Ergodic dynamical systems
Juan M. Bello-Rivas
jmb@ices.utexas.edu
Molecular Dynamics is (mostly) about computing integrals like
Z
X
f(x) π(x) dx,
where X ⊂ R
n
.
(a) If n is small (i.e., n < 7), then quadrature formulas suffice.
(b) Otherwise, we need other techniques: Monte Carlo and
variants.
In essence, Monte Carlo is about noticing that
Z
X
f(x) π(x) dx = E[f]
and then using the Central Limit Theorem:
E[f] =
1
N
N
X
n=1
f(x
i
) + Error(N )
where the x
i
are random samples of the probability µ with p.d.f.
π(x) and
Error(N) ∼ Normal(0,
1
N
Var(f)
| {z }
variance
) as N → ∞.
The problem then shifts to obtaining random samples of µ. One
possible avenue is to use the Metropolis algorithm:
Markov chain T
p.d.f. π of µ
)
⇒ sample paths from a Markov chain A,
where A is s.t. lim
n→∞
A
n
(x, ·) = π(·).
Introduction
Ergodicity in a nutshell: Let t ∈ [0, ∞) 7→ x(t) ∈ X , then
lim
T →∞
1
T
Z
T
0
f(x(t)) dt =
Z
X
f(x) π(x) dx.
This yields another approach to sampling.
How is it even possible? The integrand in the LHS is 1-dimensional
whereas the one in the RHS is n-dimensional!
Introduction
Space-filling curve
Dynamical systems
Let X be a set that we will refer to as the state space.
Example
Let X be the two-dimensional torus. A point in X corresponds to
a pair of angles, (x
1
, x
2
), determining a point on the torus.
(x
1
, x
2
)
Dynamical systems
Let µ be a probability measure defined on X . This means that to
each C ⊆ X we can assign a value µ(C) between 0 and 1
Example
x
2
x
1
Dynamical systems
Example (cont.)
x
2
x
1
C
Let C be a (measurable) subset of the torus. We define
µ(C) =
x
C
dx
1
dx
2
4π
2
.
Dynamical systems
Example (Flow on the torus)
Let x = (x
1
, x
2
) ∈ [0, 2π) × [0, 2π) and ω = (ω
1
, ω
2
) ∈ R
2
.
Consider the trajectories ϕ
t
(x) = x + t ω of a free particle started
at x with constant velocity ω.
Dynamical systems
In general, we may consider a family of mappings ϕ
t
: X → X
parameterized by t ≥ 0 so that, for all x ∈ X , we have
ϕ
0
(x) = x
and
ϕ
t+s
(x) = ϕ
s
(ϕ
t
(x)).
Dynamical systems
Definition
A dynamical system is a tuple (X , µ, ϕ).
Example
X = [0, 2π) × [0, 2π), µ(C) =
x
C
dx
1
dx
2
4π
2
, ϕ
t
(x) = x + tω.
Dynamical systems
Example (Microcanonical ensemble)
Let X = R
2N
, H : X → R, and
µ(C) =
R
C
δ(H(x) − E)dx
R
X
δ(H(x
0
) − E)dx
0
,
where x = (q
0
, p
0
) ∈ X and ϕ
t
(x) is the solution of the system of
ODEs
dq
dt
(t) = ∇
p
H(q(t), p(t)),
dp
dt
(t) = −∇
q
H(q(t), p(t)),
q(0) = q
0
,
p(0) = p
0
.
Dynamical systems
Definition
Let f : X → R. The time average of f (starting at x) is
¯
f(x) = lim
T →∞
1
T
Z
T
0
f(ϕ
t
(x)) dt.
Example
For ϕ
t
(x
1
, x
2
) = (x
1
+ tω
1
, x
2
+ tω
2
), we have
¯
f(x
1
, x
2
) = lim
T →∞
1
T
Z
T
0
f(x
1
+ tω
1
, x
2
+ tω
2
) dt.
Dynamical systems
Definition
Denote by π the probability mass (or density) function of the
probability measure µ. The space average of the function f is
hfi =
Z
X
f(x) π(x)dx.
Example
Let X = [0, 2π) × [0, 2π), then for f = f(x, y),
hfi =
2π
Z
0
2π
Z
0
f(x, y)
dx dy
4π
2
.
Dynamical systems
Definition
We say that (t, x) 7→ ϕ
t
(x)
preserves the measure µ for
the set C ⊆ X if
Z
ϕ
t
(C)
π(x) dx =
Z
C
π(x) dx
for all C ⊆ X and t ≥ 0.
x
2
x
1
C
ϕ
t
(C)
Dynamical systems
Definition
A set C ⊆ X is said to be invariant if,
ϕ
t
(x) ∈ C for all x ∈ C and all t ≥ 0.
In general, we write ϕ
t
(C) = C.
Ergodic theorem
Theorem (Birkhoff-Khinchin (1931))
Consider the dynamical system (X , µ, ϕ) such that ϕ preserves the
measure µ. Let C ⊆ X be an invariant set and let f : C → R be
any integrable function. Then, the limit
¯
f(x) = lim
T →∞
1
T
Z
T
0
f(ϕ
t
(x)) dt
exists almost everywhere in C. Moreover,
¯
f(x) is independent of
the choice of the initial point x (so we write
¯
f from now on).
Ergodic theorem
Definition
If C ⊆ X cannot be written as
C = C
1
∪ C
2
such that ϕ
t
(C
i
) = C
i
and
µ(C
i
) > 0, for i = 1, 2, then we
say that X is metrically
indecomposable.
Reproduced from V. I. Arnold and A. Avez, Ergodic
problems of classical mechanics, W. A. Benjamin, Inc.,
New York-Amsterdam, 1968.
Ergodic theorem
Theorem
Let C ⊆ X be an invariant set. The following is true: C is
metrically indecomposable if and only if
¯
f = hfi.
Ergodic theorem
¯
f 6= hfi
Ergodic theorem
¯
f = hfi
Ergodic theorem
Theorem (Strong law of large numbers)
Let f : X → R and let x
0
, x
1
, x
2
, . . . be a sequence of independent
and identically distributed random variables. If h|f|i < +∞, then
lim
N→∞
1
N
N−1
X
n=0
f(x
n
) = E[f]
Ergodic theorem
If ϕ
t
(x
0
) is the solution of the system of ODEs
(
˙x(t) = v(x(t), t),
x(0) = x
0
.
and if x
n
= ϕ
t+nτ
(x
0
) for some constant τ > 0 could be regarded
as a sequence of i.i.d. random variables, then the Ergodic theorem
is a generalization of the SLLN:
¯
f(x) = lim
N→∞
1
N
N−1
X
n=0
f(x
n
) = E[f] = hfi.
Questions?
