6
L. Rondoni
Fig. 1.1 Plots of f (x) = 10 −7 sin x at various resolutions O(10 −k ), k = 0, −4, −6, −9. For many
levels of description, f can be treated as a straight horizontal line; for other resolutions it can
be treated as several vertical straight lines (only one of which is reported). Only at intermediate
resolutions does f look like a sinusoidal function. All representations are correct, but their usefulness
depends on the scale of interest
our eyes at a speed that allows us to grasp just a tiny fraction of the information in
them. Selecting what is essential and what is not, however, may be a hard task.
Which are the variables and the scales described by Thermodynamics? What
is relevant to it? What is not relevant? What happens when scales are changed?
Investigating these questions, which are particularly interesting for systems that are
not of thermodynamic interest, is the subject of the present paper.
1.2 Continuum Description
Consider the simplest of the non-equilibrium thermodynamic relations, the mass
continuity equation. This relation is derived considering mass as a continuum, which
moves flowing through space, and one computes how much mass occupies a given
volume in space by knowing how much mass is in it, how much flows in, and how
much flows out, cf. Fig.1.2. Indeed, at sufficiently low energies that nuclear reactions
are negligible, mass is a locally conserved quantity.
4
The mass entering per unit time from direction x is
ρ v x
x
(1.1)
while that exiting is
ρ v x
x+
(1.2)
4 As discussed in the Prologue, this statement makes sense only once the scale of interest has been
specified, because nucelar reactions cannot be completely excluded.
L. Rondoni
Fig. 1.1 Plots of f (x) = 10 −7 sin x at various resolutions O(10 −k ), k = 0, −4, −6, −9. For many
levels of description, f can be treated as a straight horizontal line; for other resolutions it can
be treated as several vertical straight lines (only one of which is reported). Only at intermediate
resolutions does f look like a sinusoidal function. All representations are correct, but their usefulness
depends on the scale of interest
our eyes at a speed that allows us to grasp just a tiny fraction of the information in
them. Selecting what is essential and what is not, however, may be a hard task.
Which are the variables and the scales described by Thermodynamics? What
is relevant to it? What is not relevant? What happens when scales are changed?
Investigating these questions, which are particularly interesting for systems that are
not of thermodynamic interest, is the subject of the present paper.
1.2 Continuum Description
Consider the simplest of the non-equilibrium thermodynamic relations, the mass
continuity equation. This relation is derived considering mass as a continuum, which
moves flowing through space, and one computes how much mass occupies a given
volume in space by knowing how much mass is in it, how much flows in, and how
much flows out, cf. Fig.1.2. Indeed, at sufficiently low energies that nuclear reactions
are negligible, mass is a locally conserved quantity.
4
The mass entering per unit time from direction x is
ρ v x
x
(1.1)
while that exiting is
ρ v x
x+
(1.2)
4 As discussed in the Prologue, this statement makes sense only once the scale of interest has been
specified, because nucelar reactions cannot be completely excluded.