• R. Kubo, K. Matsuo and K. Kitahara (1973) Fluctuation and relaxation of
macrovariables. Journal of Statistical Physics9:51–96
We know that the study of physical phenomena always starts from a phenomenological approach, reflecting the initial stage of organization of experimental facts. This stage often leads to a mathematical model in terms of differential equations where the variables of interest are macro- scopic variables, in which the microscopic fluctuations are neglected (av- eraged over), resulting in a deterministic theory. Examples include Ohm’s law, chemical rate equations, population growth dynamics,etc.
At the next, more fundamental (mesoscopic) level, the fluctuations are taken into account – by the master equation (or the Fokker-Planck equation, if it applies) for stationary Markov processes. As the latter de- termines the entire probability distribution, it must be possible to derive from it the macroscopic equations as an approximation for the case that the fluctuations are negligible.
LetY be a physical quantity with Markov character, taking the value y0 at t = 0; i.e., P(y, t|y0,0) → δ(y−y0) as t → 0. For definiteness,
say the system is closed and isolated. Then we have from equilibrium statistical mechanics that the probability distributionp(y, t) tends toward some equilibrium distribution peq(y) ast→ ∞,
lim
t→∞p(y, t) =peq(y),
(see figure 6.2). We know from experience that the fluctuations remain small during the whole process, and so p(y, t) for each t is a sharply- peaked function. The location of this peak is a fairly well-defined number,
having an uncertainty of the order of the width of the peak, and it is to be identified with the macroscopic value ¯y(t). Usually, one takes,
¯ y(t) =hYit= ∞ Z −∞ yp(y, t)dy. (6.12)
As t increases, the peak slides bodily along the y-axis from its initial location ¯y(0) = y0 to its final location ¯y(∞) = yeq; the width of the
distribution growing to the equilibrium value. Bearing this picture in mind, we have:
d dty¯(t) = ∞ Z −∞ y∂ ∂tp(y, t)dy.
Using either the master equation or the Fokker-Planck equation, we have, d dty¯(t) = ∞ Z −∞ a1(y)p(y, t)dy=ha1(y)it, (6.13)
which we identify as the macroscopic equation.
Example: One-step process (see p. 62). The discrete master equation reads,
d
dtpn=rn+1pn+1+gn−1pn−1−(rn+gn)pn. (6.14) Multiplying both sides by n, and summing over all n ∈ (−∞,∞), we obtain the evolution equation for the average n,
dhni
dt =hgni − hrni. (6.15) For example, withrn=βnandgn=α, Eq. 6.15 reduces to:
dhni
dt =α−βhni.
Note in this last example, we hadhrnit=hr(n)it=r(hnit) and simi-
larly for gn. The situation illustrated by this example holds in general:
Proposition: If the function a1(y) is linear in y, then the
macroscopic variable ¯y(t) satisfies the macroscopic equation, d
dty¯(t) =a1(¯y), (6.16) which follows exactly from the master equation.
If, however, a1(y) isnonlinear in y, then Eq. 6.166= 6.13, and the macro-
scopic equation is no longer determined uniquely by the initial value of ¯y. Not only, but one must also specify what “macroscopic equation” means.
In the literature, one finds Eq. 6.16 used even whena1(y)is nonlinear; the
(usually-not-stated-) assumption is that since the fluctuations are small, anda1(y) is smooth, we may expand it abouthyit= ¯y,
a1(y) =a1(¯y) +a01(¯y)·(y−y¯) + 1 2a 00 1(¯y)·(y−y¯) 2+. . .
Taking the average, we have,
ha1(y)it=a1(¯y) + 1 2a 00 1(¯y)· h(y−y¯) 2i t+. . . ha1(y)it≈a1(¯y),
on the grounds that the fluctuations h(y−y¯)2it are small. In that case,
we read Eq. 6.16 as an approximation, d
dty¯(t)≈a1(¯y), (6.17) ifa1(y) is nonlinear – this is the “meaning” that is assigned to the macro-
scopic equation.
It is also possible to deduce from the master equation (or the Fokker- Planck equation) an approximate evolution for the width of the distribu- tion; first, one shows that
dhy2i
t
dt =ha2(y)it+ 2hya1(y)it, from which it follows that the varianceσ2(t) =hy2i
t− hyi2t obeys,
dσ2
dt =ha2(y)it+ 2h(y−y¯)a1(y)it. (6.18) Again, if a1anda2 arelinear in y, then Eq. 6.18 is identical with,
dσ2
dt =a2(¯y) + 2σ
2a0
1(¯y), (6.19)
though in general, Eq. 6.19 will be only an approximation. This equation for the variance may now be used to compute corrections to the macro- scopic equation, Eq. 6.17,
d dty¯(t) =a1(¯y) + 1 2σ 2a00 1(¯y), (6.20) dσ2 dt =a2(¯y) + 2σ 2a0 1(¯y). (6.21)
Note that, by definition, a2(y)>0, and for the system to evolve toward
a stable steady state, one needsa01(y) <0. It follows that σ2 tends to increase at the rate a2, but this tendency is kept in check by the second
term; hence,
σ2→ a2
2|a01|, (6.22)
and so the condition for the approximate validity of Eq. 6.17 – namely that the 2ndterm in Eq. 6.20 be small compared to the first – is given by,
a2 2|a01| 1 2|a 00 1| |a1|, (6.23)
which says that it is the second derivative ofa1(responsible for the depar-
ture from linearity) which must be small. The linear noise approximation, which we shall consider in Chapter 4 (Section 5.1) provides a more satis- fying derivation of the macroscopic evolution from the master equation, proceeding as it does from a systematic expansion in some well-defined small parameter.
6.3.1
Coefficients of the Fokker-Planck Equation
The coefficients of the Fokker-Planck equation, ∂ ∂τp(y, τ)≈ − ∂ ∂y[a1(y)p(y, τ)] + 1 2 ∂2 ∂y2[a2(y)p(y, τ)], (6.24)
are given by,
a1(y) = ∞ Z −∞ ∆yw(y,∆y)d∆y a2(y) = ∞ Z −∞ ∆y2w(y,∆y)d∆y.
In theory, they can be computed explicitly sincew(y,∆y) is known from an underlying microscopic theory. In practice, that is not always easy to do, and an alternative method would be highly convenient. The way one usually proceeds is the following:
Letp(y, t|y0, t0) be the solution of Eq. 6.24 with initial conditionδ(y−
y0). According to Eq. 3.4, i.e.,
p(y2, t2) = ∞
Z
−∞
p(y2, t2|y1, t1)p(y1, t1)dy1,
one may construct a Markov process with transition probabilityp(y2, t2|y1, t1)
whose one-time distributionp(y1, t1) may still be chosen arbitrarily at one
initial time t0. If one chooses the “steady-state” solution of Eq. 6.24,
ps(y) = constant a2(y) exp 2 y Z a 1(y0) a2(y0) dy0 , (6.25)
then the resulting Markov process is stationary. This, of course, is only possible ifpsis integrable. For closed systems, Eq. 6.25 may be identified
with the equilibrium probability distribution peq known from statistical
mechanics.
To “derive” the coefficients of the Fokker-Planck equation, we proceed as follows. From Eq. 6.24, the macroscopic equation is,
d
dthyit=ha1(y)it≈a1(hyit),
(neglecting fluctuations). Since this equation must coincide with the equation known from the phenomenological theory, the function a1 is de-
termined. Next, we know peq from statistical mechanics, identified with
Eq. 6.25; hencea2is also determined. Note the various (for the most part
uncontrolled) assumptions made in this derivation.
The simplest of all procedures, however, is the derivation based upon the Langevin equation. Recall that this begins with the dynamical de- scription,
dy¯
dt +βy¯=f(t) (6.26)
with assumptions about the statistics of the fluctuating force f(t),
hf(t)i= 0,hf(t1)f(t2)i= 2Dδ(t1−t2). (6.27)
Integrating Eq. 6.26 over a short time interval,
∆¯y=−βy¯∆t+
t+∆t
Z
t
so that, a1(¯y) = lim ∆t→0 1 ∆th∆¯yi=−βy¯+ lim∆t→0 1 ∆t t+∆t Z t :0 hf(ξ)idξ Similarly, D (∆¯y)2E=β2y¯2(∆t)2+ t+∆t Z t Z hf(ξ)f(η)idξdη,
from which it follows – using Eq. 6.27– that a2(¯y) = 2D,
and the corresponding Fokker-Planck equation reads, ∂p ∂t =β ∂ ∂y(yp) +D ∂2p ∂y2. (6.28)
This coincides with the appropriate Fokker-Planck equation for the posi- tion of a Brownian particle in the absence of an external field, as derived by Ornstein and Uhlenbeck.
The identification ofa1(y) as the macroscopic rate law is really only
valid when a1(y) is a linear function of y – as it is in the Ornstein-
Uhlenbeck process described above. For nonlinear systems, derivation of the Fokker-Planck equation in the manner described here can lead to serious difficulties, and so the systematic expansion scheme described in Section 5.1 is indispensable. See N. G. van Kampen (1965) Fluctuations in Nonlinear Systems.