1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
69
by taking long τ , because the transient FR refers to the initial state, however long
τ might be. Nevertheless, under certain conditions that also guarantee convergence
to the steady state, the transient FR turns into the steady state FR. The following
derivation explains how this happens. Using the conservation of probability in phase
space, that states
μ
(t)
(E) = μ
(0)
(S
−t E) , for any measurable set E ∈ M
(1.244)
we may advance the initial distribution μ
(0) up to time t, take the log of Eq. (1.242)
and divide it by τ , which yields:
1
τ
ln
μ
(t)
(
(0)
0,τ ∈ A
+
δ )
μ (t) (
(0)
0,τ ∈ A
−
δ )
= −
1
τ
ln
e
−
(0)
0,t · e
−
(0)
t,t+τ
· e
−
(0)
t+τ ,2t+τ
(0)
(0)
t,t+τ ∈A
+
δ
= A + (δ, t, A, τ ) −
1
τ
ln
e
−
(0)
0,t · e
−
(0)
t+τ ,2t+τ
(0)
(0)
t,t+τ ∈A
+
δ
(1.245)
where (δ, t, A, τ ) ≤ δ, and ··
(0)
(0)
t,t+τ ∈A
+
δ
denotes the average with respect to the initial
distribution μ
(0) , under the condtion
(0)
t,t+τ (() ∈ A
+
δ
(1.246)
This relation is exact, like Eq. (1.242) is. Taking t → ∞, one should obtain the
asymptotic expression for the steady state distribution μ ∞ . Then, letting τ → ∞
should eliminate the condtional average, producing the steady state FR. There is
however a difficulty: the t → ∞ limit does not need to exist: the exponentials inside
the conditional average could diverge, because they contain integrals from 0 to t of
(0)
t,t+τ . On the other hand, one cannot take the τ → ∞ limit first, because infinitely
long time averages may collapse on a single value; that value then gets probability
1, and all the others get probability 0, making nonsensical the FR expression. What
makes sense is the steady state probability, if any, of the finite time averegaes of
(0)
t,t+τ ,
which can be positive; therefore, t has to grow first. When the μ ∞ probabilities exist,
one may ask what their ratio does, for longer and longer observation times τ .
A common way to proceed, at this point, is to search for the conditions under
which
M(A, δ, τ , t) = ln
e
−
(0)
0,t · e
−
(0)
t+τ ,2t+τ
(0)
(0)
t,t+τ ∈A
+
δ
(1.247)
does not diverge when t grows without bounds. Unfortunately, typically one obtains
sufficient conditions, which may be too restrictive to be physically relevant. One
example of that is the Anosov property, which can hardly be verified by systems of
physical interest. The drawback of this approach is that one may miss the mechanism
leading to the validity of the desired result, mingling it with unnecessary ingredients.
A different path has been followed in Ref. [43], where necessary conditions have
69
by taking long τ , because the transient FR refers to the initial state, however long
τ might be. Nevertheless, under certain conditions that also guarantee convergence
to the steady state, the transient FR turns into the steady state FR. The following
derivation explains how this happens. Using the conservation of probability in phase
space, that states
μ
(t)
(E) = μ
(0)
(S
−t E) , for any measurable set E ∈ M
(1.244)
we may advance the initial distribution μ
(0) up to time t, take the log of Eq. (1.242)
and divide it by τ , which yields:
1
τ
ln
μ
(t)
(
(0)
0,τ ∈ A
+
δ )
μ (t) (
(0)
0,τ ∈ A
−
δ )
= −
1
τ
ln
e
−
(0)
0,t · e
−
(0)
t,t+τ
· e
−
(0)
t+τ ,2t+τ
(0)
(0)
t,t+τ ∈A
+
δ
= A + (δ, t, A, τ ) −
1
τ
ln
e
−
(0)
0,t · e
−
(0)
t+τ ,2t+τ
(0)
(0)
t,t+τ ∈A
+
δ
(1.245)
where (δ, t, A, τ ) ≤ δ, and ··
(0)
(0)
t,t+τ ∈A
+
δ
denotes the average with respect to the initial
distribution μ
(0) , under the condtion
(0)
t,t+τ (() ∈ A
+
δ
(1.246)
This relation is exact, like Eq. (1.242) is. Taking t → ∞, one should obtain the
asymptotic expression for the steady state distribution μ ∞ . Then, letting τ → ∞
should eliminate the condtional average, producing the steady state FR. There is
however a difficulty: the t → ∞ limit does not need to exist: the exponentials inside
the conditional average could diverge, because they contain integrals from 0 to t of
(0)
t,t+τ . On the other hand, one cannot take the τ → ∞ limit first, because infinitely
long time averages may collapse on a single value; that value then gets probability
1, and all the others get probability 0, making nonsensical the FR expression. What
makes sense is the steady state probability, if any, of the finite time averegaes of
(0)
t,t+τ ,
which can be positive; therefore, t has to grow first. When the μ ∞ probabilities exist,
one may ask what their ratio does, for longer and longer observation times τ .
A common way to proceed, at this point, is to search for the conditions under
which
M(A, δ, τ , t) = ln
e
−
(0)
0,t · e
−
(0)
t+τ ,2t+τ
(0)
(0)
t,t+τ ∈A
+
δ
(1.247)
does not diverge when t grows without bounds. Unfortunately, typically one obtains
sufficient conditions, which may be too restrictive to be physically relevant. One
example of that is the Anosov property, which can hardly be verified by systems of
physical interest. The drawback of this approach is that one may miss the mechanism
leading to the validity of the desired result, mingling it with unnecessary ingredients.
A different path has been followed in Ref. [43], where necessary conditions have