I'm not sure to completely understand what you mean by "self-similarity". However, I can tentatively answer the following. Dynamics is a deterministic science dealing with masses, forces, inertia and motion. The size of the system has no importance. Thermodynamics concerns only macroscopic systems, and so its "determinism" is of statistical origin. Cardinal thermodynamic concepts such as temperature, heat, and entropy are statistical quantities. Even pressure is of statistical origin. Temperature or heat have no meaning in dynamics. The first law in thermodynamics for instance must include the heat exchanged. In dynamics, thermal energy is disregarded; if nevertheless there is some heat exchanged, then the system is just called "dissipative", which means "non-conservative".

I completely agree with Michel.

Now, just for the sake of our conversation, let's please not forget the workings of non-equlibrium thermodynamics and auqntum chaos. This does enlarge the scope from the macrosopic to the microscopic world.

Thats what the whole mistery of the universe is about..!! For example if we take a gaseous system (thermodynamic), if we observe the collisions of individual molecules, they're according to Newton's law hence they're completely reversible. But when we observe the collective behavior of all the molecules, its just not reversible. So, there is a time arrow. We have never seen a shattered egg re-assemble again and sit back on a table (though its perfectly possible as the individual collisions of molecules is reversible), So in reality it is Newton's law + Time arrow, for Macroscopic systems. That is nothing but the Second law.

The so-called "time arrow" is the consequence of the evolution of macroscopic systems. If everything were reversible, there would be no "time arrow", no past and future. Dynamical equations such as Newton's or Lagrange's are reversible and are valid at any scale. However, macroscopic systems seem to tend to states called "equilibrium states". But we know that these are not the end. Sooner or later (rather later !), the system in equilibrium will awake, and very, very, improbable states will happen. This is Poincaré's return theorem. This was used by Zermelo as an objection to Boltzmann's H-theorem. Boltzmann's answer was that Poincaré's return times were so fantastically enormous that they exceed the age of the universe. So, everything seems to occurs AS IF true equilibrium states really happens.

Dear Michel,

Self-similarity in the theoretical structures means that a macro-theoretical structure is similar to the micro-theoretical structure of itself. Here I borrowed the term “Self-similarity” from theory of chaos. Please see:

http://en.wikipedia.org/wiki/Self-similarity

For a same law of physics, such as conservation of energy, there should be a self-similar theoretical structure in macro and micro scales, particularly for the master equations, but now they are not complete self-similar, that is a question I think.

The equation of the first law of thermodynamics is also reversible, similar to dynamical equations such as Newton's or Lagrange's, these equations do not seem to involve the question of irreversibility.

I was not sure that by "self-similarity" you meant that, i.e. fractals. Self-similarity is an important concept in mathematics, but there is nothing self similar in nature : it is only approximately realized for a limited range of scales. This is a quality only possible for continuum. But matter is not. It is composed of atoms. Turning to theory, it is right that some "laws" of physics are similar at the micro- and macroscale : for instance, yes, the laws of mechanics (but only from atoms to stars, and not ad infinitum !). Furthermore, many "laws" (including the second law and the master equations) are not, and should not be.

Dear Michel,

Maybe the term self-similarity I borrowed is not accurate enough, what I meant was that, for instance, the law of conservation of energy is the same in different scales, why the theoretical structure/ master equation of the first law of thermodynamics is not similar to the law in dynamics?

The law of conservation of energy can be understood as expressing invariance under time *translations*, i.e. the absence of any privileged time instant, for the system in question-it can't detect its ``age''.

While the first law of thermodynamics may not *appear* to make the same statement, upon closer look, it should be possible to realize that it does-in a different language, but the *meaning* should be the same. Of course one way may be easier to grasp or, more importantly, to generalize, than others.

Yes, thermodynamics and dynamics now express the physical laws in different languages, the question is that thermodynamic language and dynamical language cannot translate each other,

In dynamics, a collective behaviour is the sum of its basic dynamical processes, the equation of the sum can be applied to all scales, but " this equation of the sum” cannot translate into thermodynamic language, even for the law of conservation of energy / the first law. Due to thermodynamic language can only be applied to the collective behaviour of macroscopic system, but cannot applied to the basic dynamical processes, the theoretical structures of the two now are different. Why?

For the question of irreversibility, I think there has no a similar theoretical structure in dynamics, so we are not sure whether the basic dynamical processes are reversible. We can see T-symmetry of dynamical equations, Poincaré's return theorem, some similar conclusions in the equation of the first law of thermodynamics.

Thermodynamics and dynamics of course can translate into each other-this is known since Boltzmann's work. Thermodynamics can be seen as a ``macroscopic'' description of the dynamics of microscopic degrees of freedom, that are treated in a statistical way. And the experimental studies of atoms in the beginning of the 20th century by Jean Perrin showed that the degrees of freedom used by Boltzmann did have a physical basis-which was also part of the work of Einstein in his thesis.

For further discussion on the second law, in particular, cf. J. Lebowitz,

``Time's arrow and Boltzmann's entropy'',

http://www.scholarpedia.org/article/Time%27s_arrow_and_Boltzmann%27s_entropy

Dear Stam,

Thanks.

The theoretical assumptions of these theories are different, the valid range of statistical mechanics seems to be less than thermodynamics and both dynamics due to the postulate of the equal a priori probability does not need to be considered for thermodynamics and both dynamics, so the “translation” is not complete.

For example, can you describe a free falling body, a real process of the potential energy to be converted into heat energy in statistical mechanics?

The theoretical foundations are, now, well established for equilibrium processes and much is known for non-equilibrium processes.

Indeed it is possible to describe particle motion in statistical terms, through the path integral formalism-where fluctuations are described by interaction with an appropriate heat bath.

Dear Stam,

The key question is that the valid ranges of these theories are different, so they cannot translate each other completely.

The driving force of the freely falling body comes from the potential energy, and the driving force and both the potential energy cannot be described by the postulate of the equal a priori probability. In this example, we can see that the valid range of statistical mechanics is less than thermodynamics and dynamics.

1) Thermodynamics can be applied to macroscopic system, but now cannot be applied to basic dynamical processes.

2) Dynamics can be applied to basic dynamical processes and both macroscopic system but does not involve the theoretical structure of irreversibility.

3) Statistical mechanics is restricted to the postulate of the equal a priori probability, but this postulate does not need to be considered in thermodynamics and dynamics.

Yes they can and the formalism that unites them is the study of probability measures in the space of configurations. The three assertions are incorrect: there does exist a thermodynamic formalism for dynamical systems, it is possible to understand why microscopic dynamics is reversible and statistical mechanics is *not* ``restricted'' to any such postulate, since what matters is the equilibrium measure. All these issues, insofar as they involve equilibrium processes are well-known, well-established and the subject of courses in statistical mechanics and quantum field theory. Non-equilibrium phenomena are, also, well-studied and the subject of many books and courses. So there isn't any ``foundational'' issue, in this sense. There are many issues regarding non-equilibrium steady states and measures of configurations in this case.

Dear Stam,

I agree that equilibrium thermodynamics is well established as an applied theory and only consider applied theory itself, except for one or a little more small point. But as a pure theory and considering the relation with dynamics, there still exist some fundamental issues.

I think “it is possible to understand why microscopic dynamics is reversible” for known dynamical laws, but I cannot sure whether or not a real dynamical process is reversible, due to all of the criterion for reversible can also be applied to the equation of the first law of thermodynamics, the first law of thermodynamics is also a reversible law, as we know, the first law of thermodynamics does not involve the question of irreversibility, and similar to known dynamical laws.