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Statistical mechanics of a dissipative dynamical system with small noise
PHYSICA ELSEVIER
Physica D 122 (1998) 105-116
Statistical mechanics of a dissipative dynamical system with small noise Melvin E. Stem 1 Department of Oceanography, Florida State University, Tallahassee, FL 32306-4320, USA Received 19 February 1998; received in revised form 5 May 1998; accepted 8 May 1998 Communicated by EH. Busse
Abstract
A "wagon" rolls up a slightly inclined plane (effective gravity acceleration = g*), and then it rolls down until it collides inelastically (coefficient of restitution = 1 - e) with a piston oscillating parallel to the plane between ~ = 0 and ~ = ~rnax and with velocity vf(cot), where v is amplitude, co is frequency, and f is a specified waveform (e.g., sinusoidal). In the high frequency limit (rio~g* ~ oo) there are very many periodic solutions, but in the presence of a small noise level (e.g., the standard deviation in e) none of these are realizable because, when 09 ~ oo, this noise is assumed to introduce a random phase o~tn --- ~bn(mod 27r) of the piston each time (tn) the wagon arrives at ~ ----~m~xmoving down-hill with velocity V. For mathematical simplicity we confine our attention to the case of a square wave oscillator ( f = +1), and derive an evolution equation for the probability distribution function Pn+l (V). Numerical solutions reveal the approach to a unique statistically steady state (n ~ oo) independent of the assumed initial distribution P1 (V), and independent of the noise level. © 1998 Elsevier Science B.V. Keywords: Bouncing ball problem; Small noise asymptotics
1. Introduction
Highly nonlinear fluid dynamical systems may have very many solutions satisfying the Navier-Stokes equations with the same boundary conditions, and the number of such solutions may increase with the control parameter (e.g., the Reynolds number or the Rayleigh number). In the differentially rotating cylinder experiment [1], for example, more than one solution can be realized by manipulating initial conditions. In view of this, and the initial value structure of the Navier-Stokes equations, the fundamental question arises as to why a unique statistically steady state is realized in fully turbulent flow [2,3], when the control parameter becomes very large. One notion is that small and unpredictable "noise" disturbances always present (e.g., at the boundaries) in a real flow are randomly amplified in locally unstable regions, thereby producing finite amplitude effects which enable the system to "forget its history". The purpose of this paper is to illustrate the effect in a simpler but realizeable dynamical system (Fig. l(a)), in which the small noise produces statistical uniqueness. 1E-mail: [email protected]. 0167-2789/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S0167-2789(98)00175-4
106
M.E. Stern/Physica D 122 (1998) 105-116
•
(b)
(c)
m
t= tn
F~=O
t='i'n
~=0
t="~n
(d)
t= tn
"4~'
~=0
(0
t='~n
Fig. 1. Schematic diagram. (a) A wagon rolling down an inclined plane is about to collide with an oscillating piston. (b) The wagon is shown arriving at the outer range (~m) of the displacement (0 < ~ < ~m) of the piston executing very high frequency (to) oscillations with fixed maximum velocity v. At this time the phase of the piston is torn,its uphill velocity is vf(totn), and Vn is the downhill velocity. (c) Shortly thereafter (t = tn) the wagon collides with the piston at ~ = ~, and Vn~ denotes the resulting uphill velocity, which depends on the coefficient of restitution (1 - e). (d) For this arrival at ~m the piston is moving downhill at tn. (e) The collision is rear-end. (f) Vn~ may be so small or negative that a second collision (see text) occurs before the wagon reaches ~ = ~m. Otherwise the wagon in Fig. l(c) rolls uphill and downhill with acceleration g*, and arrives at the outer range with velocity Vn+l = Vrn at time tn+l. The main statistical assumption is that if g*(vto) -1 ~ oo, with a fixed noise level (.4) however small, then the arrival phase tOtn (mod 2zr) in Fig. l(b) is equally likely to lie anywhere between 0 and 2zr.
M.E. Stern/Physica D 122 (1998) 105-116
107
The statistical problem (Fig. l(a)), partially considered previously [4], consists of a wagon rolling on an inclined plane and colliding intermittently and inelastically with a piston executing oscillations with very high frequency (w), very small maximum displacement ~m, and specified velocity
vf(wt),
(1)
where t is time, f denotes the waveform of period 2zr (e.g., f(wt) = sin o90 for a sinusoidal oscillator, and f = ±1 for a square wave oscillator); the average coefficient of restitution is (1 - e). An essentially equivalent problem is that of a ball bouncing on an oscillating horizontal plate, which for small ~o leads to a deterministic chaotic oscillator [5]; and statistical aspects of the dissipative collisional process have also been investigated [6,7], but not in the context 2 of predicting the wagon motion for deterministic forcing (1) with small noise. Before turning to this problem it is important to review [4] the dense degeneracy of the deterministic solutions. Let V, denote the downhill velocity of the wagon immediately before the nth collision with the piston at the time ~, so that fin + vf(COtn) is the relative velocity, and immediately after the collision (1 - e)(r/n + vf ) + v f is the absolute uphill velocity of the wagon. Consider a steady solution f'n which is larger than v, so that the wagon will roll uphill and return to the piston at collision n + 1 with the same velocity that it left. Then the fixed points in the phase space If',, ~Otn(mod(Zzr))] are given by Vn+l = f'n = U, and O)tn+ 1 -- ogtn =
2JrN,
N = integer,
(2)
or
cotn = 2rr Nn + or,
(3)
where a is an arbitrary constant and n is an integral constant. The steady collision dynamics require U = (1 - e)U + (2 - e)vf(~otn), and the time to roll uphill and downhill is tn+l - tn = 2U/g*. From these it follows that
rre ( g ~ - ~ ) N = f ( 2 z r N n + o t ) = f ( o t ) .
(4a)
2-e Since ff(~)[ < 1 (n.b. the sinusoidal oscillator where f ( c 0 = sinot) we see that if R =- vw/g* >> 1, there are a very large number of integers N, and corresponding phase angles a ( N ) which satisfy Eq. (4a), and the wagon velocity for each such solution is U = v((2 - e)/e)f(ot). Eq. (4a) shows that the number of discrete solutions for ot becomes dense when o~ --+ oo. We now recognize the physical complexity of an actual collision by introducing (say) a small standard deviation A for the coefficient of resitution, so that the uncertainty of the time interval between two collisions is of order yeA/g*, and the uncertainty of the piston phase before collision is of order cove A/g* = ReA >> 2~r when R ~ with fixed Zi << e. Under these conditions it is clear that none of the foregoing phase locked deterministic solutions are realizable, and that we must formulate a statistical problem; the same is true if the fluctuations occur in the high frequency (w). Let ~ -----~max denote the "outer range" of the piston displacement (Fig. 1(b)), and let Vn denote the downhill velocity of the wagon when its nose arrives at the outer range for the nth time; this occurs at time tn, and ~bn = wtn (mod 270 denotes the phase. When R ~ o~, with fixed v, it is assumed, as in previous work [4,7], that ~b,, is randomly and uniformly distributed in 0 < 4~ < 2re. Thus we see that because the piston executes a very large number of oscillation between each arrival of the wagon at the outer range and because of the "noise", the uncertainty of the piston phase is very large. The nearly elastic (e << 1) problem was previously considered for any periodic 2 In the context of the problem of oscillating granular particles the authors of [7] assume a Gaussian velocity distribution for Vn, and confine themselves to computing the average dissipation rate.
M.E. Stern/Physica D 122 (1998) 105-116
108
.0
~o+ 5 I
+
+
, I
+ +
~o+~ f---1
+
I
--
Zn
--
8 ~ = 2 -
Zn(1- ~:)
~o+ 5
I
I
4 - 2E
I
I
+
+
+
+
+
+
+
+
+
+
f=+l
V n Zn= --
V
2-e
3-e
2-e
1-E
1-e
e
Fig. 2. Regime diagram for Vn/v, and phase (ordinate). In the lower half ( f = -1) of the diagram only head-on collisions (+) occur. In the upper half (f = -1), 1 - ~/(~0 + 8) is the fractional distance of the piston from its origin (~ = 0) at the time when the wagon is at outer range (~0 + ~). Single rear-end collisions occur in the region marked with (-), and double collisions occur in the triangular region labeled "dbr'. At any Zn -- Vn/v the probability of each type of collision is proportional to the length of the vertical line segment lying in each of the three regions (see text). waveform f , and it was shown [4, p. 425] that in the statistically steady state the ensemble average mean square arrival velocity is given by 2zr
(v~)
4V2 f = - j
2 dq~ f (~b)~-.
(4b)
g 0
Attention is now directed to the question of statistical uniqueness at finite e, and for this purpose it seems sufficient to consider the mathematically simplest problem of the square wave oscillator with f either equal to + 1 or - 1; in this case the random phase assumption is equivalent to: all ~0 are equally probable if f = + l ,
(5)
where ~0 is the displacement (Fig. l(b)) of the piston when the wagon arrives at ~m; the piston is also equally likely to be moving uphill or downhill at this tn. The elementary calculations in Sections 2 and 3 wiU give the phase space diagram (Fig. 2) from which we obtain the transition probabilities and the evolutionary equation (Section 4) for the probability Pn ( V ) d V that the wagon
M.E. Stern/Physica D 122 (1998) 105-116
109
velocity lies between V and V + dV on the nth downhill arrival at tm. The recursion equation for Pn will be solved numerically for e = 0.1, and for different initial conditions (P1), in order to answer the questions as to whether a single P ~ (V) exists independent of the assumed Pl (V).
2. Collisions for the square wave piston oscillator For reasons mentioned previously, f = 4-1 in Fig. l(b) and all values of t0 are equally probable, so that the piston is equally likely to be moving uphill with velocity = + v or downhill with velocity = - v . Let f denote the value of f and q~n = torn the value of the phase (mod 2:r) at collision (Fig. l(c)). In the oJ ~ c~ limit the time interval tn - tn is so small that the fractional change in the downhill wagon speed Vn is negligible compared to the change (~ - t0) in t , and therefore the relative downhill wagon speed immediately before collision is Vn + v f . Immediately after collision the relative uphill speed is (1 - s)(Vn + v f ) , and thus the absolute uphill speed is V,I = (1 - e)Vn + (2 - e ) v f .
(6)
If this is sufficiently large compared to v (as specified in Section 3) there will be no second ("double") collision before the wagon crosses tm, in which case it moves uphill, reverses, and returns to the outer range (tin) at time tn+l with downhill speed Vn+l = Wn, i.e., V~+I = (1 - e)V, + (2 - s ) v f
(for no double collision).
(7a)
Unlike f, f is correlated with Vn and non-uniformly distributed, as is readily seen (Figs. l(d) and (e)) by the fact that if f = - 1 the downhill moving piston might reverse before collision, giving f = + 1; thus head-on collisions are more likely than rear-enders. Let us now consider the double collisions which can occur if Vn > v and f = - 1. After the first collision we have V~ = (1 - e)Vn + (2 - e ) v ( - 1 ) ,
(7b)
and if V~ < v, then the piston might reverse and overtake the wagon before it reaches tin. In this case the relative speed of the wagon before the second collision is v - Vnr > 0. After this collision the relative uphill speed is ( 1 - e ) (v - Vn~), and the absolute uphill speed (Fig. 1(f)) of the wagon is V~' = - ( 1 - e ) 2 gn -~-( 1 q- g) [ (3 -- g ) 1)] Jr- I). Since this exceeds v no further collision occurs, and therefore the downhill speed at the next "arrival" is
Vn+l = - ( 1 - e)Zvn + 4v - 4ev + eZv
(for a double collision at Vn > v).
(8)
3. The head-on, rear-end, and double collision domains If G < (2 - e)v/e for any n (say the initial value) then (7a) implies Vn+l < Vn and the state of the entire system is determined by a point in the two-dimensional space: 0 < V < (2 - e)v/e, 0 < ~ n ~ 27g. I f 0 < ~n < ~ , the piston is moving uphill and only a head-on collision can occur, as denoted by the + in the lower half of the regime diagram (Fig. 2). The same is true for Vn < v at Jr < ~n < 2rr because the downhill moving piston ( f = - 1) must reverse before it collides with the slower wagon. It only remains to consider the values of f when z =- Vn/v > 1 and when f = - 1; in which case it is convenient to express the ordinate in the upper half of Fig. 2 in terms of -- tmax - t0 (rather than q0, with all values of 3/(t0 + 8) being equally probable. Consider first the "farfield" region 3-e
Vn > 1 - e
(9)
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M.E. Stern/ Physica D 122 (1998) 105-116
For either f = - 1 or f = + 1 Eqs. (6), (7b) and (9) imply that after the first collision v~' > v, so that no second collision can occur. As shown in Fig. 2 only head-on ( + ) and rear-end ( - ) collisions can occur for Vn > (3 - e)/(1 - e). In this region the probability of a rear-end collision ( f = - 1) equals 1/2 times the probability that the piston does not reverse before the impending collision. The latter condition requires that the downhill-distance V(tn - tn) traveled by the piston before collision is less than t0 (see Fig. l(e)). During this time interval the constant relative velocity Vn - v > 0 causes the separation between piston and wagon to decrease from 8 = ~m - t0 to zero. Since 8 = (Vn - v)(tn - tn), the condition for non-reversal of the piston is v[8/(Vn - v)] < t0, or 8/(50 + 8) < 1 - v / V n . Since all values of 8/(50 + 8) are equally probable, and since all values of 8/(50 + 8) between zero and (1 - v / V n ) give f = - 1, we conclude that (1 - v / V n ) is the conditional probability for f = - 1, and the absolute probability of a rear-end collision at Vn is r = 1/2(1 - v / V n )
for f = - 1 .
(10a)
This collision occurs below the curve 8/(50 + 8) = 1 - 1/Zn (Fig. 2) and above 8 = 0. At all values of Vn the region above the curve is occupied by head-on collisions, so that the absolute probability of the latter is r = 1/2(1 + v / V ~ )
for j3 = +1 (all Vn > v).
(10b)
Consider next the region 1 < Vn <
2-e 1-e
v,
f(~b)=-l.
(11)
Since the first collision is rear-end, Eq. (7b) gives the absolute uphill velocity V~' = (1 - e)V~ - (2 - e)v < 0, so that the wagon is actually moving downhill, and a double collision must occur. Eq. (8) then gives Vn+l, and the probability of this is given by the relative length of a vertical line (Fig. 2) in the "dbl" region between 8 = 0 and ~/(~o + 8) = 1 - 1/zn.
It only remains to consider the region Vn ~
2-e 1--e
i),
f(~b) = - - 1 ,
(12)
in which either a rear-end or double collision can occur. As previously mentioned tn - tn = 8/(Vn - v) > 0 gives the time interval for the first collision (at g), after which the uphill velocity is Vn' = (1 - e) Vn - (2 - e)v > 0. The necessary and sufficient condition that this be a single rear-end collision is that the wagons reach ~ = ~m = to + in a time ((to + 8) - ~ ) / V n' shorter than that required for the piston to go downhill from ~ to ~ = 0, and to then go uphill to ~m = t0 + 8. Thus the latter time (~ + (t0 + 8 ) ) / v must exceed the former time, and therefore a single rear-end collision requires (1
+
V~)
+(50+8)
(~
1)
Vn' > 0 .
In Figs. l(d) and (e) we see that at the first collision to - g = v ( ~ - t~), = vS/(V,, - v). By eliminating g and Vn~ = (1 - e)Vn - (2 - e ) v from the preceding inequality, and by simplifying we get 8
- -
50+8
< 2
(4 - 2 e ) v (1 -- E)V n
(single rear-end).
(13)
The bounding curve (Fig. 2) for this, obtained by replacing the inequality sign in (13) by an equal sign, intersects the 8/(50 + 8) = 1 - v / V n curve at V ~ / v = (3 - e)/(1 - e) which is the boundary of Eq. (9). Thus, we conclude that the "dbl" region between these two curves is occupied by double collisions, while the region below (13) and
M.E. Stern/Physica D 122 (1998) 105-116
111
above 8 = 0 is occupied by rear-end collisions. The probability of each kind of collision is given by the relative length o f the intercepted vertical line. The value of Vn+l in each region is obtained from Eq. (7) or (8).
4. P r o b a b i l i t y
evolution
equation
Consider a "range ordered" ensemble in which Pn (V) dV denotes the total number of wagons with downhill velocity between V and V + dV that arrive at ~m for the nth time. A given value of z = Vn+l/v can occur either by a head-on collision originating from some ZH -------Vn/v, or by a single rear-end collision originating from ZR = Vn/v, or by a double collision originating from ZD = Vn/v; where these respective z-values obtained from (7a) and (8) are z - (2 - e) 1 -e '
dz dZH -- 1 - e
> 0,
(14)
ZR --
z + (2 - e) 1 --e '
dz dZR -- 1 - - e > 0,
(15)
ZD=
--z + 4 - - 4e + e 2 (l--e) 2 '
dz I dZDI-- ( l - - e ) 2 > 0
ZH
-
( Z D > 1).
(16)
If TR (ZR), TH (ZH), TO (ZD) denote the respective transition probabilities to z, and if Pn (z) dz is the fractional number of wagons in the ensemble at collision n that are between z and z + dz, then the distribution function at collision n+lis
Pn+l (Z) dz = TH(EH)Pn(ZH)dZH + TR(ZR)Pn(ZR) dZR + TD(ZD)Pn(ZD)I dZD[,
(17)
TH(ZH)Pn(ZH) TR(ZR)Pn(ZR) TD(ZD)Pn(ZD) + + 1 -- e 1 -- e (1 -- e ) 2
(18)
Pn+l(Z) -
If z > 3 - 2e then the value of ZD in Eq. (16) is smaller than unity, and therefore double collisions cannot produce such large z; the only ones that can are head-on and rear-end collisions. For these the transition probabilities, as mentioned in Section 3, are given by the relative lengths of vertical lines intercepted by the curves in Fig. 2, i.e., for3-2e
(2-e)/e 1
1 ZH > 1
-, TH(ZH) = ~ + -2ZH TR(ZR) - -
5 - 3e
1
1
2
2ZR'
ZR > - - ,
TD = 0.
1 -- e
(19)
For smaller z the three transition probabilities obtained from Fig. 2 are
TH(ZH)
/0'
0
/
ZH<0,
1,
1,
0, 1 TR(ZR) = ~
(20a)
I
2
4--2e ZR(1 -- e ) '
1 -
--,
1
ZR
2--e -1 ---
3--e
- -
1 --e
2--e 1--e 3 --e < 1- -- e ' -5--3e
< ZR < - - ,
1 -e
(20b)
112
M.E. Stern/Physica D 122 (1998) 105-116
2--e
1 1--
ZD 1
TD(ZD) = ~
1-e
I
--
1
1 -- - - -ZD
(2
ZD(1--e)]'
- -
l--e--
(20C)
< ZD < - -
-- 1 - - e '
3--e ZD> 1 - - e
0,
We may, without loss of generality, restrict the initial P1 (Z) to values 0 < z < (2 - e)/e. Then the value of PE(z) at any z is obtained by first computing (14)-(16), then Eqs. (19) and (20) and then (18). Note that for any Vn -- v( the transition probabilities in (20) are conservative in the sense TH(() + TR(() + TD(¢) = 1 so that the number of wagons is the same from one n to the next, i.e., the solution of (18) satisfies (2-e)/e
f
(2-e)/e
Pn+l(()d(=
0
f
Pn(()d(.
0
5. A n a l y t i c s o l u t i o n f o r t h e s t e a d y
Pn w h e n
e --~ 0
According to (4b) the rms z is O(e-1/2), and it is of interest to examine the probability density at these large velocities for which TD = 0. The steady state (Pn+l (z) = Pn(z) -- P(Z)) version of Eqs. (18)-(20) for z >_ 3 - 2e reduces to 0= -P(z) + 2
-l~-e
+ ~Pn
z-(2-~)
i
1- e
e
z+(2_e)e
]-~-
j.
(21)
Anticipating z = 0(e-1/2), the e -+ 0 expansion of these terms, is
p
z-2+e
P
- 1-e
T-~-
=P(z+ez-2+O(e))=P(z)+(ez-2)P'(z)+-~P"(z)+O(e3/2),
= P(z+ez+2+O(e))
(1 -- e)/2 p l / Z - 2 + e ' ~ z-(2-e)
~,]l-e
1/2(1 + O(e3/2)) . . . .
. . . .
_ ( 1+- e()2/ -2 z e ) P ( Z - ~ 2 - e ) - - e
= P(z)+(ez+2)P'(z)+-=-P"(z)+O(e3/2),
(z-)
-~
trl.z) - 2P'(z) + O(e)],
-1/2(l+O(e3/2))[P(z)-2P'(z)+O(e)]'(z +2)
so that Eq. (21) becomes 2P(z)
p,(z)(
0 = ezP'(z) -t- 2P"(z) + eP(z) + z2 ----4
1
~-2
+ ~
1 )
+ O(e3/2)"
(22)
Now let z = ( e -1/2, retaining these leading [O(e)] terms, to obtain the e -+ 0 limit
o=
-
,".
(23)
M.E. Stern/Physica D 122 (1998) 105-116
113
For the solution of this steady-state asymptotic equation we require that P --+ 0 for z/,-1/2 >> 1 and for z / , 1/2 << 1, i.e., OO gl
P(O) = P(
/
(24)
1.
d~'P(~')
o As may be verified directly by substitution, the only one of the two solutions of (23) that can satisfy the latter condition is p(~) = ~e-¢2/4. 2
(25)
If the normalized energy (E = g2/2) distribution function is 12(E), i.e. I2(E) dE --= P(g) dg, then I2(E) = l e - e / 2 . The means that the asymptotic spectrum CX)
maximizes
[ 12(E) In I2 dE,
(26)
t/
0
with
fo
EI2(E) dE = 4, and
fo t2(E) d e
= 1.
6. F i n i t e , evolution
For a n y , and any initial condition (Pl(V)) Eqs. (14)--(20) can be used to compute P2, P3 . . . . . but an exact solution must involve the distribution in the entire z-continuum, whereas a numerical approximation only involves a finite number of grid points zj (J = 1, 2, 3 . . . . . Jmax) separated by a small distance S. Although the initial distribution P1 (z j) may be exactly specified at such points, the computation of P2(zs) requires data at the values of ZH, ZR, ZO (Eqs. (14)-(16)) which are not integral multiples of S, and for which P is not defined. Therefore we approximate these values by those that are defined at the next highest zj. For any z j = z in Eq. (16) let .
.{zj
JH=l+mt~
- (2- ,)\ S(1-,))'
. .{zg
,,=l+mt~
JD = l + I n t ( - z J S(1 + 4 -~e--'~-4" +'2) ,
+ ( 2 - ,)'~ S0-e))'
(27)
(ZJH, ZJR, ZJD) -~ S(JH, JR, Jo)-
These are the approximate values of (ZH, ZR, ZD) from which Pn+l (zg) is computed from Eqs. (18)-(20). First we consider an initial distribution (P1) in the form of an isosceles triangle: z
Pl(Z)
Zm~x/2
for z < Zmax/2,
1
for z > Zmax/2,
=
(28) z - (Zmax/2) Zmax/2
for Zmax = (2 - , ) / , , , = 0.1, S = 1/18. The arbitrary amplitude (100) is such that the total number of wagons, summed over all J, is 1.71000 x 104. In the subsequent evolution (Fig. 3) this total remains constant to four significant figures, and the main numerical source of error is in the "binning" process (Eq. (27)). The ensemble average z, which is proportional to the average time interval between two successive arrivals at the outer range (essentially the inverse collision frequency), decreases from 9.5 at n = 1, and at n = 21 (Fig. 3) a virtual steady state is reached, as indicated by the plots of both (z/and (z 2} in Fig. 5.
M.E. Stern/Physica D 122 (1998) 105-116
114 150.0
10010
N=20,21
50.0
.0 .o
s.o
o.o
5.o
2o.
z =V/v Fig. 3. The probability distribution Pn (z) for e = 0.1 obtained from the finite difference equation using Eq. (28) as initial condition. The absolute value of the ordinate has no significance. The curves for collision N = 20 and N = 21 coincide.
In order to examine the question of uniqueness, the calculation was repeated using an extremely different initial state in which P1 (z) was constant in 1 < z < 2, and zero elsewhere. For small n (not shown) a very "gappy" and ragged Pn (z) develops. As expected, the binning error due to the discontinuities is significant at small n, so that the number of wagons is not conserved, but for n > 16 the function Pn(z) starts to smooth and the number of wagons (1.733 x 104) then remains constant to four figures (Fig. 4). The values of (z) and (z 2) for this run are seen (Fig. 5) to be approaching the same asymptotes as the corresponding values in the previous run, and this strongly suggests that the solution of (14)-(18) for P ~ ( z ) is unique and independent of initial conditions. We estimate that for e = 0.1, (z) = 5.36 4- 0.02; also note that the value of (z 2) - 36 is somewhat less than the asymptotic (e - * 0) value (z 2) = 4 / e = 40 using e = 0.1. It is worth noting that the maximum entropy property (26) does not hold for e = 0.1, because the steady P is not • --z2/a 2 • 3 2 similar t o E q . (25). I f i t was, i.e., if P(z) = z e for some a, then lt would follow that (z) = a ~/-~/4, ( z ) = a4/2, and (2z2)3/4(z) -1 = 2.25. On the other hand, our numerical calculations for Figs. (4) and (5) at n = 21 give the much large value of 4.65 so that the corresponding 12 (E) is not a simple exponential function, and the entropy is not maximal if the energy is fixed. Perhaps some more general variational principle for the dissipative system might be found.
M.EI Stern/Physica D 122 (1998) 105-116
150.0
115 I
I
I
l
l
l
100.0
50.0
N=22,23,24,25
.0 .0
5.0
10.0
15.0
20.0
z:V/v
Fig. 4. Same as Fig. 3 except that P1 (z) is constant in 1 < z < 2 and zero elsewhere. The distribution for collisions 22-25 are shown.
7. Conclusion The rolling on an inclined plane of a wagon which intermittently and inelastically collides with a sinusoidally oscillating piston of very high frequency (vto/g* >> 1) is characterized by a dense distribution of phase-locked deterministic solutions, but none of these could be realized in an actual experiment due to the presence of small amplitude noise. This can be produced by many factors, such as a "small but finite" standard deviation (A) in the coefficient of restitution (e) which results in large uncertainties in the piston phase ~n = ogtn [mod(2rr)] at each (n) arrival of the wagon at the "outer range". Thus we assumed that in the high frequency limit this phase should be uniformly random in 0 _< ~ _< 2zr. In order to show that the statistically steady solution is unique we solved the problem for the mathematically simpler case of a square wave oscillator ( f = 4-1). For e << 1 the solution of the general evolutionary equation (18) yields a steady probability distribution function which maximizes the entropy for the average energy, but the main new result of this paper is the numerical solution (Figs. 3 and 4) for finite e which yields a statistically steady distribution function independent of initial conditions (as well as the small noise level (A)). Fig. 5 gives the evolution of the mean square wagon energy (proportional to (z2)), and the collision frequency (proportional to (z)). From this the ensemble statistics can be converted to long time averages for a single realization (Fig. l(a)). In this sense our system behaves ergodically.
M.E. Stern/Physica D 122 (1998) 105-116
116 6.0
I
'
I
'
I
'
I
'
I
'
I
i
I
I
i
5.5
5.0
4.5
4.0
~"-3.5
I
I
9
I
I
11
I
I
13
t
I
I
15
I
17
i ,.,---'-'t-----'T~
19
I
21
23
25
N Fig. 5. Plots of (z) (upper two curves) and (z2) as a function of N for the PN(Z) used in Figs. 3 and 4. The results for Fig. 3 approach the asymptote from above, while the results for Fig. 4 approach the asymptote from below. The agreement of the asymptotes for the two different initial conditions suggest statistical uniqueness.
Acknowledgements I gratefully acknowledge the financial support for this work provided by the National Science Foundation under grants OCE-9216319 and OCE-9529261. References [1] [2] [3] [4] [5] [6] [7]
D.J. Coles, Fluid Mech. 21 (1965) 385. A.J. Roshko, Fluid Mech. 10 (1961) 345. R. Krishnamurti, J. Fluid Mech. 42 (1970) 309. M.E. Stem, J. Math. Phys. (SIAM) XLVII (1968) 416. P.J. Holmes, J. Sound and Vibration 84 (1982) 173. L.A. Wood, K.P. Byrne, J. Sound and Vibration 78 (1981) 329. S. Warr, J.M. Huntley, Phys. Rev. E 52 (1995) 5596.
Physica D 122 (1998) 105-116
Statistical mechanics of a dissipative dynamical system with small noise Melvin E. Stem 1 Department of Oceanography, Florida State University, Tallahassee, FL 32306-4320, USA Received 19 February 1998; received in revised form 5 May 1998; accepted 8 May 1998 Communicated by EH. Busse
Abstract
A "wagon" rolls up a slightly inclined plane (effective gravity acceleration = g*), and then it rolls down until it collides inelastically (coefficient of restitution = 1 - e) with a piston oscillating parallel to the plane between ~ = 0 and ~ = ~rnax and with velocity vf(cot), where v is amplitude, co is frequency, and f is a specified waveform (e.g., sinusoidal). In the high frequency limit (rio~g* ~ oo) there are very many periodic solutions, but in the presence of a small noise level (e.g., the standard deviation in e) none of these are realizable because, when 09 ~ oo, this noise is assumed to introduce a random phase o~tn --- ~bn(mod 27r) of the piston each time (tn) the wagon arrives at ~ ----~m~xmoving down-hill with velocity V. For mathematical simplicity we confine our attention to the case of a square wave oscillator ( f = +1), and derive an evolution equation for the probability distribution function Pn+l (V). Numerical solutions reveal the approach to a unique statistically steady state (n ~ oo) independent of the assumed initial distribution P1 (V), and independent of the noise level. © 1998 Elsevier Science B.V. Keywords: Bouncing ball problem; Small noise asymptotics
1. Introduction
Highly nonlinear fluid dynamical systems may have very many solutions satisfying the Navier-Stokes equations with the same boundary conditions, and the number of such solutions may increase with the control parameter (e.g., the Reynolds number or the Rayleigh number). In the differentially rotating cylinder experiment [1], for example, more than one solution can be realized by manipulating initial conditions. In view of this, and the initial value structure of the Navier-Stokes equations, the fundamental question arises as to why a unique statistically steady state is realized in fully turbulent flow [2,3], when the control parameter becomes very large. One notion is that small and unpredictable "noise" disturbances always present (e.g., at the boundaries) in a real flow are randomly amplified in locally unstable regions, thereby producing finite amplitude effects which enable the system to "forget its history". The purpose of this paper is to illustrate the effect in a simpler but realizeable dynamical system (Fig. l(a)), in which the small noise produces statistical uniqueness. 1E-mail: [email protected]. 0167-2789/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S0167-2789(98)00175-4
106
M.E. Stern/Physica D 122 (1998) 105-116
•
(b)
(c)
m
t= tn
F~=O
t='i'n
~=0
t="~n
(d)
t= tn
"4~'
~=0
(0
t='~n
Fig. 1. Schematic diagram. (a) A wagon rolling down an inclined plane is about to collide with an oscillating piston. (b) The wagon is shown arriving at the outer range (~m) of the displacement (0 < ~ < ~m) of the piston executing very high frequency (to) oscillations with fixed maximum velocity v. At this time the phase of the piston is torn,its uphill velocity is vf(totn), and Vn is the downhill velocity. (c) Shortly thereafter (t = tn) the wagon collides with the piston at ~ = ~, and Vn~ denotes the resulting uphill velocity, which depends on the coefficient of restitution (1 - e). (d) For this arrival at ~m the piston is moving downhill at tn. (e) The collision is rear-end. (f) Vn~ may be so small or negative that a second collision (see text) occurs before the wagon reaches ~ = ~m. Otherwise the wagon in Fig. l(c) rolls uphill and downhill with acceleration g*, and arrives at the outer range with velocity Vn+l = Vrn at time tn+l. The main statistical assumption is that if g*(vto) -1 ~ oo, with a fixed noise level (.4) however small, then the arrival phase tOtn (mod 2zr) in Fig. l(b) is equally likely to lie anywhere between 0 and 2zr.
M.E. Stern/Physica D 122 (1998) 105-116
107
The statistical problem (Fig. l(a)), partially considered previously [4], consists of a wagon rolling on an inclined plane and colliding intermittently and inelastically with a piston executing oscillations with very high frequency (w), very small maximum displacement ~m, and specified velocity
vf(wt),
(1)
where t is time, f denotes the waveform of period 2zr (e.g., f(wt) = sin o90 for a sinusoidal oscillator, and f = ±1 for a square wave oscillator); the average coefficient of restitution is (1 - e). An essentially equivalent problem is that of a ball bouncing on an oscillating horizontal plate, which for small ~o leads to a deterministic chaotic oscillator [5]; and statistical aspects of the dissipative collisional process have also been investigated [6,7], but not in the context 2 of predicting the wagon motion for deterministic forcing (1) with small noise. Before turning to this problem it is important to review [4] the dense degeneracy of the deterministic solutions. Let V, denote the downhill velocity of the wagon immediately before the nth collision with the piston at the time ~, so that fin + vf(COtn) is the relative velocity, and immediately after the collision (1 - e)(r/n + vf ) + v f is the absolute uphill velocity of the wagon. Consider a steady solution f'n which is larger than v, so that the wagon will roll uphill and return to the piston at collision n + 1 with the same velocity that it left. Then the fixed points in the phase space If',, ~Otn(mod(Zzr))] are given by Vn+l = f'n = U, and O)tn+ 1 -- ogtn =
2JrN,
N = integer,
(2)
or
cotn = 2rr Nn + or,
(3)
where a is an arbitrary constant and n is an integral constant. The steady collision dynamics require U = (1 - e)U + (2 - e)vf(~otn), and the time to roll uphill and downhill is tn+l - tn = 2U/g*. From these it follows that
rre ( g ~ - ~ ) N = f ( 2 z r N n + o t ) = f ( o t ) .
(4a)
2-e Since ff(~)[ < 1 (n.b. the sinusoidal oscillator where f ( c 0 = sinot) we see that if R =- vw/g* >> 1, there are a very large number of integers N, and corresponding phase angles a ( N ) which satisfy Eq. (4a), and the wagon velocity for each such solution is U = v((2 - e)/e)f(ot). Eq. (4a) shows that the number of discrete solutions for ot becomes dense when o~ --+ oo. We now recognize the physical complexity of an actual collision by introducing (say) a small standard deviation A for the coefficient of resitution, so that the uncertainty of the time interval between two collisions is of order yeA/g*, and the uncertainty of the piston phase before collision is of order cove A/g* = ReA >> 2~r when R ~ with fixed Zi << e. Under these conditions it is clear that none of the foregoing phase locked deterministic solutions are realizable, and that we must formulate a statistical problem; the same is true if the fluctuations occur in the high frequency (w). Let ~ -----~max denote the "outer range" of the piston displacement (Fig. 1(b)), and let Vn denote the downhill velocity of the wagon when its nose arrives at the outer range for the nth time; this occurs at time tn, and ~bn = wtn (mod 270 denotes the phase. When R ~ o~, with fixed v, it is assumed, as in previous work [4,7], that ~b,, is randomly and uniformly distributed in 0 < 4~ < 2re. Thus we see that because the piston executes a very large number of oscillation between each arrival of the wagon at the outer range and because of the "noise", the uncertainty of the piston phase is very large. The nearly elastic (e << 1) problem was previously considered for any periodic 2 In the context of the problem of oscillating granular particles the authors of [7] assume a Gaussian velocity distribution for Vn, and confine themselves to computing the average dissipation rate.
M.E. Stern/Physica D 122 (1998) 105-116
108
.0
~o+ 5 I
+
+
, I
+ +
~o+~ f---1
+
I
--
Zn
--
8 ~ = 2 -
Zn(1- ~:)
~o+ 5
I
I
4 - 2E
I
I
+
+
+
+
+
+
+
+
+
+
f=+l
V n Zn= --
V
2-e
3-e
2-e
1-E
1-e
e
Fig. 2. Regime diagram for Vn/v, and phase (ordinate). In the lower half ( f = -1) of the diagram only head-on collisions (+) occur. In the upper half (f = -1), 1 - ~/(~0 + 8) is the fractional distance of the piston from its origin (~ = 0) at the time when the wagon is at outer range (~0 + ~). Single rear-end collisions occur in the region marked with (-), and double collisions occur in the triangular region labeled "dbr'. At any Zn -- Vn/v the probability of each type of collision is proportional to the length of the vertical line segment lying in each of the three regions (see text). waveform f , and it was shown [4, p. 425] that in the statistically steady state the ensemble average mean square arrival velocity is given by 2zr
(v~)
4V2 f = - j
2 dq~ f (~b)~-.
(4b)
g 0
Attention is now directed to the question of statistical uniqueness at finite e, and for this purpose it seems sufficient to consider the mathematically simplest problem of the square wave oscillator with f either equal to + 1 or - 1; in this case the random phase assumption is equivalent to: all ~0 are equally probable if f = + l ,
(5)
where ~0 is the displacement (Fig. l(b)) of the piston when the wagon arrives at ~m; the piston is also equally likely to be moving uphill or downhill at this tn. The elementary calculations in Sections 2 and 3 wiU give the phase space diagram (Fig. 2) from which we obtain the transition probabilities and the evolutionary equation (Section 4) for the probability Pn ( V ) d V that the wagon
M.E. Stern/Physica D 122 (1998) 105-116
109
velocity lies between V and V + dV on the nth downhill arrival at tm. The recursion equation for Pn will be solved numerically for e = 0.1, and for different initial conditions (P1), in order to answer the questions as to whether a single P ~ (V) exists independent of the assumed Pl (V).
2. Collisions for the square wave piston oscillator For reasons mentioned previously, f = 4-1 in Fig. l(b) and all values of t0 are equally probable, so that the piston is equally likely to be moving uphill with velocity = + v or downhill with velocity = - v . Let f denote the value of f and q~n = torn the value of the phase (mod 2:r) at collision (Fig. l(c)). In the oJ ~ c~ limit the time interval tn - tn is so small that the fractional change in the downhill wagon speed Vn is negligible compared to the change (~ - t0) in t , and therefore the relative downhill wagon speed immediately before collision is Vn + v f . Immediately after collision the relative uphill speed is (1 - s)(Vn + v f ) , and thus the absolute uphill speed is V,I = (1 - e)Vn + (2 - e ) v f .
(6)
If this is sufficiently large compared to v (as specified in Section 3) there will be no second ("double") collision before the wagon crosses tm, in which case it moves uphill, reverses, and returns to the outer range (tin) at time tn+l with downhill speed Vn+l = Wn, i.e., V~+I = (1 - e)V, + (2 - s ) v f
(for no double collision).
(7a)
Unlike f, f is correlated with Vn and non-uniformly distributed, as is readily seen (Figs. l(d) and (e)) by the fact that if f = - 1 the downhill moving piston might reverse before collision, giving f = + 1; thus head-on collisions are more likely than rear-enders. Let us now consider the double collisions which can occur if Vn > v and f = - 1. After the first collision we have V~ = (1 - e)Vn + (2 - e ) v ( - 1 ) ,
(7b)
and if V~ < v, then the piston might reverse and overtake the wagon before it reaches tin. In this case the relative speed of the wagon before the second collision is v - Vnr > 0. After this collision the relative uphill speed is ( 1 - e ) (v - Vn~), and the absolute uphill speed (Fig. 1(f)) of the wagon is V~' = - ( 1 - e ) 2 gn -~-( 1 q- g) [ (3 -- g ) 1)] Jr- I). Since this exceeds v no further collision occurs, and therefore the downhill speed at the next "arrival" is
Vn+l = - ( 1 - e)Zvn + 4v - 4ev + eZv
(for a double collision at Vn > v).
(8)
3. The head-on, rear-end, and double collision domains If G < (2 - e)v/e for any n (say the initial value) then (7a) implies Vn+l < Vn and the state of the entire system is determined by a point in the two-dimensional space: 0 < V < (2 - e)v/e, 0 < ~ n ~ 27g. I f 0 < ~n < ~ , the piston is moving uphill and only a head-on collision can occur, as denoted by the + in the lower half of the regime diagram (Fig. 2). The same is true for Vn < v at Jr < ~n < 2rr because the downhill moving piston ( f = - 1) must reverse before it collides with the slower wagon. It only remains to consider the values of f when z =- Vn/v > 1 and when f = - 1; in which case it is convenient to express the ordinate in the upper half of Fig. 2 in terms of -- tmax - t0 (rather than q0, with all values of 3/(t0 + 8) being equally probable. Consider first the "farfield" region 3-e
Vn > 1 - e
(9)
110
M.E. Stern/ Physica D 122 (1998) 105-116
For either f = - 1 or f = + 1 Eqs. (6), (7b) and (9) imply that after the first collision v~' > v, so that no second collision can occur. As shown in Fig. 2 only head-on ( + ) and rear-end ( - ) collisions can occur for Vn > (3 - e)/(1 - e). In this region the probability of a rear-end collision ( f = - 1) equals 1/2 times the probability that the piston does not reverse before the impending collision. The latter condition requires that the downhill-distance V(tn - tn) traveled by the piston before collision is less than t0 (see Fig. l(e)). During this time interval the constant relative velocity Vn - v > 0 causes the separation between piston and wagon to decrease from 8 = ~m - t0 to zero. Since 8 = (Vn - v)(tn - tn), the condition for non-reversal of the piston is v[8/(Vn - v)] < t0, or 8/(50 + 8) < 1 - v / V n . Since all values of 8/(50 + 8) are equally probable, and since all values of 8/(50 + 8) between zero and (1 - v / V n ) give f = - 1, we conclude that (1 - v / V n ) is the conditional probability for f = - 1, and the absolute probability of a rear-end collision at Vn is r = 1/2(1 - v / V n )
for f = - 1 .
(10a)
This collision occurs below the curve 8/(50 + 8) = 1 - 1/Zn (Fig. 2) and above 8 = 0. At all values of Vn the region above the curve is occupied by head-on collisions, so that the absolute probability of the latter is r = 1/2(1 + v / V ~ )
for j3 = +1 (all Vn > v).
(10b)
Consider next the region 1 < Vn <
2-e 1-e
v,
f(~b)=-l.
(11)
Since the first collision is rear-end, Eq. (7b) gives the absolute uphill velocity V~' = (1 - e)V~ - (2 - e)v < 0, so that the wagon is actually moving downhill, and a double collision must occur. Eq. (8) then gives Vn+l, and the probability of this is given by the relative length of a vertical line (Fig. 2) in the "dbl" region between 8 = 0 and ~/(~o + 8) = 1 - 1/zn.
It only remains to consider the region Vn ~
2-e 1--e
i),
f(~b) = - - 1 ,
(12)
in which either a rear-end or double collision can occur. As previously mentioned tn - tn = 8/(Vn - v) > 0 gives the time interval for the first collision (at g), after which the uphill velocity is Vn' = (1 - e) Vn - (2 - e)v > 0. The necessary and sufficient condition that this be a single rear-end collision is that the wagons reach ~ = ~m = to + in a time ((to + 8) - ~ ) / V n' shorter than that required for the piston to go downhill from ~ to ~ = 0, and to then go uphill to ~m = t0 + 8. Thus the latter time (~ + (t0 + 8 ) ) / v must exceed the former time, and therefore a single rear-end collision requires (1
+
V~)
+(50+8)
(~
1)
Vn' > 0 .
In Figs. l(d) and (e) we see that at the first collision to - g = v ( ~ - t~), = vS/(V,, - v). By eliminating g and Vn~ = (1 - e)Vn - (2 - e ) v from the preceding inequality, and by simplifying we get 8
- -
50+8
< 2
(4 - 2 e ) v (1 -- E)V n
(single rear-end).
(13)
The bounding curve (Fig. 2) for this, obtained by replacing the inequality sign in (13) by an equal sign, intersects the 8/(50 + 8) = 1 - v / V n curve at V ~ / v = (3 - e)/(1 - e) which is the boundary of Eq. (9). Thus, we conclude that the "dbl" region between these two curves is occupied by double collisions, while the region below (13) and
M.E. Stern/Physica D 122 (1998) 105-116
111
above 8 = 0 is occupied by rear-end collisions. The probability of each kind of collision is given by the relative length o f the intercepted vertical line. The value of Vn+l in each region is obtained from Eq. (7) or (8).
4. P r o b a b i l i t y
evolution
equation
Consider a "range ordered" ensemble in which Pn (V) dV denotes the total number of wagons with downhill velocity between V and V + dV that arrive at ~m for the nth time. A given value of z = Vn+l/v can occur either by a head-on collision originating from some ZH -------Vn/v, or by a single rear-end collision originating from ZR = Vn/v, or by a double collision originating from ZD = Vn/v; where these respective z-values obtained from (7a) and (8) are z - (2 - e) 1 -e '
dz dZH -- 1 - e
> 0,
(14)
ZR --
z + (2 - e) 1 --e '
dz dZR -- 1 - - e > 0,
(15)
ZD=
--z + 4 - - 4e + e 2 (l--e) 2 '
dz I dZDI-- ( l - - e ) 2 > 0
ZH
-
( Z D > 1).
(16)
If TR (ZR), TH (ZH), TO (ZD) denote the respective transition probabilities to z, and if Pn (z) dz is the fractional number of wagons in the ensemble at collision n that are between z and z + dz, then the distribution function at collision n+lis
Pn+l (Z) dz = TH(EH)Pn(ZH)dZH + TR(ZR)Pn(ZR) dZR + TD(ZD)Pn(ZD)I dZD[,
(17)
TH(ZH)Pn(ZH) TR(ZR)Pn(ZR) TD(ZD)Pn(ZD) + + 1 -- e 1 -- e (1 -- e ) 2
(18)
Pn+l(Z) -
If z > 3 - 2e then the value of ZD in Eq. (16) is smaller than unity, and therefore double collisions cannot produce such large z; the only ones that can are head-on and rear-end collisions. For these the transition probabilities, as mentioned in Section 3, are given by the relative lengths of vertical lines intercepted by the curves in Fig. 2, i.e., for3-2e
(2-e)/e 1
1 ZH > 1
-, TH(ZH) = ~ + -2ZH TR(ZR) - -
5 - 3e
1
1
2
2ZR'
ZR > - - ,
TD = 0.
1 -- e
(19)
For smaller z the three transition probabilities obtained from Fig. 2 are
TH(ZH)
/0'
0
/
ZH<0,
1,
1,
0, 1 TR(ZR) = ~
(20a)
I
2
4--2e ZR(1 -- e ) '
1 -
--,
1
ZR
2--e -1 ---
3--e
- -
1 --e
2--e 1--e 3 --e < 1- -- e ' -5--3e
< ZR < - - ,
1 -e
(20b)
112
M.E. Stern/Physica D 122 (1998) 105-116
2--e
1 1--
ZD 1
TD(ZD) = ~
1-e
I
--
1
1 -- - - -ZD
(2
ZD(1--e)]'
- -
l--e--
(20C)
< ZD < - -
-- 1 - - e '
3--e ZD> 1 - - e
0,
We may, without loss of generality, restrict the initial P1 (Z) to values 0 < z < (2 - e)/e. Then the value of PE(z) at any z is obtained by first computing (14)-(16), then Eqs. (19) and (20) and then (18). Note that for any Vn -- v( the transition probabilities in (20) are conservative in the sense TH(() + TR(() + TD(¢) = 1 so that the number of wagons is the same from one n to the next, i.e., the solution of (18) satisfies (2-e)/e
f
(2-e)/e
Pn+l(()d(=
0
f
Pn(()d(.
0
5. A n a l y t i c s o l u t i o n f o r t h e s t e a d y
Pn w h e n
e --~ 0
According to (4b) the rms z is O(e-1/2), and it is of interest to examine the probability density at these large velocities for which TD = 0. The steady state (Pn+l (z) = Pn(z) -- P(Z)) version of Eqs. (18)-(20) for z >_ 3 - 2e reduces to 0= -P(z) + 2
-l~-e
+ ~Pn
z-(2-~)
i
1- e
e
z+(2_e)e
]-~-
j.
(21)
Anticipating z = 0(e-1/2), the e -+ 0 expansion of these terms, is
p
z-2+e
P
- 1-e
T-~-
=P(z+ez-2+O(e))=P(z)+(ez-2)P'(z)+-~P"(z)+O(e3/2),
= P(z+ez+2+O(e))
(1 -- e)/2 p l / Z - 2 + e ' ~ z-(2-e)
~,]l-e
1/2(1 + O(e3/2)) . . . .
. . . .
_ ( 1+- e()2/ -2 z e ) P ( Z - ~ 2 - e ) - - e
= P(z)+(ez+2)P'(z)+-=-P"(z)+O(e3/2),
(z-)
-~
trl.z) - 2P'(z) + O(e)],
-1/2(l+O(e3/2))[P(z)-2P'(z)+O(e)]'(z +2)
so that Eq. (21) becomes 2P(z)
p,(z)(
0 = ezP'(z) -t- 2P"(z) + eP(z) + z2 ----4
1
~-2
+ ~
1 )
+ O(e3/2)"
(22)
Now let z = ( e -1/2, retaining these leading [O(e)] terms, to obtain the e -+ 0 limit
o=
-
,".
(23)
M.E. Stern/Physica D 122 (1998) 105-116
113
For the solution of this steady-state asymptotic equation we require that P --+ 0 for z/,-1/2 >> 1 and for z / , 1/2 << 1, i.e., OO gl
P(O) = P(
/
(24)
1.
d~'P(~')
o As may be verified directly by substitution, the only one of the two solutions of (23) that can satisfy the latter condition is p(~) = ~e-¢2/4. 2
(25)
If the normalized energy (E = g2/2) distribution function is 12(E), i.e. I2(E) dE --= P(g) dg, then I2(E) = l e - e / 2 . The means that the asymptotic spectrum CX)
maximizes
[ 12(E) In I2 dE,
(26)
t/
0
with
fo
EI2(E) dE = 4, and
fo t2(E) d e
= 1.
6. F i n i t e , evolution
For a n y , and any initial condition (Pl(V)) Eqs. (14)--(20) can be used to compute P2, P3 . . . . . but an exact solution must involve the distribution in the entire z-continuum, whereas a numerical approximation only involves a finite number of grid points zj (J = 1, 2, 3 . . . . . Jmax) separated by a small distance S. Although the initial distribution P1 (z j) may be exactly specified at such points, the computation of P2(zs) requires data at the values of ZH, ZR, ZO (Eqs. (14)-(16)) which are not integral multiples of S, and for which P is not defined. Therefore we approximate these values by those that are defined at the next highest zj. For any z j = z in Eq. (16) let .
.{zj
JH=l+mt~
- (2- ,)\ S(1-,))'
. .{zg
,,=l+mt~
JD = l + I n t ( - z J S(1 + 4 -~e--'~-4" +'2) ,
+ ( 2 - ,)'~ S0-e))'
(27)
(ZJH, ZJR, ZJD) -~ S(JH, JR, Jo)-
These are the approximate values of (ZH, ZR, ZD) from which Pn+l (zg) is computed from Eqs. (18)-(20). First we consider an initial distribution (P1) in the form of an isosceles triangle: z
Pl(Z)
Zm~x/2
for z < Zmax/2,
1
for z > Zmax/2,
=
(28) z - (Zmax/2) Zmax/2
for Zmax = (2 - , ) / , , , = 0.1, S = 1/18. The arbitrary amplitude (100) is such that the total number of wagons, summed over all J, is 1.71000 x 104. In the subsequent evolution (Fig. 3) this total remains constant to four significant figures, and the main numerical source of error is in the "binning" process (Eq. (27)). The ensemble average z, which is proportional to the average time interval between two successive arrivals at the outer range (essentially the inverse collision frequency), decreases from 9.5 at n = 1, and at n = 21 (Fig. 3) a virtual steady state is reached, as indicated by the plots of both (z/and (z 2} in Fig. 5.
M.E. Stern/Physica D 122 (1998) 105-116
114 150.0
10010
N=20,21
50.0
.0 .o
s.o
o.o
5.o
2o.
z =V/v Fig. 3. The probability distribution Pn (z) for e = 0.1 obtained from the finite difference equation using Eq. (28) as initial condition. The absolute value of the ordinate has no significance. The curves for collision N = 20 and N = 21 coincide.
In order to examine the question of uniqueness, the calculation was repeated using an extremely different initial state in which P1 (z) was constant in 1 < z < 2, and zero elsewhere. For small n (not shown) a very "gappy" and ragged Pn (z) develops. As expected, the binning error due to the discontinuities is significant at small n, so that the number of wagons is not conserved, but for n > 16 the function Pn(z) starts to smooth and the number of wagons (1.733 x 104) then remains constant to four figures (Fig. 4). The values of (z) and (z 2) for this run are seen (Fig. 5) to be approaching the same asymptotes as the corresponding values in the previous run, and this strongly suggests that the solution of (14)-(18) for P ~ ( z ) is unique and independent of initial conditions. We estimate that for e = 0.1, (z) = 5.36 4- 0.02; also note that the value of (z 2) - 36 is somewhat less than the asymptotic (e - * 0) value (z 2) = 4 / e = 40 using e = 0.1. It is worth noting that the maximum entropy property (26) does not hold for e = 0.1, because the steady P is not • --z2/a 2 • 3 2 similar t o E q . (25). I f i t was, i.e., if P(z) = z e for some a, then lt would follow that (z) = a ~/-~/4, ( z ) = a4/2, and (2z2)3/4(z) -1 = 2.25. On the other hand, our numerical calculations for Figs. (4) and (5) at n = 21 give the much large value of 4.65 so that the corresponding 12 (E) is not a simple exponential function, and the entropy is not maximal if the energy is fixed. Perhaps some more general variational principle for the dissipative system might be found.
M.EI Stern/Physica D 122 (1998) 105-116
150.0
115 I
I
I
l
l
l
100.0
50.0
N=22,23,24,25
.0 .0
5.0
10.0
15.0
20.0
z:V/v
Fig. 4. Same as Fig. 3 except that P1 (z) is constant in 1 < z < 2 and zero elsewhere. The distribution for collisions 22-25 are shown.
7. Conclusion The rolling on an inclined plane of a wagon which intermittently and inelastically collides with a sinusoidally oscillating piston of very high frequency (vto/g* >> 1) is characterized by a dense distribution of phase-locked deterministic solutions, but none of these could be realized in an actual experiment due to the presence of small amplitude noise. This can be produced by many factors, such as a "small but finite" standard deviation (A) in the coefficient of restitution (e) which results in large uncertainties in the piston phase ~n = ogtn [mod(2rr)] at each (n) arrival of the wagon at the "outer range". Thus we assumed that in the high frequency limit this phase should be uniformly random in 0 _< ~ _< 2zr. In order to show that the statistically steady solution is unique we solved the problem for the mathematically simpler case of a square wave oscillator ( f = 4-1). For e << 1 the solution of the general evolutionary equation (18) yields a steady probability distribution function which maximizes the entropy for the average energy, but the main new result of this paper is the numerical solution (Figs. 3 and 4) for finite e which yields a statistically steady distribution function independent of initial conditions (as well as the small noise level (A)). Fig. 5 gives the evolution of the mean square wagon energy (proportional to (z2)), and the collision frequency (proportional to (z)). From this the ensemble statistics can be converted to long time averages for a single realization (Fig. l(a)). In this sense our system behaves ergodically.
M.E. Stern/Physica D 122 (1998) 105-116
116 6.0
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I
'
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i
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i
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9
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11
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13
t
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15
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i ,.,---'-'t-----'T~
19
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21
23
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N Fig. 5. Plots of (z) (upper two curves) and (z2) as a function of N for the PN(Z) used in Figs. 3 and 4. The results for Fig. 3 approach the asymptote from above, while the results for Fig. 4 approach the asymptote from below. The agreement of the asymptotes for the two different initial conditions suggest statistical uniqueness.
Acknowledgements I gratefully acknowledge the financial support for this work provided by the National Science Foundation under grants OCE-9216319 and OCE-9529261. References [1] [2] [3] [4] [5] [6] [7]
D.J. Coles, Fluid Mech. 21 (1965) 385. A.J. Roshko, Fluid Mech. 10 (1961) 345. R. Krishnamurti, J. Fluid Mech. 42 (1970) 309. M.E. Stem, J. Math. Phys. (SIAM) XLVII (1968) 416. P.J. Holmes, J. Sound and Vibration 84 (1982) 173. L.A. Wood, K.P. Byrne, J. Sound and Vibration 78 (1981) 329. S. Warr, J.M. Huntley, Phys. Rev. E 52 (1995) 5596.