By W. C. Reynolds, R. W. MacCormack

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**Additional resources for 7th Int'l Conference on Numerical Methods in Fluid Dynamics**

**Example text**

In the limit of large g this function takes the simple form f (z) = zN . 1 + zN This result is known as the Hill equation and was originally introduced as a ﬁt to the experimental curve for the ﬁlling fraction. The Ising model is thus in qualitative agreement with the experimentally measured oxygen saturation curve for hemoglobin. Let us now turn to the deoxyribonucleic acid or DNA molecule. 5. The physical phenomena which we would like to describe is the “melting” of DNA, that is the breaking up of the hydrogen bonds as the temperature increases.

Proof. Consider an arbitrary irreversible thermodynamic transformation, tI , which transforms a system from a state A into a state B. Find a reversible thermodynamic transformation, tR , which transforms the system back from B to A. Note that we assume that such a reversible transformation is always possible to ﬁnd. By the Clausius inequality, we have B A,tI δQ + T A B,tR δQ ≤ 0. 9 Sketch of the entropy function for ﬁxed N . If the system is thermally isolated as it undergoes the irreversible transformation tI , then we have Q = 0 during the transformation, and so S(B) − S(A) ≥ B A,tI δQ =0 T which gives S(B) ≥ S(A).

For a gas of molecules the dynamical state involved specifying 6N variables for N > 1020 molecules. The thermodynamic state was described by a small number of variables such as the gas temperature, the gas pressure, and the gas volume. From our remarks regarding the interpretation of temperature and internal energy it follows that many dynamical states correspond to the same thermodynamic state since clearly a gas of molecules of ﬁxed energy can be in many different dynamical conﬁgurations. For instance, a molecule moving along the x axis with momentum p has the same kinetic energy as a molecule moving with momentum p in the y direction.