Contrary to a common misconception, thermodynamics is not restricted to macroscopic systems.
Thermodynamics though in standard textbooks is valid only for the macroscopic limit –often incorrectly called the thermodynamic limit–, a formal limit where surface effects and others phenomena are ignored.
For small systems as cellular micelles with a large ratio surface/volume or for very large systems –such as stars–, where gravitational effects are important, an extended formulation is necessary, because the macroscopic limit is not applicable.
Macroscopic (or classical) thermodynamics is the science arising when the limit is applied.
Quantum thermodynamics is to classical thermodynamics as quantum electrodynamics is to classical electrodynamics.
For instance, the entropy for an atom is
with ρ the generalized quantum state.
Nanothermodynamics lives –so say that– in the region between classical and quantum thermodynamics. As its name indicates, nanothermodynamics studies systems on the nanoscale (nm), that is, at 10(\sup –9) m; there is ten millions of nm on a single centimeter.
In the last few years, our experimental capabilities for investigating small systems as molecular clusters, microemulsions, biological membranes, and others has increased.
Some research groups on the entire world have recently claimed theoretical and experimental verifications of violations of thermodynamics on the nanoscale. Some authors claim violation of the inequality in nanoscale electric circuits whereas others argue –in basis to theoretical work– that via sophisticated 'molecular bridges' some systems can violate the classical law of thermal equilibrium. Others measure thermal properties in colloids and even smaller systems as atomic nuclei. Contrary to an extended belief, thermodynamical laws were not violated. Those groups have just found that macroscopic thermodynamics conclusions are not applicable everywhere.
Those works prove nothing about thermodynamics as a whole. Classical thermodynamics does not apply on the small system and very low temperature regime; thus, application of classical thermodynamics outside its range of validity is not acceptable.
The Center has established some recent advances in the formulation of non-macroscopic systems, such as a systematic expansion outside the macroscopic limit, a rule for transformation of intensive macroscopic parameters to the nanoregime or an extended version of the classical second law of thermodynamics.
From thermal canonical science, we can obtain the law of equilibrium for temperatures in the non-macroscopic regime
When taking the macroscopic limit N → ∞, the classical limit ε → 0, or considering the high temperature regime, for instance T ~ 10(\sup 3), we obtain the equilibrium law of macroscopic thermodynamics
The non-macroscopic law also reduces to the classical law in the limit T → ∞. The new thermal phenomena is mainly associated to very low temperatures, small amounts of matter, and quantum interchanges.
Traditionally, it has been thought that spontaneous evolutions to states characterized by higher values of entropy imply the evolution to a final homogeneous state. For example, as seen above macroscopic thermodynamics predicts the homogeneity of temperatures at the final equilibrium state.
This fact of classical thermodynamics has perplexed many researchers during decades, since the history of our universe is the history of an observed increase of order: from an initial soap of elementary particles to the current observed cosmos with galactic structures, planets, and highly organized living structures. If the evolution of our universe is from order to disorder then why do we see more order each day? Physicists as wait to explain the evolution of the universe through frozen accidents
by some kind of self-organization mechanism was compatible with the classical law in some way, but their whole approach remains untested. The new canonical theory reinterprets the molecular disorder and its relation to overall observables.
The canonical science modifications to classical classical laws are in good agreement with theoretical and experimental works on the topic. Above we see how the equality of temperatures at equilibrium is not so universal as was generally believed before now. Our conclusion is in good agreement with results obtained from independent groups. As another example, the correction to the equality on pressures for a 1000-molecules gaseous system coincides with results obtained by others groups. We would wait also departures from classical thermodynamics in the very large regime also; see the discussion on gigathermodynamics below.
Canonical nanothermodynamics shows how an increasing of molecular disorder may be compatible with the generation of 'visual' structures.
The generalized second law predicts the formation of structures at equilibrium –see figure below–. These epsilon-structures may be an important key in our understanding of the origin of life and of self-organization in the Universe.
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epsilon thermal structure in a non-macroscopic system at equilibrium (left). Absence for macroscopic systems (right). The new laws allow spatial inhomogeneity out of the macroscopic limit. |
The behavior of very large systems may also be radically different from usual macroscopic systems. The application of macroscopic thermodynamics to the astronomical scale always was problematic and has generated some 'paradoxes'. The difficulty arises from the long range of gravitational interactions. The new theory explains us why.
From above canonical expression, we notice that deviations from classical thermodynamics are proportional to ε/N(\sub A). since particle correlations are short ranged and N ~ 10(\sup 23) for typical macroscopic systems the term can be dropped. When scaling down the system, the number of particles is smaller and the deviation term gains more and more importance. For thousands of particles, the ε/N(\sub A) term is typically 10(\sup 20) times larger than for a macroscopic system.
The number of particles for systems on the astronomical scale is much larger. Geologists estimate the Earth consists of roughly 10(\sup 50) atoms; astronomers estimate there are roughly 10(\sup 57) particles in the Sun. The traditional view stated that classical thermodynamics also applies to stars, or even to the universe as a whole. However, we know that 'exotic' phenomena are common on astronomical objects –e.g. negative heat capacities–. Apparently, canonical science can also explain this.
For both macroscopic and nanomatter ε is of molecular size because gravity can be avoided. However, when gravitational effects are present, ε may be large enough. The term ε/N(\sub A) does not vanish at the giga scale even if the number of particles is very large.
Gigathermodynamics studies systems on the gigascale meter (Gm), that are 10(\sup 9) m.
We wait that the gigathermodynamics derived from canonical science can explain the formation of large-scale structures on the universe without need for the ad-hoc introduction of new laws of thermodynamics, such as the fourth law postulated by .
The Center also addresses the relationships between finite-size thermodynamics and the recurrence 'paradox', the link with other nanothermodynamics (, , and ...) and the link with the time symmetry of quantum theory.
An unexpected field of investigation opened by thermal canonical science has been the discovering of a relationship between the stoichiometric vectors of the canonical theory and the non-standard analysis worked out by mathematicians.
The continuum limit can be alternatively derived from the condition ε ≠ 0, verifying ε(\sup 2) = 0. There is not adequate real number satisfying both conditions, and this open an intriguing relationship with the non-standard topology of the real line and with that mathematicians call hiperreal numbers.
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