10 1. HEURISTICS

where

a+

= a ∨ 0 = max{a, 0} denotes the positive part of a real number a. In

particular, if U(y) U(x), then V (x, y) = 0, i.e. going downhill in the landscape

does not require work. If, however, x and y are in different potential wells, we have

(recall U(0) = 0)

(1.5) V (x, y) = −2U(x).

This equation reflects the fact that the minimal work to do to pass to y consists in

reaching the saddle 0, since then one can just go downhill.

Rudiments of the following arguments can also be found in the explanation of

stochastic resonance by McNamara and Wiesenfeld [74]. The main ingredient is

the exit time law by Freidlin and Wentzell [40] (see also Eyring [37], Kramers [65]

and Bovier et al. [14]). For y ∈ R, ε 0 the first time y is visited is defined by

τy

ε

= inf{t ≥ 0: Xt

ε

= y}.

If Px denotes the law of the diffusion (Xt ε)t≥0 started at x ∈ R, the exit time law

states that for any δ 0, x ∈ R we have

(1.6) Px e

V (x,y)−δ

ε

≤ τy

ε

≤ e

V (x,y)+δ

ε

→ 1

as ε → 0.

In other words, in the limit ε → 0, the process started at x takes approximately

time exp(

V (x,y)

ε

) to reach y, or more roughly

ε ln τy

ε

∼

=

V (x, y)

as ε → 0. As a consequence, one finds that as ε → 0, on time scales T (ε) at least

as long as exp(

V (x,y)

ε

) or such that

ε ln T (ε) V (x, y),

we may expect with Px-probability close to 1 that the process XtT ε

(ε)

has reached

y by time 1. Remembering (1.4) and (1.5) one obtains the following statement

formulated much more generally by Freidlin. Suppose

(1.7) lim

ε→0

ε ln T (ε) 2 max{−U(−1), −U(1)},

and U(−1) U(1). Then the Lebesgue measure of the set

(1.8) t ∈ [0, 1]: |XtT

ε

(ε)

− (−1)| δ

tends to 0 in Px-probability as ε → 0, for any δ 0.

In other words, the process

Xε,

run in a time scale T (ε) large enough, will

spend most of the time in the deeper potential well. Excursions to the other well

are exponentially negligible on this scale, as ε → 0. The picture is roughly as

deployed in Figure 1.11.

1.2.3. Periodic step potentials and quasi-deterministic motion. As a

rough approximation of temporally continuously varying potential functions we may

consider periodic step function potentials such as

(1.9) U(t, ·) =

U1(·), t ∈ [k, k +

1

2

),

U2(·), t ∈ [k +

1

2

, k + 1), k ∈ N0.

We assume that both U1 and U2 are of the type described above, that U1(x) =

U2(−x), x ∈ R, and that U1 has a well of depth

V

2

at −1, and a well of depth

v

2

at