1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
5
After the observables of interest have been identified, and the range of measurements has been delimited, a theory will connect the sets of measurements performed
within those scales, hopefully constituting a satisfactory explanation of the observed
connections between sets of data concerning different observables. If the scales are
changed, hence the tools and the protocols, and perhaps also the observables are
changed, a new theory will be needed. This new theory could be found to match with
the previous one (finer or coarser that it may be) at the border between the two, or
could be contained or contain the previous one. Often, however, the different theories
do not mathematically reduce to each other [3], extra assumptions, alien to a given
theory, are required to make the new theory mathematically agree with the old. After
all, the same happens with our senses: a cathedral is meangfully described with a
language that differs from the one we might use to describe its stones; a movie is more
meaningfully described in words that tell a story, than with the frequencies of the
colors of the pixels of all its frames. It is not impossible or wrong to adopt a picture
based on stones and pixels: the cathedral is made of stones, and the movie is made of
pixels. But so much detail overshadows what we intend as the beauty or the meaning
of the phenomenon of interest. In other words, our (present) knowledge of stones
does not immediately reveal what makes a cathedral beautiful; what we (presently)
know about pixels does not directly convey the message of a movie. Analogously,
the (present) mathematical structures with which we describe atomic dynamics do
not immediately lead to the second law of thermodynamics.
A fundamental point to understand is that a given theory is successfull if the range
of scales to which it properly applies is not too narrow, so that different people, bound
to perform measurements in different ways, can still get compatible results. It will not
be required that an agreement is found on all possible scales, indeed it will even be
acceptable that on sufficiently separate scales, results are in some sense “orthogonal”
to each other. We can illustrate that with the function
f (x) = 10
−7 sin x, x ∈ [0:2π]
assuming it represents a certain measurable quantity. Figure 1.1 shows that within a
wide range of observations f is corectly represented by a straight horizontal line, it
looks wavy within certain scales, and eventually it appears as a set of several vertical
lines on still other scales. All these representations of f are legitimate and objective;
the one that is also useful, or meaningful, depends on the scale we want to explore.
Truly, the choice of space and time scales, that of the observables of interest, and
the choice of measurement tools and protocols are fundamental for the development
of a physical theory. But this does not imply any subjectivity in the corresponding
results. It should also be clear now that understanding better a given phenomenon
does not mean to increase the resolution of measurements, in order to gain ever more
detailed information. One should rather discard a large fraction of information, in
order to highlight the relevant information. Galileo used to say: “difalcare gli impedimenti”, i.e. get rid of inessential details, in order to highlight the pure phenomenon
of interest. Clearly, the message of a movie is not given by an accurate analysis of the
pixels and sound bits of all frames of the film: it emerges when the frames pass before
5
After the observables of interest have been identified, and the range of measurements has been delimited, a theory will connect the sets of measurements performed
within those scales, hopefully constituting a satisfactory explanation of the observed
connections between sets of data concerning different observables. If the scales are
changed, hence the tools and the protocols, and perhaps also the observables are
changed, a new theory will be needed. This new theory could be found to match with
the previous one (finer or coarser that it may be) at the border between the two, or
could be contained or contain the previous one. Often, however, the different theories
do not mathematically reduce to each other [3], extra assumptions, alien to a given
theory, are required to make the new theory mathematically agree with the old. After
all, the same happens with our senses: a cathedral is meangfully described with a
language that differs from the one we might use to describe its stones; a movie is more
meaningfully described in words that tell a story, than with the frequencies of the
colors of the pixels of all its frames. It is not impossible or wrong to adopt a picture
based on stones and pixels: the cathedral is made of stones, and the movie is made of
pixels. But so much detail overshadows what we intend as the beauty or the meaning
of the phenomenon of interest. In other words, our (present) knowledge of stones
does not immediately reveal what makes a cathedral beautiful; what we (presently)
know about pixels does not directly convey the message of a movie. Analogously,
the (present) mathematical structures with which we describe atomic dynamics do
not immediately lead to the second law of thermodynamics.
A fundamental point to understand is that a given theory is successfull if the range
of scales to which it properly applies is not too narrow, so that different people, bound
to perform measurements in different ways, can still get compatible results. It will not
be required that an agreement is found on all possible scales, indeed it will even be
acceptable that on sufficiently separate scales, results are in some sense “orthogonal”
to each other. We can illustrate that with the function
f (x) = 10
−7 sin x, x ∈ [0:2π]
assuming it represents a certain measurable quantity. Figure 1.1 shows that within a
wide range of observations f is corectly represented by a straight horizontal line, it
looks wavy within certain scales, and eventually it appears as a set of several vertical
lines on still other scales. All these representations of f are legitimate and objective;
the one that is also useful, or meaningful, depends on the scale we want to explore.
Truly, the choice of space and time scales, that of the observables of interest, and
the choice of measurement tools and protocols are fundamental for the development
of a physical theory. But this does not imply any subjectivity in the corresponding
results. It should also be clear now that understanding better a given phenomenon
does not mean to increase the resolution of measurements, in order to gain ever more
detailed information. One should rather discard a large fraction of information, in
order to highlight the relevant information. Galileo used to say: “difalcare gli impedimenti”, i.e. get rid of inessential details, in order to highlight the pure phenomenon
of interest. Clearly, the message of a movie is not given by an accurate analysis of the
pixels and sound bits of all frames of the film: it emerges when the frames pass before