# A brill - noether theory for k-gonal nodal curves by Ballico E.

By Ballico E.

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**Example text**

As far as thermodynamic macroscopic observables are concerned, making the link with a thermodynamic potential is enough, because all the other observables can be worked out using the standard macroscopic relations (such as Maxwell’s relations). The usual approach is to deﬁne entropy in the microcanonical ensemble as the logarithm of phase space volume. Then one shows that the proposed deﬁnition has all the properties that entropy is expected to have. Then one proceeds by considering a large-N subsystem of a larger system (to be called thermostat), so that the total energy of the subsystem is no longer a constant, and by working out the ensemble density in the phase space of the subsystem, one is led to deﬁne a canonical ensemble where the basic mathematical object, the partition function, is directly related to the Helmholtz free energy.

50) where v = V/N is the speciﬁc volume (inverse density), and where we have used P (v) = −(1/N )(∂F/∂v) for the pressure of the system. It is an experimental fact that (∂P/∂v) ≤ 0 holds always true. 50) we have σN /N → ∞ as N → ∞. 48), we ﬁnally obtain the Helmholtz free energy, from which all the other thermodynamic functions can be derived. 6 Phase Transitions Phase transitions involve abrupt major changes of the physical properties of macroscopic objects when a thermodynamic parameter is even slightly varied across a critical value.

Nevertheless, as we shall see at the end of the present chapter, the mixing characteristic time can be a nontrivial function of the energy of the system, so that also very slow relaxations (even apparent freezing) of time to ensemble averages can be observed. 5 From Micro to Macro: The Link with Thermodynamics We have seen that the time averages of a physical observable can be replaced by static ensemble averages under rather generic physical conditions. As far as thermodynamic macroscopic observables are concerned, making the link with a thermodynamic potential is enough, because all the other observables can be worked out using the standard macroscopic relations (such as Maxwell’s relations).