Temperature as a derived quantity

We have already mentioned that for small systems with a finite number of degrees of freedom, the relative fluctuations have much more impact and cause broader distributions. Calculating canonical averages cancels out relevant information about the system s thermodynamic behavior. Therefore, it might be useful to decouple the temperature from the heat bath and introduce it as a structural property of the system itself The idea is that a conformational phase and transitions between different such phases can then be described entirely by the change of the configurational/conformational entropy S upon a 

As it has already been discussed in Section 2.2.3, the microcanonical entropy is given by S E)=kQ i g E). In this context, the (microcanonical) temperature T E) becomes a defined quantity, which is associated to the entropic change caused by a variation of macrostate energy. We strictly introduce it via 

What is now the advantage of the microcanonical temperature compared to its canonical counterpart Since ( ) is directly derived from the fundamental system quantities S and E, its curve E) should contain all information about the system behavior if it undergoes a macroscopic change such as a cooperative transition. It will turn out that transitions occur if /3( ) responds least sensitively to changes in system energy. By means of this least-sensitivity principle, it is possible to identify transitions uniquely, even in small systems. This also means that a transition point can be uniquely assigned a single transition temperature - this is typically not possible in the canonical formalism, where the transition point depends on the fluctuation extremum of the chosen order parameter. In contrast to the canonical counterpart, the microcanonical temperature is a system property and not an external parameter that can be controlled. 

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