adiabatic accessibility
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Adiabatic accessibility denotes a certain relation between two
equilibrium state Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
s of a
thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that are ...
(or of different such systems). The concept was coined by
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
in 1909 ("adiabatische Erreichbarkeit") and taken up 90 years later by
Elliott Lieb Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis. Lieb i ...
and J. Yngvason in their axiomatic approach to the foundations of thermodynamics. It was also used by R. Giles in his 1964 monograph.Robin Giles: "Mathematical Foundations of Thermodynamics", Pergamon, Oxford 1964


Description

A system in a state ''Y'' is said to be adiabatically accessible from a state ''X'' if ''X'' can be transformed into ''Y'' without the system suffering transfer of energy as heat or transfer of matter. ''X'' may, however, be transformed to ''Y'' by doing work on ''X''. For example, a system consisting of one kilogram of warm water is adiabatically accessible from a system consisting of one kilogram of cool water, since the cool water may be mechanically stirred to warm it. However, the cool water is not adiabatically accessible from the warm water, since no amount or type of work may be done to cool it.


Carathéodory

The original definition of Carathéodory was limited to reversible,
quasistatic process In thermodynamics, a quasi-static process (also known as a quasi-equilibrium process; from the Latin ''quasi'', meaning ‘as if’), is a thermodynamic process that happens slowly enough for the system to remain in internal physical (but not ne ...
, described by a curve in the manifold of equilibrium states of the system under consideration. He called such a state change adiabatic if the infinitesimal 'heat' differential form \delta Q=dU-\sum p_idV_i vanishes along the curve. In other words, at no time in the process does heat enter or leave the system. Carathéodory's formulation of the
Second Law of Thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
then takes the form: "In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state." From this principle he derived the existence of
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
as a state function S whose differential dS is proportional to the heat differential form \delta Q, so it remains constant under adiabatic state changes (in Carathéodory's sense). The increase of entropy during irreversible processes is not obvious in this formulation, without further assumptions.


Lieb and Yngvason

The definition employed by Lieb and Yngvason is rather different since the state changes considered can be the result of arbitrarily complicated, possibly violent, irreversible processes and there is no mention of 'heat' or differential forms. In the example of the water given above, if the stirring is done slowly, the transition from cool water to warm water will be quasistatic. However, a system containing an exploded firecracker is adiabatically accessible from a system containing an unexploded firecracker (but not vice versa), and this transition is far from quasistatic. Lieb and Yngvason's definition of adiabatic accessibility is: A state Y is adiabatically accessible from a state X, in symbols X\prec Y (pronounced X 'precedes' Y), if it is possible to transform X into Y in such a way that the only net effect of the process on the surroundings is that a weight has been raised or lowered (or a spring is stretched/compressed, or a flywheel is set in motion).


Thermodynamic entropy

A definition of thermodynamic entropy can be based entirely on certain properties of the relation \prec of adiabatic accessibility that are taken as axioms in the Lieb-Yngvason approach. In the following list of properties of the \prec operator, a system is represented by a capital letter, e.g. ''X'', ''Y'' or ''Z''. A system ''X'' whose extensive parameters are multiplied by \lambda is written \lambda X. (e.g. for a simple gas, this would mean twice the amount of gas in twice the volume, at the same pressure.) A system consisting of two subsystems ''X'' and ''Y'' is written (X,Y). If X \prec Y and Y \prec X are both true, then each system can access the other and the transformation taking one into the other is reversible. This is an equivalence relationship written X \overset Y. Otherwise, it is irreversible. Adiabatic accessibility has the following properties: *Reflexivity: X \overset X *Transitivity: If X \prec Y and Y \prec Z then X \prec Z *Consistency: if X \prec X' and Y \prec Y' then (X,Y) \prec (X',Y') *Scaling Invariance: if \lambda > 0 and X \prec Y then \lambda X \prec \lambda Y *Splitting and Recombination: X\overset((1-\lambda)X,\lambda X) for all 0 < \lambda < 1 *Stability: if \lim_ X,\epsilon Z_0) \prec (Y,\epsilon Z_1)/math> then X \prec Y The entropy has the property that S(X)\leq S(Y) if and only if X\prec Y and S(X)= S(Y) if and only if X \overset Y in accord with the Second Law. If we choose two states X_0 and X_1 such that X_0 \prec X_1 and assign entropies 0 and 1 respectively to them, then the entropy of a state ''X'' where X_0 \prec X \prec X_1 is defined as: :S(X) = \sup (\lambda : ((1-\lambda)X_0, \lambda X_1) \prec X)


Sources


References

translated from André Thess: ''Das Entropieprinzip - Thermodynamik für Unzufriedene'', Oldenbourg-Verlag 2007, . A less mathematically intensive and more intuitive account of the theory of Lieb and Yngvason.


External links

* A. Thess
''Was ist Entropie?''
{{DEFAULTSORT:Adiabatic Accessibility Equilibrium chemistry Thermodynamic cycles Thermodynamic processes Thermodynamic systems Thermodynamics