In full equilibrium, not just thermal equilibrium, the system must reside in the lowest minimum of this function with respect to the variable. 1 actually correspond to the equilibrium value of the Helmholtz free energy, and so Next, we recognize that the first two terms on the r.h.s. Physically, this corresponds to a situation where we have thermal equilibrium in place but not the mechanical equilibrium, unless is so happens that the current value is at the stable minimum of. It will be most practical to consider a reduced form of the thermodynamic potential from the preceding Chapter, where we leave the volume as a variable while setting the energy at its likeliest value for each individual value of volume : Let us now consider an alternative situation where the pertinent thermodynamic potential has two minima. Non-withstanding these fluctuations, the system characterized by a single minimum has only one thermodynamic state, or “phase”, to speak of. For instance, the average of the total energy of a system of size scales proportionally with, while the fluctuation of the total energy go only as, a much slower function of. In reality, local fluctuations of extensive variables such as the energy, volume, etc., are substantial but largely cancel out when added together. One should not be misled by the fact that the fluctuations of extensive variables are relatively small, for large systems. These fluctuations are thus intrinsic physically but are also inherent to the very definition of a thermodynamic state. Consistent with this notion, a thermodynamic state corresponds to a vast set of rather similar microstates that are explored as the system fluctuates. The length and the tension of the spring may change in the process, yet microscopically, the material is much the same as in the absence of external load and will return back to its initial state once the load is off. Changing the volume and/or energy of a system with just one minimum is much like compressing/stretching a mechanical spring. We have so far dealt largely with systems whose free energy has just one minimum.
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