numbers that defines the state of the whole fluid. For example,
the vorticity equation
∂∇ 2ψ
+ J (ψ ,∇ 2ψ ) = ν∇4ψ
∂t
(1.2)
governs two-dimensional Navier-Stokes flow, where ψ(x,y,t) is the streamfunction for the
flow, and
J ( A, B) ≡
∂ ( A,B )
∂ ( x,y )
(1.3)
is the horizontal Jacobian operator. If the flow is unbounded and 2π-periodic, then the
Fourier transform of (1.2) is
dψ k
= ∑ Ak,p,qψ pψ q − ν kψ k ,
dt
p, q
(1.4)
where ψk(t) is the Fourier transform of ψ(x,t),
ψ (x,t ) = ∑ψ k (t )eik ⋅x .
(1.5)
k
Here, x=(x,y), k=(kx,ky), νk=νk2, k=|k|, and the interaction coefficients are given by
Ak,p,q =
1
2
(p × q)( q2 − p2 )k −2 δp +q = k ,
(1.6)
where p×q≡pxqy-pyqx, and δ denotes the Kronecker delta. Eqn. (1.4) fits the form of
(1.1) with i≡(α,k), and α=1(2) denoting the real (imaginary) part of ψk. In this case, the
indices in (1.1) are triplets.
As a second example, we consider two-dimensional flow bounded by a smooth,
simply-connected curve in the (x,y) plane, and expand
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ψ ( x, y,t ) = ∑ yi (t ) ki −1φi ( x, y )
(1.7)
i
in the orthonormal eigenfunctions φi(x,y) defined by
∇ 2 φi ( x, y ) = −k i2 φi ( x,y ) ,
(1.8)
the boundary condition φi=0, and the normalization,
∫∫ dx φ φ
i
j
= δij .
(1.9)
Now ki2 is the eigenvalue corresponding to eigenfunction φi(x). The ki-1-factor in the
definition (1.7) of yi(t) is a matter of later convenience. Substituting (1.7) into (1.2), and
using (1.8) and (1.9), we obtain an equation of the form (1.1) with coupling coefficients
Aijm
(k
=
2
j
− km 2 )
2kik jk m
∫∫ dx φ J (φ ,φ ) .
i
j
(1.10)
m
In both of these examples, the set
{y1,y2,y3,......}
(1.11)
is countably infinite. If, on the other hand, the flow were spatially unbounded (and
nonperiodic), we would replace (1.5) by
ψ (x,t ) = ∫∫ dk ψ k (t )eik ⋅x ,
(1.12)
and obtain an equation like (1.4) in which integrals replace the sums over discrete
indices. Then k varies continuously, and the set analogous to (1.11) is uncountably
infinite. However, there is little practical difference between these countably and
uncountably infinite sets. Even in the case of a bounded or periodic flow, the continuousk representation (1.12) is a frequently useful idealization, because the higher
wavenumbers differ by relatively small increments. This agrees with the physical notion
that small-scale eddies are unaffected by distant boundaries.
On the other hand, it will be very important to distinguish between systems with
infinitely many degrees of freedom and systems in which the number N of dependent
variables
{y1,y2,y3,....,yN}
(1.13)
is finite. The latter frequently arise as numerical approximations to the differential
equations governing fluid motion. For example, (1.4) with the sums truncated to exclude
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all the wavenumbers larger than a prescribed cutoff kc is a respectable numerical model
of (1.2). Other examples include finite-difference approximations to (1.2), in which the
yi(t) are the values of ψ (x,t) at the N gridpoints.
Whether N is infinite or not, the dynamics (1.1) lead to a unclosed hierarchy of
equations for the averages. The average of (1.1) yields N dynamical equations,
d
y = ∑ Aijk y j yk − νi yi ,
dt i
j, k
(1.14)
for the N first moments <yi> in which the second moments <yjyk> also occur. Similarly,
the equation,
(
)
d
y y = ∑ Aikm y j yk ym + A jkm yi yk ym − (ν i + ν j ) yi y j ,
dt i j
k ,m
(1.15)
for the second moments also contains the third moments <yjykym>, and so on. This
hierarchy is mathematically unclosed, because the equations for the n-th moments always
contain the (n+1)-th moments.
Many theorists regard the difficulties associated with this unclosed hierarchy of
equations as being equivalent to the general closure problem of turbulence. While this
viewpoint may be too narrow, it provides a useful departure point from which to explore
general statistical ideas that have been applied to turbulence. In any case, we shall see
that the difficulties associated with the infinite hierarchy of equations are not
mathematical difficulties in the technical sense, but inevitably involve additional physical
hypotheses which at least partly amount to a refinement in the meaning of averaging.
2. The eddy-damped Markovian model
Let the equations of fluid motion be
dyi
= ∑ Aijk y j yk − ν i yi ,
dt
j ,k
(2.1)
with initial conditions
yi (0) = yi0 .
(2.2)
With no loss in generality, we assume that the interaction coefficients are symmetric in
their last two indices,
Aijk = Aikj .
(2.3)
We also assume that Aijk vanishes whenever two of its indices are equal,
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Aijj = A jij = A jji = 0 .
(2.4)
The interaction coefficients (1.6) and (1.10) both have this property. In fact, (2.4) is a
rather general property of fluid dynamical equations.2 Finally, we assume that
Aijk + A jki + Akij = 0 ,
(2.5)
so that when ν=0, the dynamics (2.1) conserve the energy
E ≡ ∑ yi 2 .
(2.6)
i
For example, the energy,
∫∫ dx ∇ψ ⋅∇ψ ,
(2.7)
of two-dimensional flow takes the form (2.6) when the yi are defined by (1.7-9). The
interaction coefficients (1.10) obey (2.5), and hence (1.2) conserves (2.7) when ν=0.
The eddy-damped quasi-normal Markovian model (hereafter EDM) is an
approximation to the moment hierarchy (1.14-15, etc.) arising from (2.1) that closes the
hierarchy at the level of the second moments. We shall study EDM as a representative
theory of nonequilibrium statistical mechanics.3
The EDM equations take a relatively simple form if the the turbulence is statistically
homogeneous, that is, if the statistics do not depend on location in the flow. If the
turbulence is homogeneous, then <ψ(x)> is a constant (conveniently assumed to be
zero), and <ψ(x)ψ(x')> depends only on x-x'. For definiteness, we now regard (2.1) as
the Fourier transformation of the equation (1.2) governing 2π-periodic, two-dimensional,
homogeneous turbulence. The yi(t) are the coefficients of the functions
sin(nx + my)
and
cos(nx + my)
(2.8)
in the expansion of ψ(x,y,t), and the subscript i denotes (m,n) and whether cosine or sine.
Since the turbulence is homogeneous, it follows that <yi(t)>=0 and
yi (t ) y j (t ) = Yi (t )δij .
(2.9)
Hence the second-moment equations (1.15) take the form,
⎛d
⎞
⎜ + 2ν i ⎟ Yi (t ) = 2∑ Aijk yi y j yk .
⎝ dt
⎠
j, k
(2.10)
For the third moments, we obtain
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⎛⎜ d + ν + ν + ν ⎞⎟ y y y
i
j
k
⎝ dt
⎠ i j k
{
= ∑ Aimn ym yn y j y k + A jmn ym y n yi y k + Akmn y m yn yi y j
m ,n
}
(2.11)
EDM closes (2.10-11) by treating the {yi} on the right-hand side of (2.11) as
Gaussian random variables. If A, B, C, and D are Gaussian random variables with zero
means, then4
ABC = 0
(2.12)
ABCD = AB CD + AC BD + AD BC .
(2.13)
and
Factoring the right-hand side of (2.11) in the same way as (2.13), making use of (2.9),
and assuming (for simplicity) that the triple moments <yiyjyk> all vanish at the initial
time t=0 (as if the initial conditions were completely random), we solve (2.11) and
substitute the result into (2.10). The result is a closed equation,
t
−( t − s ) (ν i +ν j +ν k )
⎛⎜ d
⎞
+ 2ν i ⎟ Yi (t ) = ∑ ∫0ds e
{4Aijk 2Yj (s)Yk ( s) + 8Aijk A jikYk ( s)Yi (s)},
⎝ dt
⎠
j,k
(2.14)
for the covariances {Yi(t)}.
Note that the assumptions leading to (2.14) have been applied somewhat
inconsistently. If the {yi} were really Gaussian random variables, then according to
(2.12), the right-hand side of (2.10) would vanish at all times, just as if the nonlinear
terms in the original dynamical equations (2.1) had been entirely absent. Thus it is
critical that the Gaussianity-assumption is applied only to the higher-order equations
(2.11).
Unfortunately, numerical integrations of (2.14) lead to large negative Yi when the
Reynolds number is very large. Since Yi=<yi2> must be positive, this is clearly an
unreasonable result. However, we can understand this calamity, and use the
understanding to repair (2.14).
To understand why (2.14) predicts negative Yi, suppose that the initial conditions
{Yi(0)} correspond to a sharp peak in the wavenumber spectrum at t=0 (Figure 5.1). At
small t>0, we expect the right-hand side of (2.14) to become large and negative for Yk
inside the peak, and positive for Yk outside the peak. At later times, the curly bracket in
the integrand of (2.14) should decrease in magnitude. However, since the timeintegration in (2.14) runs all the way back to t=0, the large initial tendency for the
spectral peak to decrease is never completely forgotten. The persistence of this large
initial tendency is especially strong in the limit of very large Reynolds number
(corresponding to small viscosity, ν i , ν j ,ν k → 0 ) because the values of the curly bracket
in the distant past (s<<t) then weigh equally with those of the near past (s≈t) in the
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integrand of (2.14). In the limit of very large Reynolds number, the large initial
tendencies are not forgotten, and consequently the spectral peak plunges right through
zero.
Clearly, this sling-shot behavior is caused by the unrealistically long memory in
(2.14) as the {νi}→0. However, in real turbulent flows, past states are forgotten not
because of the molecular viscosity, but because of the nonlinear-interaction terms in the
equations of motion. The fluid forgets its past over roughly the same interval as future
predictions become unreliable, and for much the same reason. At large Reynolds
number, this memory time should be independent of the molecular viscosity, in
contradiction with (2.14).
These thoughts suggest that we can repair (2.14) by replacing the molecular
viscosities νi, νj, νk in (2.11) with much larger values µi, µj, µk that parameterize the
rapid memory loss of past states at high Reynolds number. These µi differ from the usual
eddy viscosity coefficients in that they then enter (2.11) but not (2.10) and thus do not
directly cause energy to decrease. The resulting covariance equation,
t
−( t − s ) ( µi + µ j +µ k )
⎛⎜ d
⎞
+ 2ν i ⎟ Yi (t ) = ∑ ∫0ds e
{4 Aijk 2 Yj ( s)Yk ( s) + 8Aijk A jikYk ( s)Yi (s)},
⎝ dt
⎠
j,k
(2.15)
differs from (2.14) only in that µi replace the νi on the right-hand side.
Once again, the µi parameterize the decrease in the triple correlations <yiyjyk> caused
by the scrambling effect of the flow on itself. Hence the {µi} ought to increase with the
Reynolds number. That is, the more turbulent the flow, the more rapidly triple
correlations are destroyed. Therefore, at very large Reynolds number (large µi), most of
the contribution to the integral in (2.15) should come from s near t. This in turn suggests
that, at large Reynolds number, we can replace the curly-bracket term in (2.15) with its
value at s=t. The result is
⎛⎜ d
⎞
+ 2ν i ⎟ Yi (t ) = ∑θ ijk{4Aijk 2 Yj (t )Yk (t ) + 8Aijk Ajik Yk (t )Yi (t )}
⎝ dt
⎠
j,k
(2.16)
where
t
θ ijk ≡ ∫0ds e
− (t − s ) ( µi + µ j + µk )
.
(2.17)
Eqn. (2.16) is the eddy-damped quasi-normal Markovian approximation (EDM). The
words eddy-damped refer to the introduction of the µi. Markovian refers to the
evaluation of the curly-bracket term in (2.15) at s=t. Normal refers to the use of (2.13).
The approximation is termed quasi-normal because we have applied the Gaussian
assumption inconsistently. Again, if the {yi} were indeed Gaussian with zero means,
then <yiyjyk> would vanish in (2.10).
It is easy to see that EDM does not allow the negative-Yi calamity. For, if Yi=0 for
some particular i, with all other {Yj} positive, then, since the right-hand side of (2.16) is
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positive, Yi must be increasing. Using (2.5), one can show that when νi=0 EDM
conserves the average energy,
E = ∑ Yi .
(2.18)
i
We come back to this at length.
We can regard µi-1 as the time interval over which yi remains correlated with itself,
and θijk as the average persistence time for the product yiyjyk. According to (2.17), at
short times θijk increases linearly with time, as the triple correlations build up from their
assumed zero initial value. At long times,
θ ijk ~
1
µi + µ j + µk
(t → ∞ ) ,
(2.19)
assuming that the µi are constants or change slowly (on a time scale much longer than
µi-1).
Our derivation of EDM leaves the µi unspecified. However, by the reasoning of
Chapter 4, the time scale Tk for the distortion of an eddy of size k -1 is given by
k
Tk −2 ~ ∫0 k ′2 E( k ′)dk ′ ,
(2.20)
where E(k) is the energy spectrum. If we suppose that
µk ∝ Tk −1 ,
(2.21)
then the EDM equations corresponding to three-dimensional Navier-Stokes turbulence
are consistent with Kolmogorov’s theory and predict an inertial-range spectrum E(k)~k 5/3.5
Readers familiar resonant wave-interaction theory will notice a strong
resemblance between EDM and the equations for weakly interacting waves. Suppose we
generalize (2.1) to
dyi
+ iω i yi = ∑ Aijk y j yk − ν i y i ,
dt
j ,k
(2.22)
so that, in the limit of infinitesimal {yi}, the solutions are linear waves. Here, ωi is the
(constant) wave frequency associated with mode yi, and i (where not a subscript) is the
square root of -1. An equation of the form (2.22) arises from the generalization,
∂∇ 2ψ
∂ψ
+β
+ J (ψ ,∇ 2ψ ) = ν∇4ψ ,
∂t
∂x
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(2.23)
of (1.2) to include Rossby waves.6 By the same steps as before, we again obtain (2.16),
but with
θ ijk ~
(µ
µi + µ j + µ k
+ µ j + µ k ) + (ω i + ω j + ω k )
2
i
2
,
t→∞
(2.24)
instead of (2.19). In the limit of strong turbulence (µ>>ω), (2.24) reduces to (2.19).
However, in the limit of weak turbulence (µ<<ω), (2.24) reduces to
θ ijk ~ π δ (ω i + ω j + ω k ) ,
(2.25)
in agreement with resonant wave-interaction theory. In this limit, only triads (yi,yj,yk)
whose three frequencies sum to zero can interact. However, (2.24) shows that even weak
nonlinearity relaxes this criterion, allowing slightly off-resonant triads to transfer energy
between modes. In fact, it seems very doubtful that one is ever justified in setting {µi}≡0
and eliminating the off-resonant interactions completely.
3. Stochastic model representation
Thus far we have regarded EDM as an approximation to the moment hierarchy
corresponding to the exact dynamics (2.1). Now we take a different viewpoint. We show
that EDM is the exact result of averaging stochastic differential equations that resemble
the exact dynamics. From this new viewpoint, the EDM approximation consists of
replacing the exact dynamical equations with stochastic-model equations that are easier
to analyze.7
Again we consider homogeneous turbulence governed by
dyi
+ νi yi = ∑ Aijk y j yk ,
dt
j ,k
(3.1)
with initial conditions yi0. Again the interaction coefficients obey (2.3-5). But now we
consider the stochastic model
dyi
+ νi yi = W (t )∑θ ijk 1/ 2 Aijk y Gj ykG ,
dt
j, k
(3.2)
in which Gaussian random variables yiG replace the yi on the right-hand side of (3.1).
These Gaussian random variables have the same (zero) means and covariances
yiG (t ) y Gj (t ) = yi (t ) y j (t ) ≡ Yi (t )δij
(3.3)
as the {yj}. W(t) is a white-noise random variable, independent of the {yiG}, with zero
mean and covariance
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W (t )W (t' ) = 2 δ (t − t' ) .
(3.4)
The presence of W(t) corresponds to the Markovianization step in the previous derivation
of EDM. The θijk are positive nonrandom numbers with the dimensions of time, fully
symmetric in their three subscripts.
Unlike (3.1), (3.2) is a linear equation, with solution
yi (t ) = yi e
0 − νi t
t
+ ∫ ds e
−ν i ( t − s )
−∞
W (s)∑ θijk
1/ 2
G
G
Aijk y j (s) yk (s) .
(3.5)
j ,k
From (3.5) and (3.2), and assuming (for simplicity) that yiG and yi0 are independent
random variables, we obtain
t
dYi
dy
= 2 yi i = −2νiYi + 2 ∫ ds e −ν i ( t −s ) W (s)W (t ) ×
dt
dt
−∞
×∑
j ,k
∑θ
θ
1/ 2
ijk
1/ 2
imn
Aijk Aimn y (s) y (s) y (t )y (t )
G
j
G
k
G
m
(3.6)
G
n
m,n
Then, using (2.3-5) and the property (2.13) of Gaussian random variables, we obtain the
covariance equation,
⎛⎜ d
⎞
+ 2ν i ⎟ Yi (t ) = ∑θ ijk{4Aijk 2 Yj (t )Yk (t )},
⎝ dt
⎠
j,k
(3.7)
for the stochastic model (3.2).
Unfortunately, (3.7) does not conserve the average energy (2.18) in the limit of
vanishing molecular viscosity νi. In fact, when νi=0, (2.18) always increases, because
the right-hand side of (3.7) is positive. The random driving terms on the right-hand side
of (3.2) cause yi2 to increase, on average, for the same reason that a particle subject to a
random forcing wanders steadily away, on average, from its initial location.
To counteract the tendency of the { y Gj } to increase the model energy, we add an
eddy-viscosity term to (3.2). That is, we replace (3.2) by the new stochastic model
dyi
+ νi yi + ηi (t ) yi = W (t )∑θ ijk 1/ 2 Aijk y Gj ykG ,
dt
j, k
(3.8)
where ηi is a (time-dependent) eddy-viscosity coefficient. The stochastic model (3.8)
leads to the covariance equation
⎛d
⎞
⎜ + 2ν i + 2ηi ⎟ Yi (t ) = ∑ θijk {4Aijk 2 Yj (t )Yk (t )} .
⎝ dt
⎠
j ,k
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(3.9)
The eddy-coefficient ηi is to be prescribed in such a way that the average energy is
conserved (apart from molecular viscosity). The choice
ηi (t ) = −4 ∑θ ijk Aijk Ajki Yk (t )
(3.10)
jk
conserves energy (when νi=0) and makes (3.9) equivalent to (2.16). Thus EDM is the
exact covariance equation for the stochastic model (3.8) with ηi(t) given by (3.10).
The existence of a stochastic model (3.8) for which EDM is exact is important,
because it guarantees that the statistics predicted by EDM are realizable, that is,
correspond to kinematically possible states of the system {yi}. In particular, EDM can
never predict negative Yi, because solutions of the EDM equations correspond to exact
averages of (3.8) for which <yi2> can never be negative.
The stochastic model also suggests an appealing alternative derivation of EDM: In
the exact equations of motion, replace the nonlinear terms by terms containing Gaussian
white-noise random variables, and add a compensating eddy viscosity. This stochasticmodel derivation gives a clearer picture of the physical contents of EDM. In particular, it
shows that EDM does include the effects of organized structures in the flow. However,
in order to complete the new derivation of EDM that begins with the introduction of the
stochastic model (3.8), we must offer a motivation for the prescription (3.10) of ηi(t).
Since energy conservation is a desirable property of the resulting covariance equations, it
is logical to require that (3.9) conserve energy in the mean. Unfortunately, this
conservation property does not uniquely determine ηi(t). The reader will verify that both
(3.10) and
ηi (t ) = 2∑ θijk A jki2 Yk (t )
(3.11)
jk
conserve energy in the mean, but only the choice (3.10) agrees with the EDM of the
previous section. Thus, from the standpoint of the stochastic-model derivation, at least
two distinct covariance equations lay equal claim to our attention, or there is yet another
criterion, besides the conservation property, that must be enforced. Fortunately, the latter
proves to be the case, and the new criterion turns out to be the irreversibility principle in
the form of the Second Law.
4. Entropy
Let the dynamics be
dyi
= ∑ Aijk y j yk − ν i yi ,
dt
j ,k
yi (0 ) = y0i ,
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(4.1)
as before, but now let the summations in (4.1) be truncated to a finite number N of
modes. We regard (4.1) as the equations for a numerical model of the fluid. We also
regard (4.1) as the equation for a trajectory in the N-dimensional phase space with
coordinates y≡(y1, y2,...., yN). Every point in the phase space corresponds to a particular
state of the whole system.
Let P(y,t) be the density of system states in phase space. That is, let P(y,t)dy be the
probability that the system is in phase-space volume dy centered on y at time t. Then,
since the moving points that represent individual realizations of (4.1) can neither be
created nor destroyed, the probability density P(y,t) obeys Liouville’s equation,
∂P
∂
( y˙i P) = 0 ,
+∑
∂t
i ∂yi
(4.2)
with initial condition,
P(y,0) = P0 (y ) ,
(4.3)
where P0(y) is the probability density of the initial conditions yi0. Here y˙ , the Ndimensional velocity in phase space, is given by (4.1).
Eqn. (4.2) is analogous to the continuity equation,
∂ρ
+ ∇ ⋅ (vρ ) = 0 ,
∂t
(4.4)
with P (the density of system-points in phase space) corresponding to ρ (the density of
molecules), y˙ corresponding to the fluid velocity v, and the summation over N phasespace dimensions corresponding to the summation over three physical-space dimensions.
Now we divide the phase-space velocity y˙ into two parts: an energy-conserving part,
c
y˙i = ∑ Aijk y j yk ,
(4.5)
j ,k
and a non-conserving (viscous) part,
y˙νi = − ν i yi .
(4.6)
Both (4.5) and (4.6) are steady velocity fields in phase space, but since Aijk vanishes
whenever two of its indices are the same (cf. (2.4)), the conserving velocity (4.5) is also
non-divergent,
∂y˙ic
∑i ∂y = 0 .
i
(4.7)
Thus if the viscosity vanishes, y˙ is non-divergent, and (4.2) reduces to the analogue
V-12
∂P
∂P
+ ∑ y˙i
=0
∂t
∂yi
i
(4.8)
of the equation
Dρ
=0
Dt
(4.9)
for incompressible flow. The non-conserving velocity (4.6) is, on the other hand,
convergent, so that if {νi}>0, then all the trajectories (4.1) eventually terminate at the
origin, y=0.
Now suppose that the non-conserving part of y˙ is negligible, either because the
viscosity actually vanishes, or because viscosity is so small that its effect can be
temporarily neglected. Then, according to (4.8), every system-point keeps its initial
value of P, and (since the phase flow is non-divergent), the volume of phase space with a
particular value of P remains the same.
From the analogy between (4.8) and the equation (4.9) for a conserved scalar, we
anticipate that the field of P(y,t) becomes steadily more complex as t increases.
Consider, for example, a two-dimensional phase-space in which P is initially uniform
within a compact region, and zero outside (Figure 5.2). As t increases, the region of
nonzero P typically spreads out by developing long filaments that gradually fill up the
accessible parts of phase space. This evolving complexity of P means that (4.8) must be
very hard to solve. However, it must also be true that exact solution of (4.8) is pointless,
because (4.8), like (4.9) and indeed (4.1), typically exhibits unbounded sensitivity to its
initial conditions. That is, the complicated structure of P(y,t) is also non-reproducible.8
However, P is typically sought for computing the average,
F = ∫∫∫ ∫ dy F (y) P(y,t ) ,
(4.10)
of relatively smooth phase functions F(y). For example, <yi2> corresponds to F=yi2.
Suppose that P actually evolves from t=0 to t=t1 as shown in Figure 5.2. While it may be
practically impossible to calculate P(y,t1) accurately from (4.8) for use in (4.10), it is also
obvious that, for any F(y) that depends smoothly on y, the average <F> at t1 can be
calculated to good accuracy by replacing P(y,t1) with a probability density function that
is uniform over the circular region in Figure 5.2. We therefore distinguish between
P(y,t), the exact solution of (4.8), and Pˆ (y,t ) , a smoothed (sometimes also called coarsegrained) version of P. Nonequilibrium statistical mechanics seeks Pˆ (y,t ) but without
first finding P(y,t). This is another way of posing the closure problem.
Clearly, the stirring of P by the phase-space velocity y˙ is what motivates the
introduction of Pˆ . However, this same stirring property imposes the consistency
requirement that Pˆ ought to be progressively more spread out at successively later
times.9 This is a qualitative statement of the Second Law. To make the statement
V-13
quantitative, we must introduce a measure of the spread of Pˆ , or, equivalently, of the
uncertainty in y(t) represented by Pˆ (y,t ) . It turns out that the entropy
[
]
S Pˆ (y,t ) = − ∫∫∫  ∫ dy Pˆ ln Pˆ
(4.11)
is the best measure of the uncertainty in y. Now we pause to explain why.
Consider a random variable ξ that can assume any one of M discrete values
{ξ1,ξ2,.....,ξM}. Let pi be the probability of ξi. We want a measure of the uncertainty in
ξ. The measure
M
S( p1 , p2 ,...., pM ) = − ∑ pi ln pi
(4.12)
i =1
has two desirable properties. First, it takes its maximum value when all the outcomes are
equally probable, pi=1/M. And second, the measure (4.12) is additive in the following
sense: If ξ is a composite random variable, for which each outcome consists of an
outcome for η and an outcome for γ, where η and γ are independent random variables,
then the uncertainty in ξ is the sum of the uncertainties in η and γ. The converse is also
true: the only measure of uncertainty with these two properties is (4.12) (to within a
constant factor).10
The definition (4.12) is useful by itself. For example, suppose we know only the first
two moments of ξ,
M
ξ ≡ ∑ ξi pi = m,
i =1
M
ξ 2 ≡ ∑ξi 2 pi = Ξ .
(4.13)
i =1
From m and Ξ we want to determine the best values of {pi} for use in estimating other
statistics of ξ. According to information theory, the most objective estimate for {pi} is
that which maximizes the entropy (4.12) subject to constraints that state everything
actually known about {pi}. Maximizing (4.12) subject to the two constraints (4.13) and
the normalization requirement
M
∑ p = 1,
i =1
(4.14)
i
we find that
⎧ M
⎫
pi = Cexp ⎨− ∑ (α ξi + β ξi 2 ) ⎬ ,
⎩ i=1
⎭
(4.15)
where the three constants α, β and C are determined from m and Ξ by (4.13) and (4.14).
V-14
Now suppose that the random variable ξ can assume any value between 0 and 1. We
are tempted to generalize (4.12) to
S[ P(ξ)] = − ∫0 dξ P (ξ ) ln P(ξ) ,
1
(4.16)
which has its maximum value when ξ is uniformly distributed on the interval [0,1].
However, we could apply the same logic to the random variable µ≡ξ2; then we would
conclude that the greatest uncertainty corresponds to a uniform distribution of µ . These
two conclusions obviously disagree: If ξ is uniformly distributed, then ξ2 cannot be.
The proper generalization of (4.16) is
S[ P(ξ)] = − ∫0 dξ m(ξ) P(ξ) ln P(ξ ) ,
1
(4.17)
where m(ξ), the measure of ξ-space, must be determined by some additional
consideration. Similarly, the uncertainty in y(t) corresponding to the probability density
Pˆ (y,t ) is
[
]
S Pˆ (y,t ) = − ∫∫∫  ∫ dy m(y,t ) Pˆ ln Pˆ ,
(4.18)
where m(y,t) is the undetermined measure of phase space. Clearly the entropy (4.11)
corresponds to the choice m(y,t)≡constant in (4.18), but what justifies this choice?
Consider the limiting case in which no smoothing of P to Pˆ occurs, either because
the velocity y˙ stirs P(y,t) ineffectively, or because the fine-grained probability density P
is actually needed to compute the average of some very wiggly phase function. In this
limiting case, P and Pˆ are the same, and (4.8) implies that (4.11) is conserved. Since no
smoothing occurs, uncertainty does not increase, and the entropy stays the same. But
(4.11) and (4.18) agree only if m(y,t) is a constant. Thus (4.7) fixes the measure of phase
space: The measure is uniform in the phase-space coordinates for which the phase-space
velocity is nondivergent.
5. The entropy principle
Now we are ready to put the definition of entropy to work. First we show how to
identify the imprecise description implicit in EDM with a particular Pˆ (y,t ) for which the
entropy (4.11) can be evaluated. Then we verify that, according to the EDM, this entropy
is a non-decreasing function of time when the viscosity vanishes.
In EDM the dependent variables are the second moments,
{ y (t )
1
2
, y2 (t ) ,......, y N (t )
2
2
}.
(5.1)
V-15
Let the moments (5.1) have the values Yi at a fixed time t. Then, according to
information theory, the most objective estimate for Pˆ (y,t ) is that which maximizes the
entropy (4.11) subject to the N constraints
yi 2 = Yi ,
(5.2)
and the normalization requirement
1 = 1.
(5.3)
This easy variational problem leads to
⎧
⎫
Pˆ (y,t ) = exp ⎨λ − ∑ α i yi 2 ⎬ ,
⎩
⎭
i
(5.4)
where {αi} and λ are the Lagrange multipliers corresponding to (5.2) and (5.3). By (5.2)
and (5.3),
⎛α ⎞
λ = 12 ∑ ln⎜ i ⎟
⎝π ⎠
i
αi =
and
1
.
2Yi
(5.5)
Then, by (4.11) and (5.4),
[
] ∑ {ln (2πY ) + 1} .
S Pˆ (y,t ) =
1
2
(5.6)
i
i
The entropy (5.6) measures the spread of the probability density (5.4). More precisely,
(5.6) measures the uncertainty associated with a partial knowledge of the system state,
namely, knowledge of the second moments alone.
Now suppose that the viscosity vanishes. Then, by the reasoning of the previous
section, the uncertainty should only increase, as the coarse-grained probability density
spreads out. Thus
dS
≥ 0.
dt
(5.7)
By (5.6), the entropy law (5.7) is equivalent to
d
dt
∑ ln Y ≥ 0 .
(5.8)
i
i
The inequality (5.8) is analogous to Boltzmann’s famous H-theorem. We shall show that
EDM implies (5.8) in the same way that Boltzmann’s collision equation implies the Htheorem. In fact, EDM and the Boltzmann equation are themselves closely analogous.
V-16
In all of this we have assumed that {νi}=0. If the viscosity is nonzero, we replace
(5.7) by
dS
≥0,
dt c
(5.9)
where d/dtc is the rate of change caused by the non-viscous (i.e. energy conserving) terms
in the EDM equation for dYi/dt. Even when viscosity is present, the energy-conserving
part of the physics stirs the probability density function P in phase space, increasing the
spread of Pˆ . On the other hand, the viscosity drives the system toward y=0,
dYi
= −2νiYi ,
dtν
(5.10)
causing the entropy (5.6) to decrease,
dS
= −2∑ ν i < 0 .
dtν
i
(5.11)
However, the general entropy law (5.9) holds whether or not the stirring actually
dominates the viscosity.
Now we verify that, when {νi}=0, EDM conserves the energy,
d
dt
∑ Y = 0,
(5.12)
i
i
and obeys the entropy law in the form (5.8). Later we show that these energy and
entropy properties virtually determine the form of EDM.
We can prove (5.8) and (5.12) by direct manipulations on EDM, using the energy
conservation property (2.5) of the exact dynamics. However, it is more illuminating to
prove (5.8) and (5.12) by means of the following alternative derivation of EDM. The
new derivation is rather formal; it is best presented as a series of steps. But the formal
derivation offers the quickest way of seeing that EDM has the energy and entropy
properties stated above.
First, expand the exact (inviscid) dynamics in a Taylor series about t=0, to obtain
yi (t ) = yi (0 ) + t ∑ Aijk y j (0 )yk (0) + t 2 ∑ ∑ Aijk Akmny j (0)ym ( 0)y n (0 ) + 
j,k
(5.13)
j,k m, n
Regard the initial conditions {yi(0)} as Gaussian random variables. Then squaring and
averaging (5.13) yields the exact equation
Yi ( t ) = Yi (0 ) + 2t 2 ∑ Aijk 2Yj (0 )Yk ( 0) + 4t 2 ∑ Aijk Ajik Yi ( 0)Yk (0) + O(t 4 ) .
j ,k
j ,k
V-17
(5.14)
Now, throw away the O(t4) terms in (5.14), differentiate the remaining terms with respect
to time, and then replace the {Yj(0)} by {Yj(t)}. We obtain
{
}
Y˙i ( t ) = ∑ t 4Aijk 2Yj (t )Yk (t ) + 8Aijk Ajik Yi (t )Yk (t ) .
j,k
(5.15)
Finally, replace the t-factor in (5.15) by the triad decorrelation time θijk. The result is
(inviscid) EDM,
dYi
= ∑ 4θ ijk Aijk (Aijk Yj + 2A jikYi )Yk .
dt
j ,k
(5.16)
It is now easy to see how EDM inherits the energy conservation property of the exact
dynamical equations. Since the exact (inviscid) dynamics have the same energy at all
times, energy conservation holds term by term for the Taylor series expansion (5.14). In
particular, (5.14) still conserves energy if all the terms except the t2-terms are dropped
from the right-hand side. The replacement of {Yj(0)} by {Yj(t)} does not spoil energy
conservation, because energy is conserved for arbitrary initial conditions {Yj(0)}. Hence
(5.15) also conserves energy. Finally, energy conservation survives the replacement of t
by θijk (provided that θijk is symmetric in its three indices) because the energy is
conserved triad-wise if at all.11
By the same reasoning, EDM conserves any invariant of the exact dynamics whose
average can be expressed as a linear combination of the Yi. For example, inviscid twodimensional Euler flow governed by (1.2) (with ν=0) conserves (twice) the energy,
E = ∫∫ dx dy ∇ ψ ⋅ ∇ψ = ∑ yi 2 ,
(5.17)
i
and the enstrophy,
Z = ∫∫ dx dy (∇ 2ψ ) = ∑ ki2 yi 2 ,
2
(5.18)
i
where yi is defined by (1.7). Hence, the corresponding EDM equations conserve
E = ∑ Yi
(5.19)
Z = ∑ ki2 Yi .
(5.20)
i
and
i
By very similar reasoning, we show that EDM obeys the H-theorem (5.8).12 By
definition,
V-18
{
}
S( t ) − S(0 ) = ∫∫∫ ∫ dy − Pˆ (t ) ln Pˆ (t ) + Pˆ (0) ln Pˆ (0) .
(5.21)
But, since the initial conditions are, by assumption, Gaussian, Pˆ (0) = P( 0) . Then,
because
∫∫∫  ∫ dy P(t ) ln P(t ) = ∫∫∫  ∫ dy P(0) ln P(0 )
(5.22)
by (4.8), (5.21) becomes
{
}
S( t ) − S(0 ) = ∫∫∫ ∫ dy − Pˆ (t ) ln Pˆ (t ) + P (t ) ln P( t ) .
(5.23)
But Pˆ (y,t ) is, by definition, the probability density that maximizes the entropy functional
(4.11) for given {Yi(t)}. Hence (5.23) implies that
S( t ) − S(0 ) ≥ 0 .
(5.24)
The exact equation (5.14) must be consistent with (5.24). But (5.15) is also exact in the
limit t→0. Thus (5.15) must obey the H-theorem (5.8). Finally, (5.8) survives the
replacement of t by θijk in (5.16) by the same reasoning as before. This proof of (5.8)
shows that the H-theorem arises from the assumption of Gaussian initial conditions in the
formal derivation of EDM.
Of course we can also prove the H-theorem directly from (5.16). Since the Htheorem holds triadwise (if at all), we need only show that the entropy of a particular
triad increases as a result of the interactions between the triad members alone. Let i, j, k
denote the three members of a particular triad. Then, using only (5.16) and the energyconservation property (2.5), we can show that
2
8θ ijk
dS d
= (ln Yi + ln Yj + lnYk ) =
AijkYj (Yi − Yk ) + A jikYi (Yj − Yk ) ≥ 0 .
dt dt
YiYjYk
[
]
(5.25)
These results have meaning whether the viscosity vanishes or not. If the viscosity is
nonzero, we can still say that the terms in EDM arising from the nonlinear-interaction
terms in the original dynamics act by themselves to conserve energy and to cause entropy
to increase. But suppose that the viscosity actually vanishes, so that the entropy
∑ lnY
(5.26)
i
i
increases for an arbitrarily long time. How big can it get?
Suppose that the energy (5.19) is the only conserved quadratic. Then the entropy
cannot exceed the value obtained by maximizing (5.26) subject to the constraint (5.19).
In this absolute equilibrium state,
V-19
Yi = E / N ,
(5.27)
and the energy is equi-partitioned among the N modes.
If however the exact dynamics (and hence EDM) have additional quadratic invariants,
then the maximum achievable entropy will generally be lower. Suppose, for example,
that the system conserves both the energy (5.19) and the enstrophy (5.20). Then the
maximum entropy, obtained by maximizing (5.26) subject to (5.19) and (5.20), occurs
when
Yi =
1
,
α + β ki 2
(5.28)
where α and β are Lagrange multipliers determined by the values of <E> and <Z>.
(Since the energy and enstrophy are conserved, these values are fixed by the initial
conditions.) According to (5.28), a linear combination of the energy and enstrophy
2
(namely α Yi + β ki Yi ) is equi-partitioned among the modes.
Two things can prevent the system from attaining the absolute equilibrium state.
First, if viscosity is present, then although the energy-conserving part of the physics still
acts to increase the entropy, the viscosity drives the system toward the (minimum
entropy) state of rest. Second, if the number of modes is infinite (N=∞), then absolute
equilibrium cannot occur in a finite time, even if the viscosity actually vanishes.
However, even in that case, the absolute-equilibrium states still represent informative
target states towards which the nonlinear terms, acting alone, would drive the flow. We
shall come back to this at length.
Now we show that the entropy principle (5.8) is all that is needed to complete the
stochastic-model derivation of EDM that began with (3.9). Assume, for simplicity, that
the energy (5.19) is the only conserved quadratic. We want to show that if
⎛⎜ d
⎞
+ 2ηi ⎟ Yi = 4∑θ ijk Aijk 2 Yj Yk ,
⎝ dt
⎠
j,k
(5.29)
then (3.10) is the only choice of eddy-viscosity coefficient ηi that is consistent with the
conservation principle (5.12) and the irreversibility principle (5.8). Since these properties
must hold for (separately) arbitrary {θijk} and {Yj}, we assume that ηi takes the form
ηi = ∑ηijkθ ijkYk ,
(5.30)
j ,k
where {ηijk} is a set of constants, to be determined. Again let i, j, k denote the three
members of a particular triad. By requiring the energy of this triad to be conserved for
arbitrary values of Yi, Yj, Yk, we obtain the three equations,
V-20
4Aijk 2 = ηkij + η jik
4A jik 2 = ηkji + ηijk
(5.31)
4Akij 2 = ηikj + η jki
Note that the last two equations are simply permutations of the first. Similarly, the
absolute-equilibrium state (5.27) is a steady solution of (5.29) if and only if
4Aijk 2 = ηijk + ηikj
4A jki 2 = η jik + η jki
(5.32)
4Akij 2 = ηkij + η kji
The sets (5.31) and (5.32) comprise 6 equations in the six unknown constants {ηijk}.
However, these six equations are not all independent; the sum of (5.31) equals the sum
of (5.32). In fact, the general solution of (5.31-32) is
ηijk = 2Aijk 2 + 2 Ajki 2 − 2Akij2 + Cε ijk ,
(5.33)
where εijk is the permutation symbol, and C is an arbitrary constant. However, unless
C=0, the solutions of (5.29) oscillate about the maximum entropy state (5.27), violating
the entropy principle (5.8). The choice C=0 corresponds to EDM, (5.16), after use of the
conservation relations (2.5). We see that the conservation principle and the irreversibility
principle virtually determine the form of the EDM approximation.
6. Equilibrium statistical mechanics
We began with the definition of entropy, a measure of the uncertainty about the state
of the physical system under consideration. Then, on account of the instability property
of the exact dynamics (that is, on account of the stirring of probability density in phase
space), we argued that the entropy ought to increase with time. Next we showed that
EDM obeys this law of entropy increase and, moreover, that the entropy law is one of
two essential ingredients (the other being conservation of energy) that largely determine
the form of EDM. Finally we discussed the absolute-equilibrium states of maximum
entropy. Acknowledging that these absolute-equilibrium states are unrealizable in the
cases of most interest to us, we argued that they nevertheless reveal the direction toward
which nonlinear interactions (that is, the stirring in phase space) drive the system. In the
next two sections, we examine these absolute-equilibrium states more closely.
Absolute equilibrium is the subject of equilibrium statistical mechanics, which
predicts the final state toward which isolated systems evolve when the number of degrees
of freedom is finite. Equilibrium statistical mechanics does not predict the path of
evolution to absolute equilibrium, nor can it predict the kind of equilibrium that results
from a balance between the external forcing of some modes in the system and the
damping of others. These are the province of nonequilibrium statistical mechanics,
V-21
exemplified by EDM. The nonequilibrium theory includes the equilibrium theory as a
limiting case. Unfortunately, the nonequilibrium theory is much more complicated than
the equilibrium theory and is in fact highly developed (i.e. non-controversial) only for
systems near absolute equilibrium, which (as we shall see) certainly excludes fluid
turbulence. However, insofar as the more easily calculable absolute-equilibrium states
offer some insight on nonequilibrium behavior, equilibrium statistical mechanics has a
considerable value of its own.
In this section, we consider equilibrium statistical mechanics as a separate theory,
emphasizing its logical simplicity and avoiding references to the nonequilibrium
theory.13 As an example, we calculate the equilibrium state of a classical ideal gas with
a fixed number of molecules, thereby repaying a debt incurred in Chapter 1. In the
following section, we consider the much more problematic equilibrium state
corresponding to a macroscopic fluid with an infinite number of modes.
Consider the general dynamical system described by
y˙i = fi (y ),
i = 1, N ,
(6.1)
where N is finite. Once again, we regard (6.1) as the equation for a trajectory in the Ndimensional phase space with coordinates
y ≡ (y1 , y2 , ..........., y N ) .
(6.2)
If the phase-space velocity (6.1) is nondivergent,
∂fi
∑ ∂y
i
= 0,
(6.3)
i
then the probability density P(y,t) of system states obeys
∂P
∂P
+ ∑ fi ( y)
=0.
∂t
∂yi
i
(6.4)
According to the fundamental principle of equilibrium statistical mechanics, in
systems governed by (6.1) and satisfying (6.3), the smoothed (or coarse-grained) version
of P(y,t) eventually becomes uniform over all the accessible parts of phase space. That
is, the dynamics stirs P until it is practically equivalent to a uniform distribution in phase
coordinates for which the phase-space velocity y˙ is nondivergent. Again we note that the
assumption of a uniform, static probability density is consistent with (6.4) only if the
phase-space motion is nondivergent; otherwise an initially uniform probability density
would tend to acquire local extrema at the points of convergence and divergence. On the
other hand, if the probability density is uniform in one set of phase-space coordinates
satisfying (6.3), then it is also uniform in any other set of coordinates for which the
motion is nondivergent, because the two coordinate sets then have a constant Jacobian of
transformation.
V-22
We must still explain the meaning of accessible. According to equilibrium statistical
mechanics, all the final states with the same value of the energy (and other conserved
quantities if present) as the initial state are considered accessible. Thus, in a system for
which the energy E(y) is the only conserved quantity, the equilibrium probability density
is
P(y) = C δ ( E(y ) − E0 ) ,
(6.5)
where δ( ) is Dirac’s delta-function, E0 is the initial energy, and C is a normalization
constant determined by
∫∫∫ ∫ dy P =1 .
(6.6)
The probability density (6.5) is called the microcanonical ensemble. (From now on, we
omit the distinction between P and Pˆ , with the understanding that we now always mean
the smoothed probability density.) If we regard the delta-function as a sequence of
functions in the usual way, then (6.5) states that the probability density is (roughly
speaking) uniform on an (N-1)-dimensional hypersurface of varying thickness.
Now, P is typically wanted for computing the averages of functions involving only a
small number of the phase coordinates. However, to compute <yi2> (for example), we
only need the partial density,
Pi ( yi ) ≡ ∫∫∫ ∫ dy1 ...dyi−1 dyi +1 ...dyN P ,
(6.7)
obtained by integrating over all the modes except yi. If P is given by (6.5), and if E(y) is
a sum of functions each depending on only a few coordinates, e.g.
E( y) = E1 ( y1 ) + E2 ( y2 ) + + EN ( y N ) ,
(6.8)
then it can be shown that (6.7) takes the approximate form
Pi ( yi ) = Ci exp[ −α Ei (yi )] ,
(6.9)
where α and Ci are constants.14 But if (6.9) holds for each partial density, and if we
want the total probability density P(y) only for evaluating the averages of sum functions
like (6.8), then we can use
P(y) = Cexp[−α E (y)]
(6.10)
as an approximation to (6.5). C is the normalization constant, and α is determined by
the average energy,
V-23
∫∫∫ ∫ dy E(y ) P( y) =E
0
.
(6.11)
The probability density (6.10), called the macrocanonical ensemble, is often also justified
by assuming that the whole system is in contact with a very large thermal bath, but the
assumption of a thermal bath is not very compatible with the applications we have in
mind.
Nowhere in the theory based upon uniform probability density in phase space does
the concept of entropy necessarily arise. However, we can define the entropy of the
equilibrium probability density to be
S = − ∫∫∫∫ dy P ln P .
(6.12)
Then the macrocanonical density (6.10) can also be obtained by maximizing the entropy
(6.12) subject to the constraints (6.6) and (6.11). Similarly, the microcanonical density
(6.5) corresponds to the maximum of (6.12) subject to (6.6), (6.11) and
∫∫∫ ∫ dy ( E(y ) − E ) P( y) = Ξ ,
2
(6.13)
0
in the limit Ξ→0. This limit corresponds to the limiting sequence of functions
represented by the delta function in (6.5).
In Section 18 of Chapter 1, we introduced Boltzmann’s definition of the entropy,
S = ln Ω ,
(6.14)
where Ω is (proportional to) the number of microstates with the given energy. Now we
show that these two definitions of entropy, (6.12) and (6.14), are essentially the same.
First suppose that probability density takes the constant value P0 over finite volume of
phase space and vanishes everywhere else. Then, bearing the normalization requirement
(6.6) in mind, we see that P0-1 is the volume of phase space in which P=P0. This volume
is proportional to the number of microstates Ω. Thus, according to (6.14),
S = ln
1
.
P0
(6.15)
If P is non-uniform in phase space, then (6.15) generalizes to
S = ln
1
= − ∫∫∫∫ dy P ln P ,
P
(6.16)
which agrees with (6.12). Thus the two definitions of entropy are essentially the same.
To illustrate the machinery of equilibrium statistical mechanics, we calculate the
absolute equilibrium state of an monatomic ideal gas inside a rigid container. The phasespace coordinates are the locations and momenta of each gas molecule,
V-24
y = {q1 ,,q3M , p1 ,, p3M } ,
(6.17)
where M is the number of molecules. The subscripts on q and p identify both the
molecule and the direction in physical space. In this case the dimension of the phase
space, N=6M, is truly finite; no artificial truncation is required. The phase-space
velocity y˙ is nondivergent because the molecules obey Hamilton’s equations,
dqi ∂H
=
,
dt ∂pi
dpi
∂H
=−
.
dt
∂q i
(6.18)
Note that (6.18) implies (6.3) but that the converse is generally untrue. However,
equilibrium statistical mechanics only demands (6.3); canonical coordinates are not
generally required.
The energy of the monatomic ideal gas is a sum function,
3M
E=∑
i=1
1 2
p .
2m i
(6.19)
The macrocanonical distribution (6.10) implies that the mean-square momenta are
pi
2
∫
=
∞
−∞
pi 2 exp{−α pi 2 / 2m}dpi
∫−∞ exp {−α pi 2 / 2m}dpi
∞
=
m
.
α
(6.20)
Hence the pressure is
p=ρ
p1 2
ρ 1
1
= n NA ,
2 =
m
mα
α
(6.21)
where n is the number of moles per unit volume and NA is Avogodro’s number. Equation
(6.21) agrees with the equation of state,
p = n R*T ,
(6.22)
for an ideal gas (where R*=kNA is the universal gas constant and k is Boltzmann’s
constant; see Chapter 1) provided that
α=
1
.
kT
(6.23)
Thus the factor α appearing in the exponent of (6.10) has the physical interpretation of
inverse temperature. For non-monatomic ideal gases, we also arrive at (6.22); the
V-25
additional degrees of freedom contribute to the energy (and heat capacity) but only the
translational degrees of freedom contribute to the pressure.
From the macrocanonical density (6.10) and the definition (6.12) of entropy, we can
calculate the fundamental relation between entropy, (average) energy, and the volume of
the gas, obtaining the result given in Section 18 of Chapter 1. As emphasized in Chapter
1, this fundamental relation tells us everything there is to know about the thermodynamic
equilibrium state of the gas.
Of course, the ideal gas is very special because its energy is a sum function, so that all
the probability integrals can be factored into simple integrals. In all the really interesting
problems studied with equilibrium statistical mechanics, the energy contains interaction
terms requiring sophisticated approximation methods. Systems undergoing phase
changes offer some of the biggest challenges.
In the following section, we re-examine the absolute-equilibrium states corresponding
to the equations for a macroscopic fluid. In the cases we consider, the energy and other
conserved quantities are also sum functions, and the calculations are therefore as easy as
in the case of an ideal gas. The fundamental difficulty turns out to be one of
interpretation — of properly understanding the behavior of the equilibrium states as the
number of modes becomes infinite.
7. The meaning of absolute equilibrium
Once again we consider the Navier-Stokes equations in the abstract form (4.1). If the
fluid is two-dimensional, the {yi} could be the coefficients in a Fourier- or eigenmodeexpansion of the streamfunction ψ(x,y,t); see (1.5) and (1.7). The subscript denotes the
wavevector or eigenmode. If the fluid is three-dimensional, then the {yi} could be the
Fourier amplitudes of each directional component of velocity. In that case the subscript
denotes the wavevector, the direction, and the real or imaginary part.
Let the exact dynamics (4.1) be truncated to a finite number of modes. Again, one
could simply discard all the modes corresponding to wavevector magnitudes larger than
some cutoff kmax. Alternatively, the truncated version of (4.1) might represent a finitedifference approximation to the fluid equations on a uniform grid, with the grid-spacing
corresponding to kmax-1. If viscosity is present, and if kmax is sufficiently large (for the
given viscosity), then the truncation in wavenumber has a negligible effect on the
dynamics, because viscosity wipes out the high-wavenumber components of the flow.
But suppose that the viscosity vanishes. Then, as can always be checked, the phasespace flow is nondivergent in the coordinates {yi}, and the methods of equilibrium
statistical mechanics apply to the truncated system. Provided that the conserved
quantities take the form of sum functions, we can work out the absolute-equilibrium
states. As already remarked, these states correspond to the final states towards which
numerical models (having a finite number of degrees of freedom) would evolve in the
absence of forcing and viscosity.
The absolute-equilibrium states indicate the general direction in which nonlinear
interactions drive numerical models, even when forcing and viscosity are present.
Whether or not this information has practical value remains to be settled. Clearly, the
assumptions of vanishing viscosity and a truncation in modes are incompatible insofar as
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the exact, untruncated dynamics is concerned. The question, then, is whether we learn
anything at all from the absolute-equilibrium states of numerical models.
In the case of inviscid, truncated three-dimensional turbulence, the only significant
invariant is the total energy,15 and equilibrium statistical mechanics predicts equipartition of the energy among the modes. In three dimensions, the number of modes with
wavevector magnitude within dk of k is proportional to k2dk. Thus, equilibrium statistical
mechanics predicts an energy spectrum of the form
E( k ) = C k 2 ,
0 < k < k max ,
(7.1)
where C is a constant determined by the total energy.
Now, suppose we increase kmax by increments, but allow the truncated system to
reach equilibrium after each change in kmax. Then the spectrum assumes the sequence of
shapes shown in Figure 5.3. As kmax increases, the energy moves into higher and higher
wavenumbers, but E(k) approaches no definite limit; there is no way to give equal
amounts of energy to an infinite number of modes, and then end up with a finite total
energy! This lack of convergence is closely related to the ultraviolet catastrophe that
occurs when equilibrium statistical mechanics is applied to the equations of classical
electrodynamics.
What then do we learn from (7.1)? Only that three-dimensional turbulence wants to
transfer its energy into very high wavenumbers. The Kolmogorov theory presented in
Chapter 4 predicts a k -5/3 energy spectrum in which the energy moves from low
wavenumbers to high wavenumbers. Now we see that any spectrum with a slope less
than that of (7.1) could be expected to transfer its energy to high wavenumbers.
Certainly this conclusion is rather tame. However, we shall see that the corresponding
absolute equilibrium states of two-dimensional and (especially) quasigeostrophic
turbulence are much more interesting and instructive. As a general rule, absoluteequilibrium theory becomes more illuminating as the corresponding dynamics (and
dynamical invariants) become richer and more complex.
The two-dimensional absolute equilibrium states are interesting because they can be
made to incorporate at least one of the dynamical invariants corresponding to the
conservation of vorticity on fluid particles. Consider the truncated, inviscid dynamical
equations corresponding to inviscid two-dimensional flow governed by
∂ 2
∇ ψ + J (ψ ,∇ 2ψ ) = 0 ,
∂t
(7.2)
in a finite domain. Let the {yi} be the eigenmode amplitudes defined by (1.7-9). Let
kmin be the lowest eigenvalue (i.e. wavenumber) in the system. Then kmin-1 is the size of
the domain. Since Aijk vanishes whenever two of its subscripts are equal, (6.3) holds, and
the flow in phase space is nondivergent. The dynamical invariants include (twice) the
energy
N
E = ∫∫ dx ∇ψ ⋅∇ψ = ∑ yi2 ,
(7.3)
i =1
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and the enstrophy
N
Z = ∫∫ dx (∇ ψ ) = ∑ ki 2 yi2 .
2
2
(7.4)
i =1
The macrocanonical ensemble based only on (7.3) and (7.4) is
⎧ N
⎫
P(y) = Cexp ⎨− ∑ (α + β ki 2 ) yi 2 ⎬ ,
⎩ i=1
⎭
(7.5)
where the constants C, α, and β are determined by
N
∑
i =1
N
∑k
yi 2 = E,
i =1
i
2
yi 2 = Z,
and
1 = 1,
(7.6)
and E, Z are the prescribed energy and enstrophy of the system. By a general theorem of
statistical mechanics,16 these constants are uniquely determined for all realizable values
of E and Z, that is, for all
E>0
and
kmin2 E < Z < kmax2 E.
(7.7)
We can regard α and β as inverse temperatures (or chemical potentials) corresponding to
the energy and enstrophy, respectively.
According to (7.5), <yi>=0, and <yi2> is given by (5.28). In two dimensions, the
number of modes with wavevector magnitude within dk of k is proportional to k dk. Thus
the absolute-equilibrium spectrum for bounded, inviscid, two-dimensional turbulence is
E( k ) =
k
,
α + βk 2
k min < k < kmax ,
(7.8)
where α and β are determined from E and Z by
kmax
∫ E (k ) dk =
kmin
1 ⎛ kmax 2 + α β ⎞
ln ⎜
⎟=E
2β ⎝ k min2 + α β ⎠
(7.9)
and
1 ⎡
α ⎛ k max 2 + α β ⎞ ⎤
2
2
∫ k E (k ) dk = 2β ⎢⎣(kmax − k min ) − β ln ⎜⎝ kmin 2 + α β ⎟⎠ ⎥⎦ = Z .
kmin
k max
2
(7.10)
In writing (7.8-10), we assume that the mode-spacing is sufficiently dense that all the
sums can be replaced by integrals. This assumption, which is least accurate for the
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modes near kmin, is an unnecessary convenience; our main conclusions also apply to the
case of perfectly discrete modes.
As the energy E and enstrophy Z vary over the range (7.7) of realizable values, the
corresponding inverse temperatures α (E,Z) and β (E,Z) take on all the values for which
α + β k2 > 0
(kmin 2 < k 2 < k max2 ) .
(7.11)
The range (7.11) includes, at its extremes, two negative temperature regimes. The first
occurs when
α < 0, β > 0,
α ≈ −β kmin 2 ,
(7.12)
and corresponds to an energy spectrum in which all the energy is concentrated at the
lowest wavenumber kmin. The second negative-temperature regime occurs when
α > 0, β < 0,
α ≈ −β kmax 2 ,
(7.13)
and corresponds to a concentration of energy at the highest wavenumbers in the truncated
system.
Now let E and Z be given constants, and consider again the limit kmax→∞, with kmin
held fixed. From (7.9) and (7.10), it follows that17
k max2
β~
2( Z − kmin 2 E )
(k max → ∞)
(7.14)
and
⎧
α
k max 2 E ⎫
2
2
~ −kmin + k max exp ⎨−
⎬
2
β
⎩ ( Z − kmin E ) ⎭
(kmax → ∞) .
(7.15)
Thus the limit of progressively finer spatial resolution always yields the state of lowest
negative α.
Let k0 be any fixed wavenumber between kmin and kmax. Then as kmax→∞ (with E,
Z, kmin, and k0 held fixed), (7.14) and (7.15) imply that
∫
k0
∫
k0
kmin
E( k ) dk → E
(7.16)
k 2 E (k ) dk → kmin2 E .
(7.17)
and
kmin
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Since (7.16) holds for all k0>kmin, it follows that, as kmax→∞, all the energy moves into
the lowest-wavenumber mode. If the energy at kmin is E, then the enstrophy at kmin must
be kmin2E. Then, because k0 can also be arbitrarily large, (7.17) implies that all the
remaining enstrophy moves toward infinite wavenumber as kmax→∞. Thus, as the
spatial resolution increases, a single domain-filling vortex absorbs all of the energy in the
system, and all the enstrophy not required by this vortex appears in eddies at the smallest
resolved lengthscales.
Of course, this equilibrium picture gives no information about the path by which the
system reaches absolute equilibrium. In Chapter 4, we noted that the energy transfer to
low wavenumbers in two-dimensional turbulence sometimes occurs through the
formation and gradual coalescence of relatively isolated axisymmetric vortices.
Equilibrium statistical mechanics cannot explain such disequilibrium behavior.18
Absolute-equilibrium theory based only on (7.6) is open to the further criticism that
the dynamics (7.2) actually conserves an infinite number of quantities — the vorticity of
each moving fluid particle. Consequently, (7.2) conserves every quantity of the form
∫∫ dx F (∇ ψ ) ,
2
(7.18)
where F( ) is an arbitrary function. Particular attention attaches to the powers,
∫∫ dx (∇ ψ )
2
n
,
(7.19)
where n is any integer. The enstrophy corresponds to n=2. Absolute-equilibrium theory
can easily accommodate the average-vorticity (n=1) invariant, but the higher (n>2)
powers lead to integrals that cannot be performed. The neglect of these higher powers is
sometimes excused by the observation that their conservation typically does not survive
the truncation in modes. However, this excuse is not so much a defense of (7.5) as it is a
criticism of the truncated dynamical equations. But suppose that the exponent in (7.5)
did include an arbitrary number of the higher powers (7.19). The resulting probability
density would be a very complicated, fine-grained function whose lower moments might
anyway be insensitive to the prescribed values of these higher powers.
Somewhat surprisingly, there is a truncated dynamics that conserves every quantity of
the form (7.18), and, moreover, exactly satisfies (7.2). This is the system composed of N
point vortices at locations {(xi(t),yi(t))}. Let Γi be the (constant) circulation around the ith vortex. In an unbounded fluid, the governing dynamical equations take the canonical
form
Γi
dx i
∂H
=−
,
dt
∂yi
Γi
dyi ∂H
=
,
dt ∂xi
(7.20)
where
V-30
H=
1
2π
∑ Γ Γ ln ( x
i>j
i
j
i
)
−xj .
(7.21)
is the Hamiltonian. Hence the motion (7.20) is nondivergent in the phase space spanned
by {xi, yi}. The truncated dynamics (7.20-21) conserves the energy H and the vorticity
on every fluid particle. This vorticity is singular on the fluid particles at xi(t), and
vanishes on every other particle. Thus the conservation of all the vorticity invariants is
built into the numbering of the vortices. The macrocanonical ensemble,
P( x1 ,…,x N , y1 ,…,y N ) = Ce−α H ,
(7.22)
thus recognizes the energy and all the vorticity invariants.
In a classic paper, Onsager (1949) considered this absolute-equilibrium state, and
predicted the clumping of like-signed vortices in solutions of (7.20). Unfortunately, the
energy (7.21) couples every vortex to every other vortex, making averages with (7.22)
very difficult to compute. The literature on the equilibrium statistical mechanics of point
vortices is large, rewarding, and incomplete.19 However, the predictions of (7.22) seem
very similar to those of (7.5), supporting the idea that the higher powers of vorticity are
of secondary importance.
In the next chapter, we apply many of the ideas in this chapter to quasigeostrophic
turbulence. Although we shall not use nonequilibrium theories like EDM, we shall make
extensive use of their two most important ingredients: the conservation principle for
energy and (potential) vorticity, and the irreversibility principle. The irreversibility
principle can take many useful forms, but we shall find the entropy principle and,
particularly, the theory of absolute equilibrium to be especially informative.
Notes for Chapter 5.
1. Leslie (1973) and McComb (1990) offer a much more extensive treatment of this
subject. See also Lesieur (1990) and the extensive reviews by Orszag (1977) and Rose
and Sulem (1978). For a very general overview of statistical turbulence theory, see
Monin and Yaglom (1971,1975).
2. Even in the cases where (2.4) do not seem to hold, it is often possible to satisfy (2.4)
by a change in the definition of the yi. For example, in the case of a finite-difference
approximation to the shallow-water equations, (2.4) hold if the yi are the gridded values
of h, hu, and hv. The property (2.4) seems to be a consequence of the underlying
Hamiltonian structure of inviscid fluid mechanics.
3. Section 2 closely follows Orszag’s (1970, 1977) derivation of EDM. One can also
regard EDM as an abridgment of Kraichnan’s (1959) direct-interaction approximation;
see Leslie (1973).
4. For a discusion of these and other properties of Gaussian random variables, see for
example Frisch (1995, Chapter 4).
5. See Lesieur (1990, p. 176 and 276). For a comparison between EDM and direct
numerical simulations, see Herring et al. (1974), Herring and McWilliams (1985), and
Herring (1990).
6. See Holloway and Hendershott (1977).
V-31
7. Section 3 is based upon the papers by Leith (1971) and Kraichnan (1971a). See also
Frisch et al. (1974).
8. Many of these ideas in were expressed by Orszag (1970,1977), who stressed that exact
solutions of the full moment hierarchy has the instability property of the original
equations and is time-reversible.
9. It would, after all, be illogical to assume that one could lose (by replacing P with Pˆ )
the information contained in the detailed structure of P(y,t) and then recover it at a later
time.
10. See, for example, Khinchin (1957, pp 9-13). This book is an excellent mathematical
introduction to information theory, a subject invented by Shannon and Weaver (1949)
and made a basis for statistical mechanics by Jaynes (1957). The textbooks by Tribus
(1961) and Katz (1967) develop statistical mechanics from the information-theory
viewpoint. This viewpoint seems ideally suited to closure theory, which is based upon an
incomplete statistical description of the flow.
11. To see this, consider the initial condition in which only the three yi corresponding to
a particular triad are nonzero.
12. See Montgomery (1976) and Carnevale et al. (1981). Carnevale (1982) demonstrated
that direct numerical simulations of two-dimensional turbulence obeyed the H-theorem.
13. For a brief mathematical introduction to statistical mechanics, see Thompson (1972).
14. The various proofs of (6.9) differ in complexity, depending on the strength of the
exact hypotheses; see Khinchin (1949), Grad (1952) or Thompson (1972).
15. Kraichnan (1973) argued that the helicity is much less important than the energy.
16. See, for example, Katz (1967, pp. 47-50)
17. See Carnevale and Frederiksen (1987).
18. However, it has been claimed that equilibrium statistical mechanics can explain the
structure of the isolated vortices that arise in two-dimensional turbulence. See Miller
(1990) and Robert and Sommeria (1991).
19. See Kraichanan and Montgomery (1980, sec. 3.3) and references therein. Kraichnan
(1975) discusses the relationship between the absolute-equilibrium states of the truncated
continuum and the system of discrete vortices.
V-32