Isodynamic path

Commonly in thermodynamics, the situation is different. In thermodynamics, the isodynamic path is fixed by the first total differential dU = 0 or some other thermodynamic function. The stability is guaranteed by the total differential of second order, d t/. If d t/ > 0 then the state is stable and if d 7 < 0, the state is not stable. In these cases for the thermodynamic analysis, the total differentials of first order and second order are sufficient. Obviously, the limiting case of d C = 0 arises when the system passes from stable to unstable. This situation can be addressed as a critical state. Here, for the thermodynamic analysis, the third derivative enters the game. As pictorially explained in Example 1.14, fluctuations are likely in critical phenomena, e.g., the critical opalescence. 

From Eq. (4.30) isodynamics can be drawn in terms of S and V, cf.. Fig. 4.1. Further, we can discuss the curve along the isodynamic path. Along the isodynamic path... 

The second total derivative along the path is positive. This means that if we are moving on a tangent away from the path, the energy will increase. Thus, the isodynamic path is a minimum with respect to a deviation in direction of any tangent touching the path. The total differential of third order is... 

For the reasons explained in Sect. 1.11.3, the energy function along the isodynamic path flattens out, as both the entropy and the volume increase. 

By the way, we can easily calculate the isodynamic path by changing the variable 5 into T. At constant volume, mol number, and total energy, we get... 

---