1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
73
if the steady state lives on a set of fractal dimension D. These times grow exponentially with N . In fact, this expresses in mathematical terms Boltzmann’s answer
to Zermelo paradox: for a system of a macroscopic number of degrees of freedom,
recurrence takes far longer than any physically imaginable time.
To build a model from known data, one looks for past states similar to the present
one, and assumes that the future will approximately repeat what happened after the
selected past state. This is the ancient method of analogues, that is translated in a
modern mathematical language thanks to the notion of recurrence. In modern times,
Maxwell noted that:
It is a metaphysical doctrine that from the same antecedents follow the same consequents.
No one can gainsay this. But it is not of much use in a world like this, in which the same
antecedents never again concur, and nothing ever happens twice. Indeed, for aught we know,
one of the antecedents might be the precise date and place of the event, in which case
experience would go for nothing. The metaphysical axiom would be of use only to a being
possessed of the knowledge of contingent events, scintia simplicis intelligentiæ degree of
knowledge to which mere omniscience of all facts, sctentiia visionis, is but ignorance. The
physical axiom which has a somewhat similar aspect is “from like antecedents follow like
consequents”. But here we have passed from sameness to likeness, from absolute accuracy
to a more or less rough approximation. There are certain classes of phenomena, as I have
said, in which a small error in the data only introduces a small error in the result. Such
are, among others, the larger phenomena of the Solar System, and those in which the more
elementary laws in Dynamics contribute the greater part of the result. The course of events
in these cases is stable.
Although he immediately added:
There are other classes of phenomena which are more complicated, and in which cases of
instability may occur, the number of such cases increasing, in an exceedingly rapid manner,
as the number of variables increases
In practice, the method of analogues assumes that the phenomena of interest are
stable in Maxwell’s sense, and proceeds as follows. Given a sequence of past events
x 1 , x 2 , . . . , x M
(1.265)
they are called “analogues” if they resemble each other in pairs with a certain degree
of accuracy , i.e. if:
x i − x j
≤
(1.266)
Then, the approximate prediction that can be made is expressed by:
x M+1 = x k+1 , if x k is an analogue of x M
(1.267)
One then models the phenomenon defining a function f such that sequences of states
are approximated by x k+1 =f(x k ) within the chosen tolerance. Now, to find analogues
one needs long series of data, as shown by Kac theorem: longer for higher accuracy
or for larger space, and exponentially long in the number of degrees of freedom. For
instance, even a relatively low accuracy such as = L/10 makes the task problematic,
73
if the steady state lives on a set of fractal dimension D. These times grow exponentially with N . In fact, this expresses in mathematical terms Boltzmann’s answer
to Zermelo paradox: for a system of a macroscopic number of degrees of freedom,
recurrence takes far longer than any physically imaginable time.
To build a model from known data, one looks for past states similar to the present
one, and assumes that the future will approximately repeat what happened after the
selected past state. This is the ancient method of analogues, that is translated in a
modern mathematical language thanks to the notion of recurrence. In modern times,
Maxwell noted that:
It is a metaphysical doctrine that from the same antecedents follow the same consequents.
No one can gainsay this. But it is not of much use in a world like this, in which the same
antecedents never again concur, and nothing ever happens twice. Indeed, for aught we know,
one of the antecedents might be the precise date and place of the event, in which case
experience would go for nothing. The metaphysical axiom would be of use only to a being
possessed of the knowledge of contingent events, scintia simplicis intelligentiæ degree of
knowledge to which mere omniscience of all facts, sctentiia visionis, is but ignorance. The
physical axiom which has a somewhat similar aspect is “from like antecedents follow like
consequents”. But here we have passed from sameness to likeness, from absolute accuracy
to a more or less rough approximation. There are certain classes of phenomena, as I have
said, in which a small error in the data only introduces a small error in the result. Such
are, among others, the larger phenomena of the Solar System, and those in which the more
elementary laws in Dynamics contribute the greater part of the result. The course of events
in these cases is stable.
Although he immediately added:
There are other classes of phenomena which are more complicated, and in which cases of
instability may occur, the number of such cases increasing, in an exceedingly rapid manner,
as the number of variables increases
In practice, the method of analogues assumes that the phenomena of interest are
stable in Maxwell’s sense, and proceeds as follows. Given a sequence of past events
x 1 , x 2 , . . . , x M
(1.265)
they are called “analogues” if they resemble each other in pairs with a certain degree
of accuracy , i.e. if:
x i − x j
≤
(1.266)
Then, the approximate prediction that can be made is expressed by:
x M+1 = x k+1 , if x k is an analogue of x M
(1.267)
One then models the phenomenon defining a function f such that sequences of states
are approximated by x k+1 =f(x k ) within the chosen tolerance. Now, to find analogues
one needs long series of data, as shown by Kac theorem: longer for higher accuracy
or for larger space, and exponentially long in the number of degrees of freedom. For
instance, even a relatively low accuracy such as = L/10 makes the task problematic,