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'''Adiabatic accessibility''' denotes a certain relation between two equilibrium states of a [[thermodynamic system]] (or of different such systems). The concept was coined by [[Constantin Carathéodory]]<ref>Constantin Carathéodory: ''Untersuchungen über die Grundlagen der Thermodynamik'', [[Mathematische Annalen|Math. Ann.]], 67:355–386, 1909</ref> in 1909 ("adiabatische Erreichbarkeit") and  taken up 90 years later by [[Elliott H. Lieb|Elliott Lieb]] and [[Jakob Yngvason|J. Yngvason]] in their axiomatic approach to the foundations of thermodynamics.<ref name="LY1999">{{cite journal |last1=Lieb |first1=Elliott H. |last2=Yngvason |first2=Jakob |year=1999 |title=The Physics and Mathematics of the Second Law of Thermodynamics |journal=Phys. Rep. |volume=310 |issue= |pages=1–96 |publisher= |doi= |url=http://de.arxiv.org/abs/cond-mat/9708200 |accessdate=7 November 2012}}</ref>
<ref name="LY2003">{{cite journal |last1=Lieb |first1=Elliott H. |last2=Yngvason |first2=Jakob |year=1999 |title=The Mathematical Structure of the Second Law of Thermodynamics |journal=arXiv |doi=10.1016/S0370-1573(98)00082-9 |year=2003|url=http://arxiv.org/abs/math-ph/0204007|accessdate=7 November 2012}}</ref> It was also used by R. Giles in his 1964 monograph.<ref>Robin Giles: "Mathematical Foundations of Thermodynamics", Pergamon, Oxford 1964</ref>
 
A system ''Y'' is said to be adiabatically accessible from ''X'' if ''X'' can be transformed into ''Y'' without heating or cooling. ''X'' may, however, be transformed to ''Y'' by  doing work on ''X''. The two systems must, of course, contain the same amount of matter. 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.
 
The original definition of Carathéodory was limited to reversible, [[quasistatic processes|quasistatic process]], 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 <math>\delta Q=dU-\sum p_idV_i</math>  
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]] 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]] as a state function <math>S</math>
whose differential <math>dS </math> is proportional to the heat differential form  <math>\delta Q</math>, 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.
 
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 <math>Y</math> is  adiabatically accessible from a state <math>X</math>, in symbols <math>X\prec Y</math>,  if it is possible to transform <math>X</math> into <math>Y</math> 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).
 
A definition of thermodynamic entropy can be based entirely on certain properties of the relation <math>\prec</math> of adiabatic accessibility that are taken as axioms in the Lieb-Yngvason approach. In the following list of properties of the <math>\prec</math> operator, a system is represented by a capital letter, e.g. ''X'', ''Y'' or ''Z''. A system ''X'' whose extensive parameters are multiplied by <math>\lambda</math>  is written <math>\lambda X</math>. (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 <math>X \prec Y</math> and <math>Y \prec X</math> 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 <math>X \overset{\underset{\mathrm{A}}{}}{\sim} Y</math>. Otherwise, it is irreversible. Adiabatic accessibility has the following properties:<ref name="LY2003"/>
 
*Reflexivity: <math>X \overset{\underset{\mathrm{A}}{}}{\sim} X</math>
*Transitivity: If <math>X \prec Y</math> and <math>Y \prec Z</math> then <math>X \prec Z</math>
*Consistency: if <math>X \prec X'</math> and <math>Y \prec Y'</math> then <math>(X,Y) \prec (X',Y')</math>
*Scaling Invariance: if <math>\lambda > 0</math> and <math>X \prec Y</math> then <math>\lambda X \prec \lambda Y</math>
*Splitting and Recombination: <math>X\overset{\underset{\mathrm{A}}{}}{\sim}((1-\lambda)X,\lambda X)</math> for all <math>0 < \lambda < 1</math>
*Stability: if <math>\lim_{\epsilon \to 0} [(X,\epsilon Z_0) \prec (Y,\epsilon Z_1)]</math> then <math>X \prec Y</math>
 
The entropy has the property that <math>S(X)\leq S(Y)</math> if and only if <math> X\prec Y</math> and <math>S(X)= S(Y)</math> if and only if <math>X \overset{\underset{\mathrm{A}}{}}{\sim} Y</math> in accord with the Second Law. If we choose two states <math>X_0</math> and <math>X_1</math> such that <math>X_0 \prec X_1</math> and assign entropies 0 and 1 respectively to them, then the entropy of a state ''X'' where <math>X_0 \prec X \prec X_1</math> is defined as:<ref name="LY2003"/>
 
:<math>S(X) = \sup (\lambda : ((1-\lambda)X_0, \lambda X_1) \prec X)</math>
 
==Sources==
<references/>
 
==References==
 
{{cite book |title=The Entropy Principle - Thermodynamics for the Unsatisfied |first=André |last=Thess |authorlink= |publisher=Springer-Verlag |location= |year= |isbn= |pages= |page= |accessdate=November 10, 2012 |url=http://www.springer.com/engineering/mechanical+engineering/book/978-3-642-13348-0 }} translated from  André Thess: ''Das Entropieprinzip - Thermodynamik für Unzufriedene'', Oldenbourg-Verlag 2007, ISBN 978-3-486-58428-8. A less mathematically intensive and more intuitive  account of the theory of Lieb and Yngvason.
 
{{cite book |title=The Entropy of Classical Thermodynamics (Princeton Series in Applied Mathematics)|last1=Lieb |first1=Elliott H. |last2=Yngvason |first2=Jakob |authorlink= |publisher=Princeton University Press |location= |year=2003|isbn= |pages=147–193 |page= |accessdate=November 10, 2012 |url=http://press.princeton.edu/titles/7688.html |editor1-first=A. |editor1-last=Greven |editor2-first=G. |editor2-last=Keller |editor3-first=G. |editor3-last=Warnecke}}
 
==External links==
* A. Thess: [http://www.tu-ilmenau.de/fakmb/fileadmin/template/fgtfd/Thess-Entropie.pdf ''Was ist Entropie?''] {{de-icon}}
 
{{DEFAULTSORT:Adiabatic Accessibility}}
[[Category:Thermodynamics]]

Revision as of 01:13, 18 January 2014

Adiabatic accessibility denotes a certain relation between two equilibrium states of a thermodynamic system (or of different such systems). The concept was coined by Constantin Carathéodory[1] in 1909 ("adiabatische Erreichbarkeit") and taken up 90 years later by Elliott Lieb and J. Yngvason in their axiomatic approach to the foundations of thermodynamics.[2] [3] It was also used by R. Giles in his 1964 monograph.[4]

A system Y is said to be adiabatically accessible from X if X can be transformed into Y without heating or cooling. X may, however, be transformed to Y by doing work on X. The two systems must, of course, contain the same amount of matter. 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.

The original definition of Carathéodory was limited to reversible, quasistatic process, 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 δQ=dU−∑pidVi 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 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 as a state function S whose differential dS is proportional to the heat differential form δQ, 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.

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≺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).

A definition of thermodynamic entropy can be based entirely on certain properties of the relation ≺ of adiabatic accessibility that are taken as axioms in the Lieb-Yngvason approach. In the following list of properties of the ≺ operator, a system is represented by a capital letter, e.g. X, Y or Z. A system X whose extensive parameters are multiplied by λ is written λX. (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≺Y and Y≺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∼AY. Otherwise, it is irreversible. Adiabatic accessibility has the following properties:[3]

The entropy has the property that S(X)≤S(Y) if and only if X≺Y and S(X)=S(Y) if and only if X∼AY in accord with the Second Law. If we choose two states X0 and X1 such that X0≺X1 and assign entropies 0 and 1 respectively to them, then the entropy of a state X where X0≺X≺X1 is defined as:[3]

S(X)=sup⁡(λ:((1−λ)X0,λX1)≺X)

Sources

  1. ↑ Constantin Carathéodory: Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann., 67:355–386, 1909
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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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  3. ↑ 3.0 3.1 3.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  4. ↑ Robin Giles: "Mathematical Foundations of Thermodynamics", Pergamon, Oxford 1964

References

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 translated from André Thess: Das Entropieprinzip - Thermodynamik für Unzufriedene, Oldenbourg-Verlag 2007, ISBN 978-3-486-58428-8. A less mathematically intensive and more intuitive account of the theory of Lieb and Yngvason.

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

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