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).