deterministic brownian motion - mcgill university
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Deterministic Brownian Motion
Michael C. Mackey
September 4, 2014
Michael C. Mackey Deterministic Brownian Motion
Brownian Motion: Kappler, 1931
Figure: 30 minute record of mirror position at 760 mm Hg (upper) and4× 10−3 mm Hg (lower).
Michael C. Mackey Deterministic Brownian Motion
Gaspard et al. 1998
Figure: The position of a 2.5µm particle in water over a 300 secondperiod with a sampling interval of 1
60 sec.
Michael C. Mackey Deterministic Brownian Motion
Theoretical treatments of Brownian motion
Thorvald Thiele: 1880
Louis Bachelier: 1900 PhD thesis on stock and option markets
Albert Einstein: one of the amazing trio of 1905 papers
Marian Smoluchowski: 1906
Predictions of Einstein & Smoluchowski were verifiedexperimentally by Jean Perrin, 1908
Paul Levy
Paul Langevin
Norbert Wiener
And the list goes on. This field is now often referred to as thestudy of stochastic processes and is a mathematicallychallenging one.
Michael C. Mackey Deterministic Brownian Motion
Introduction
We are accustomed to ‘noise’ in our world and often invokestochastic processes to deal with this
Mathematicians have abstracted this into elaborate andbeautifully developed mathematics dealing with ‘random’events
While is clear that the assumption of ‘random’ events issufficient to explain aspects of data
It is by no means clear that it is necessary
Interesting Question: “Can one produce completelydeterministic theories that have the character of randomnessthat we see in the real world?”
Michael C. Mackey Deterministic Brownian Motion
Outline
The ‘usual’ Brownian Motion
Solution properties (numerical) of a differential delay equation
Quasi-Gaussian distributions
A deterministic quasi-Brownian motion
The filtered ‘random telegraph signal’
The mathematical problems preventing analytic proof
Conclusions and problems
Michael C. Mackey Deterministic Brownian Motion
Usual Brownian motion
dx
dt= v
mdv
dt= −γv + f (t)
f is a fluctuating “force” given by
f (t) = σξ(t)
ξ =dw
dt: a ‘white noise’ (delta correlated) which is the
‘derivative’ of a Wiener process w(t)
ξ(t): normally distributed with µ = 0, σ = 1
Michael C. Mackey Deterministic Brownian Motion
A differential delay equation: Lei & MCM, 2011
dx
dt= v
dv
dt= −γv + sin(2πβv(t − 1))
v(t) = φ(t), −1 ≤ t ≤ 0
The ‘random’ force is a rapidly oscillating function withrespect to v(t − 1)
Michael C. Mackey Deterministic Brownian Motion
Sample solution
0 20 40 60 80 100−0.5
0
0.5
t
v(t)
−0.2
0
0.2
Michael C. Mackey Deterministic Brownian Motion
Statistical quantifiers
Sampling: {vn}, where vn = v(n × 103∆t) and ∆t = 0.001
Mean: µ =1
N
∑Nn=1 vn
Bound: K = maxn |vn|
Standard deviation: σ =
√1
N
∑Nn=1(vn − µ)2
Excess kurtosis: γ2 =µ4σ4− 3 where µ4 =
1
N
∑Nn=1(vn − µ)4
Michael C. Mackey Deterministic Brownian Motion
Velocity statistics vs. β (γ = 1)
−1
0
1x 10
−3
(a)
μ
0.2
0.4
0.6(b)
K
0.1
0.3(c)
σ
10 20 30 40 50−1
−0.5
0
(d)
β
Exc
ess
Kur
tosi
s
Michael C. Mackey Deterministic Brownian Motion
Velocity statistics vs. γ (β = 20)
−4
−2
0
2x 10
−4
(a)
μ
0.1
0.2
0.3(b)
K
0.02
0.04
0.06
(c)
γ
σ
0 5 10 15 20 25 30−1.5
−1
−0.5
(d)
γ
Exc
ess
Kur
osis
Michael C. Mackey Deterministic Brownian Motion
Statistics: Dependence on β, γ from numerics
Bound
K (β, γ) =1
√γ(0.68
√β + 0.60
√γ)
Standard deviation
σ(β, γ) =0.32√βγ
Excess kurtosisγ2(β, γ) = −γ
β
Michael C. Mackey Deterministic Brownian Motion
Correlation function and times
0 10 200
0.5
1
1.5
2
β
Cor
rela
tion
time
(b)
0 10 200
0.2
0.4
0.6
0.8
1
(c)
γ
Cor
rela
tion
time
0 5 10
0
0.2
0.4
0.6
0.8
1
r
C(r
)
(a)
C (r) = limT→∞
∫ T0 v(t)v(t + r)dt∫ T
0 v(t)2dt' e−r/t0 ' e−γr
Michael C. Mackey Deterministic Brownian Motion
Quasi-Gaussian distributions
Quasi-Gaussian with µ = 0 and σ = 1 defined by
p(v ; 0, 1,K0) =
{Ce−v2/2, |v | ≤ K0
0, other wise
Normalize velocities to give
ζn =vn
σ(β, γ)
Sequence {ζn} has mean µ = 0, standard deviation σ = 1,and is bounded (numerically) by
K0 =K (β, γ)
σ(β, γ)'
√β/γ
0.21√β/γ + 0.19
Michael C. Mackey Deterministic Brownian Motion
Quasi-Gaussian simulations of v from DDE
0 10 20 30 40 502
2.5
3
3.5
4
4.5
5
β/γ
K0
(a)
−4 −2 0 2 40
0.1
0.2
0.3
0.4
0.5
v/σ
Den
sity
(b) β=2β=6β=8β=10
Michael C. Mackey Deterministic Brownian Motion
Deterministic Brownian Motion
Since numerically the velocity is like a ‘quasi-Gaussian noise’
now we construct a quasi-Brownian motion
using the full systemdx
dt= v
dv
dt= −γv + sin(2πβv(t − 1))
Michael C. Mackey Deterministic Brownian Motion
Deterministic BM simulations of x
0 2000 4000 6000 8000 10000−20
−10
0
10
20
30
t
x(t)
(a)
−10 −5 0 5 100
0.1
0.2
0.3
0.4
Δ x
Den
sity
(b) t=200
t=400t=600t=800
Michael C. Mackey Deterministic Brownian Motion
A differential nonlinearity delay equation
Replace the sin function with a piecewise constant nonlinearity:
dx
dt= v
dv
dt= −γv + 2
[H(sin(2πβv(t − 1))− 1
2)
]v(t) = φ(t), −1 ≤ t ≤ 0
H is the Heavyside step function
Now the ‘random’ force is discontinuous
Solutions are piecewise exponentials, increasing and decreasing
All of the statistical results are the same!
I think that the only really important thing is the rapidlyoscillating nature of the nonlinearity
Michael C. Mackey Deterministic Brownian Motion
So ....
How can we go about understandingthese results analytically?
Michael C. Mackey Deterministic Brownian Motion
BM with ‘noise’ from a map: c.f. MCM & T-K (2006)
dx
dt= v
mdv
dt= −γv + f (t)
f (t) = mκ∑∞
n=0 ξ(t)δ(t − nτ),
ξ: a “highly chaotic” deterministic variable generated by
ξ(t) = T (ξ(t − τ)),
where T is an exact map or semi-dynamical system, e.g. thetent map on [−1, 1]
so the results of the simulation are (in principle) understoodanalytically since the numerical computations are replicating adiscrete time (high-dimensional) map
but we still can’t go to the continuous situation.
Michael C. Mackey Deterministic Brownian Motion
Random Telegraph Signal: Does this illuminate things?
Consider a signal ξ(t) that switches between +1 and −1’randomly’
+1 −→ −1 with transition probability kd∆t + o(∆t)
−1 −→ +1 with transition probability ku∆t + o(∆t)
This is the random telegraph signal
Fully characterized analytically
Michael C. Mackey Deterministic Brownian Motion
Linear dichotomous flow (LDF)
dv
dt= −γv + ξ
The ‘random’ force ξ is the random telegraph signal
Pick kd = ku ≡ αSolutions are continuous and consist of segments that arepiecewise exponentials, increasing and decreasing
State space is V =
(−1
γ,
1
γ
)Stationary density is
p∗(x) =γ
B(12 ,
αγ
) (1− γ2x2)α/γ−1 → 1√2πσ
e−x2/2σ2
,α
γ→∞
Michael C. Mackey Deterministic Brownian Motion
DDE versus LDF results
Compare numerical DDE results with the LDF results
Quantity DDE (numerical) LDF (exact)
Bound ∼ ±(√βγ + γ)−1 ±γ−1
Correlation ∼ e−γt e−γt
Mean µ ∼ 0 0
SD σ ∼ (βγ)−1/2 (αγ)−1/2
Kurtosis γ2 ∼ −γ/β −γ/αI think the result in red for the DDE is due to numerics
If β ≡ α then the exact results for the linear dichotomous flowmatch the numerical results from the differential delayequation
Michael C. Mackey Deterministic Brownian Motion
Density evolution: ODE’s
Ifdxidt
= Fi (x) i = 1, . . . , d ,
the evolution of f (t, x) ≡ Pt f0(x) is governed by the generalizedLiouville equation:
∂f
∂t= −
∑i
∂(f Fi )
∂xi
Michael C. Mackey Deterministic Brownian Motion
Density evolution in stochastic systems
For stochastic differential equations
dxidt
= Fi (x) + σ(x)ξi , i = 1, . . . , d
f (t, x) ≡ Pt f0(x) satisfies the Fokker-Planck equation:
∂f
∂t= −
∑i
∂(f Fi )
∂xi+
1
2
∑i ,j
∂2(σ2f )
∂xi∂xj
Michael C. Mackey Deterministic Brownian Motion
Stationary densities
A density f∗ such that PtS f∗ = f∗ is a stationary density (fixed
point) of PS
For the system of ordinary differentialequations, f∗ is given by the solution of∑
i
∂(f∗Fi )
∂xi= 0,
For a system of stochastic differential equations, f∗ is thesolution of
−∑i
∂(f∗Fi )
∂xi+
1
2
∑i ,j
∂2(σ2f∗)
∂xi∂xj= 0.
Michael C. Mackey Deterministic Brownian Motion
The problem with DDE’s & density evolution
If x evolves under the action of dynamics described by adifferential delay equation (DDE)
dx(t)
dt= F(x(t), xτ (t)), xτ (t) ≡ x(t − τ)
then we would like to know how some “density” of thevariable x will evolve in time
We would like to be able to write down an equation like
UNKNOWN OPERATOR (DENSITY) = 0
Unfortunately we don’t know how to do this
Why?
Michael C. Mackey Deterministic Brownian Motion
Formal ‘Transfer operator’
dx(t)
dt= F(x(t), xτ (t)), x(t ′) ≡ φ(t ′)∀t ′ ∈ [−τ, 0]
induces a flow Tt on a
phase space of continuous functions C = C ([−τ, 0],R)
{Tt : t ≥ 0} : C → C is a strongly continuous semigroup
Michael C. Mackey Deterministic Brownian Motion
Formal ‘Transfer operator’
Evolution of a density given by∫APt f (x)µ(dx) =
∫T −1t (A)
f (x)µ(dx) ∀ mbleA ⊂ C .
Writing of transfer (‘Frobenius-Perron’) operatorPt : L1(C )→ L1(C ) merely formal & highlights problems thatwe face.
What is the measure µ on the space C?What is a density f on C?What does it mean to do integration over subsets of C?How would you actually figure out what T −1
t is?
Michael C. Mackey Deterministic Brownian Motion
Densities and DDE’s: The basic problem
Differential delay equations–infinite dimensional systems
Must specify an initial function on an interval [−τ, 0]
dx(t)
dt= F(x(t), xτ (t)), x(t ′) ≡ φ(t ′)∀t ′ ∈ [−τ, 0]
How to define a density in an infinite dimensional space?
If we can figure out how to define a density on this space
how can we relate it to what we actually measure in thelaboratory?
Michael C. Mackey Deterministic Brownian Motion
What do we really measure?
0
1
Time t-1 0 1
Initial functions Corresponding solutions
x0
1
x(t)p(x,t)
Michael C. Mackey Deterministic Brownian Motion
Conclusions
From numerical experiments it appears that one can producea quasi-random process with deterministic dynamics
The numerical DDE results match the analytic predictions fora filtered random telegraph signal
However, confirmation awaits a formal proof and the problemsinvolved are legion
If true, it is no longer necessary to postulate random processesto explain ‘noise’ in data
The results suggest that all events in the natural world can beexplained using completely deterministic dynamics (i.e.deterministic dynamics are sufficient)
Implications for the interpretation of quantum mechanics and“free will” (but that is another talk–probably over a coldbeer!)
Michael C. Mackey Deterministic Brownian Motion
Research supported by
NSERC, MITACS, and Alexander von Humboldt Stiftung
Zhou Pei-Yuan Center for Applied Mathematics (TsinghuaUniversity, Beijing)
Polish State Committee for Scientific Research
Michael C. Mackey Deterministic Brownian Motion
References
J. Lei & M.C. Mackey. “Deterministic Brownian motiongenerated from differential delay equations”, Phys. Rev. E(2011) 84, 041105-1-14
M.C. Mackey & M. Tyran-Kaminska. “DeterministicBrownian Motion: The effects of perturbing a dynamicalsystem by a chaotic semi-dynamical system”, Phys. Reports(2006) 422, 167-222
Michael C. Mackey Deterministic Brownian Motion