274
Progress of Theoretical Physics Supplement No. 161, 2006
Dynamics of Coin Tossing
Tsuyoshi Mizuguchi∗) and Makoto Suwashita
Department of Mathematical Sciences,
Osaka Prefecture University, Sakai 599-8531, Japan
(Received June 29, 2005)
The motion of bouncing rigid line segment which iterates free-falls and collisions with a
flat smooth floor is investigated as a model of coin tossing process. With suitable assumptions, a map from initial conditions to the final outcome, “Head” or “Tail” is numerically
obtained. The basin structure is analyzed from the viewpoint of the length scale and the
continuity. The origin of probability is considered in connection with the basin structure.
§1.
Introduction
Coin tossing and die rolling are well-known gambling methods and are referred
to as representatives of stochastic process in many textbooks. From the viewpoint
of dynamical system, however, their outcomes, i.e., “Head” or “Tail” for toss and
the number between one and six for roll, should be deterministic. Actually, it is
empirically known that if we toss or roll them skillfully enough, the outcome is
controllable. A few deterministic approaches concerning such gambling processes
have been done: Bar tossing1)–3) and square rolling4) in the two-dimensional space
are investigated. About biased shapes, nuts throwing experiments were reported.5)
In this paper, as a model of coin tossing process, the dynamics of a rigid bar
in the two-dimensional space are numerically analyzed in a simple situation and the
basin structures concerning binary and integer outcomes are discussed.
§2.
Model
The tossed object is a rigid segment, hereinafter called “bar”, which obeys classical mechanics in the two-dimensional space. The bar iterates free-falls in vacuum
and inelastic collisions with a flat smooth floor. The x-axis is chosen in the horizontal
direction, and the y-axis is in the direction of gravitational acceleration. The region
y < 0 is the floor and y > 0 is the vacuum in which the bar moves. The length,
the mass of the bar and the gravitational acceleration are scaled as r = 2, m = 1
and g = 1, respectively. The structure of the bar is uniform, so its inertia moment
is given as I = 1/3. There are six dynamical variables, i.e., the position of center of
mass x, y and the velocity u, v, the angle θ and the angular velocity ω, where θ is
defined by the angle between x-axis and the unit vector along the bar from one side
to the other (see Fig. 1).
The outcome of one trial from the initial condition S0 = {x0 , y0 , θ0 , u0 , v0 , ω0 } is
determined via a following series of processes and conditions. i) free-fall process : The
∗)
E-mail: [email protected]
Dynamics of Coin Tossing
275
H
ea
d
bar falls and rotates freely during above the floor, i.e., the y-coordinate of the lower
edge y−A sin θ is positive, where A ≡ sgn sin θ. The dynamics are described as u̇ = 0,
v̇ = −1, ω̇ = 0. This process continues until the contact condition described below is
satisfied. ii) contact condition : When
the bar touches the floor, it changes over
m=1
from the free-fall process to the colliI
θω
sion process described below. This cony r=2
dition is represented by the equation
(x,y) (u,v)
y − A sin θ = 0. If A = +1 (−1), we
g=1
call it “left” (“right”) hand touch. iii)
x
collision process : We assume that the
floor
force the bar receives from the floor is
impulsive. Therefore, only the velocity
Fig. 1. Model.
and the angular velocity vary while the
position and the angle do not change by
this process. Namely, by setting the dynamical quantities before a collision x, y, θ,
u, v and ω, and those for after the collision x , y , θ , u , v and ω , then x = x,
y = y, θ = θ hold. The variables which characterize the collision process are
impulsive force T = (Tx , Ty ), u , v , and ω . The conditions to determine these
variables are the balance of linear and angular momentum, the collision rule for
the vertical direction characterized by a restitution coefficient e and the assumption
of smoothness, i.e., Tx = u − u, Ty = v − v, A(Tx sin θ − Ty cos θ) = I(ω − ω).
e = −(v − Aω cos θ )/(v − Aω cos θ) and Tx = 0. The last condition which is
equivalent with the condition of friction coefficient µ = 0 for the horizontal interaction of the contact point gives us two important simplifications: First, the degree
of freedom about x-direction is decoupled. We, therefore, set x = u = 0 hereafter. Second, it guarantees the energy loss for each collision, namely, E < E with
E ≡ y + v 2 /2 + Iω 2 /2. The bar goes back to the free-fall process if the termination
condition described below is not satisfied. iv) termination condition : Because the kinetic energy E decreases at each collision process, the final state Sf = {yf , vf , θf , ωf }
is stationary, i.e., vf = ωf = 0. If yf = 0 and θf = nπ with n is even (odd), the
outcome is +1 [“Head”] (−1 [“Tail”]). If yf = 1 and θf = (n + 1/2)π with an
integer n, the bar is standing, which is an unstable solution Su . Note that these
solutions are realized after infinite times collisions, so it is practically troublesome
if we wait for the realization of the stationary states in the numerical simulations.
By considering the following termination condition, however, it becomes possible to
determine the outcome during a practical calculation time for the most of the initial
conditions. If the total energy E decreases below the threshold value Ec = 1 which
corresponds to the standing states Su , the outcome is determined and we can stop
the numerical calculation. Some initial conditions, e.g., {y, v, θ0 = π/2, ω0 = 0}, do
not satisfy this termination condition during finite times collisions. Therefore, we
adopt a limitation rule for the bouncing times in the actual numerical simulations.
Let the state S ∗ = {y ∗ , v ∗ , θ∗ , ω ∗ } at the time when the total energy falls below Ec .
Then the outcome RB of this trial is determined as RB = ±1 = sgn cos θ∗ .
Ta
il
276
T. Mizuguchi and M. Suwashita
1000
6
6
y0
y0
f ()/
100
10
1
1
−π
θ0
(a)
π
0.1
10
-9
-6
10 10
-3
10
0
(b)
1
−π
θ0
π
(c)
Fig. 2. (a) Cross section of the basin structure in the initial condition space. White (black) denotes
that the outcome RB is +1(−1). Other parameters are fixed as v0 = 0 and ω0 = 0. (b) Initial
condition disturbance vs f ()/, where f () is the outcome change probability. The solid line
denotes the numerical result and the dashed line represents 1/(2). (c) Basin structure of the
integer outcome RZ ∈ Z (defined in §4). Light gray, dark gray and black denote RZ = 0, 1 and
2, respectively. White indicates other values.
§3. Binary outcome
Figure 2(a) shows the distribution of the outcome RB in the cross section of the
initial condition space, i.e., the basin structure. The following two properties are
observed numerically. (1) The characteristic scale of the basin structure, namely,
the interval between the basin boundaries, decreases non-uniformly as the initial
energy increases. (2) Smooth structure is observed if we magnify the complicated
region enough. Therefore, the fractality is not observed. These observations can
be interpreted in connection with the origin of probability as follows: If the initial
error of the initial condition is sufficiently smaller than the interval l of the basin
structure at that point, the outcome is predictable or controllable. On the other
hand, if is much larger than l, the outcome is “probabilistic”.
In order to characterize this structure, the outcome sensitivity to the initial
condition is measured.6) First, a subspace Γ in the initial condition space is chosen
and an initial condition S1 is randomly selected in Γ . Next, S2 which locates near
S1 with a distance in some direction is chosen. The outcome change probability
f () is defined as {# of RB (S1 )RB (S2 ) < 0}/{# of trials}. Typical numerical results
show that the length scale is divided into three regions using exponent ν in the form
f ()/ ∝ −ν , i.e., the random region (ν = 1), the fractal region (0 < ν < 1) and
the flat region (ν = 0) as shown in Fig. 2 (b). The length scales which separate the
regions depend on Γ . It represents the non-uniformity of the basin structure.
Note that this process is not chaotic by considering the fact that the final state
is stationary, i.e., either laying {y = 0, θ = nπ} or standing {y = 1, θ = (n +
1/2)π}. The complex structure is produced by the orbit instability during finite
times collisions.
Dynamics of Coin Tossing
§4.
277
Discussion
Now, we discuss each outcome in detail. In stead of the binary information RB ,
i.e., “Head” or “Tail”, we focus on the integer information RZ , a rotation number
of the final states define by RZ = [θ∗ /π + 1/2], where [θ] means a maximum integer
which does not exceed θ. Figure 2 (c) shows the basin structure concerning the
outcome RZ = 0, 1, 2 and others. If we focus on the region RZ = 1, it is surrounded
only by the region RZ = 0 and RZ = 2.
In order to interpret these numerical results we spotlight a continuity about
the initial condition (CIC). The CIC means that when two initial condition S1 and
S2 are close each other infinitesimally, the outcomes RZ (S1 ) and RZ (S2 ) are the
same. There is, at least, one obvious and important situation in which this CIC is
broken, i.e., S1 and S2 sandwich the stable manifold of trivial standing solution Su .
In this case, no matter how S1 and S2 approach each other, their outcomes RZ are
different by unity and there is a basin boundary between them. Next, if there is
no other situation in which CIC is broken, all the basin boundary is necessarily the
stable manifold of the unstable solutions Su , i.e., the inverse image of Su . Moreover,
because the system is invariant under the transformation θ → θ + π, RZ → RZ + 1,
all of the basin boundaries are a superposition of infinite number of nπ shift toward
θ direction of the inverse image of the single unstable solution Su with θ = π/2.
Is the unstable solution Su the only situation in which CIC is broken? For actual
tosses, the answer for this question is expected to be yes. In this model (or other
collisional models with impulsive force), however, the proof of having this property is
not easy. Except Su there are at least three situations in which CIC may be broken,
i.e., the contact condition is satisfied or not, the contact hand is left or right, and the
termination condition is satisfied or not.∗) Especially, the difference of the contact
hand (left or right) complicates the problem because multiple collisions should be
treated. The analytic proof of CIC remains open even for this simple model.
Finally, we note that there are regions in which the initial conditions do not
show definite outcome because the simulation is terminated by the bouncing number
limitation before the energy criterion is satisfied. The ratio of area of this region is
a few tenth of a percent. The condition for this situation is not clarified.
Acknowledgements
We would like to thank Professor T. Chawanya, Professor H. Daido, Professor
H. Nishimori and Professor T. Konishi for fruitful discussions.
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Moreover, if we consider the friction force along the horizontal direction, another discontinuity
may arise from the difference between static and dynamic friction.