1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
61
interested in the marginals of singular phase space measures, on spaces of sufficiently
lower dimension, which are usually regular [35, 36]. These facts can be briefly
recalled as follows. Ruelle showed that the effect of a perturbation δ F(t) = δ F (t) +
δ F ⊥ (t) on the response of a generic (smooth enough) observable O is given by:
O t − −O 0 =
t
0
R
(O)
(t − τ )δ F (τ )dτ +
t
0
R
(O)
⊥ (t − τ )δ F ⊥ (τ )dτ (1.217)
where the subscript 0 denotes averaging with respect to μ, R
(O)
may be expressed
in terms of correlation functions evaluated with respect to μ, while R
(O)
⊥ depends on
the dynamics along the stable manifold, hence it may not.
Let us adopt the point of view of Ref. [34]. For a d-dimensional dissipative dynamical system consider, for simplicity, an impulsive perturbation → + δ, such
that all components of δ vanish except one, denoted by δ i . The probability distribution μ is correspondingly shifted by δ, and turns into a non-invariant distribution
μ 0 , whose evolution μ t tends to μ in the t → ∞ limit. For every measurable set
E ⊂ M, μ 0 (E) is given by μ(E − δ),
21 and μ t (E) is computed as explained in
Sec. 1.6. Taking O(() = i , one obtains:
i t − − i 0 =
i dμ t (() −
i dμ(()
(1.218)
Approximate the singular μ by means of piecewise constant distributions, introducing
an -partition made of a finite set of d-dimensional hypercubes k () of side and
centers k . We define an -approximation of μ and of μ t in terms of the probabilities
P k () and P t,k (; δ) of the hypercubes k ():
P k () =
k ()
dμ(() , P t,k () =
k ()
dμ t (() .
(1.219)
This yields the coarse grained invariant density ρ((; ):
ρ((; ) =
k
ρ k ((; ) , with ρ k ((; ) =
P k ()/
d if x ∈ k ()
0
e l s e
(1.220)
If Z i is the number of one-dmensional bins of form
(q)
i − /2, ,
(q)
i + /2
, q ∈
{1, 2, . . . , Z i }, in the i-th direction, marginalizing the approximate distribution yields
the following quantities:
21 The set E − δ is defined by { ∈ M : + δ ∈ E}.
61
interested in the marginals of singular phase space measures, on spaces of sufficiently
lower dimension, which are usually regular [35, 36]. These facts can be briefly
recalled as follows. Ruelle showed that the effect of a perturbation δ F(t) = δ F (t) +
δ F ⊥ (t) on the response of a generic (smooth enough) observable O is given by:
O t − −O 0 =
t
0
R
(O)
(t − τ )δ F (τ )dτ +
t
0
R
(O)
⊥ (t − τ )δ F ⊥ (τ )dτ (1.217)
where the subscript 0 denotes averaging with respect to μ, R
(O)
may be expressed
in terms of correlation functions evaluated with respect to μ, while R
(O)
⊥ depends on
the dynamics along the stable manifold, hence it may not.
Let us adopt the point of view of Ref. [34]. For a d-dimensional dissipative dynamical system consider, for simplicity, an impulsive perturbation → + δ, such
that all components of δ vanish except one, denoted by δ i . The probability distribution μ is correspondingly shifted by δ, and turns into a non-invariant distribution
μ 0 , whose evolution μ t tends to μ in the t → ∞ limit. For every measurable set
E ⊂ M, μ 0 (E) is given by μ(E − δ),
21 and μ t (E) is computed as explained in
Sec. 1.6. Taking O(() = i , one obtains:
i t − − i 0 =
i dμ t (() −
i dμ(()
(1.218)
Approximate the singular μ by means of piecewise constant distributions, introducing
an -partition made of a finite set of d-dimensional hypercubes k () of side and
centers k . We define an -approximation of μ and of μ t in terms of the probabilities
P k () and P t,k (; δ) of the hypercubes k ():
P k () =
k ()
dμ(() , P t,k () =
k ()
dμ t (() .
(1.219)
This yields the coarse grained invariant density ρ((; ):
ρ((; ) =
k
ρ k ((; ) , with ρ k ((; ) =
P k ()/
d if x ∈ k ()
0
e l s e
(1.220)
If Z i is the number of one-dmensional bins of form
(q)
i − /2, ,
(q)
i + /2
, q ∈
{1, 2, . . . , Z i }, in the i-th direction, marginalizing the approximate distribution yields
the following quantities:
21 The set E − δ is defined by { ∈ M : + δ ∈ E}.