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In the two fluid model a sound wave in He II must mean a sinusoidal oscillation of Pn and Ps in phase with each other, because only when they oscillate in phase can the total mass density vary sinusoidally We can also imagine a ne\\ mode of oscillation, however, in which the normal fluid and the superfluid oscillate out of phase by 180 This would not be a sound wave, for the total mass density would be constant throughout the liquid, but it would represent a sinusoidal variation of the entropy per unit mass, because the superfluid has no entropy Therefore, we may hope to excite this new mode of oscillation by local heating of He II The temperature gradient established would not propagate b~ diffusion, as in heat conduction, but would propagate like a wave, with a characteristic velocity This phenomenon is called the second sound.

All these phenomena are observed in He II Applying purely thermodynamic considerations, we can describe them quantitatively in terms of the two-fluid model In the two-fluid model the relative amount of normal fluid and superfluid present may be deduced from an experiment of Andronikashvili A pile of disks spaced 02 mm apart are mounted on a shaft, and the assembly is made to rotate in He II, as shown in Fig 135 The moment of inertia of the assembly is measured as a function of the temperature, when the He II in equilibrium with its own vapor By assuming that the superfluid remains completely unaffected by the rotation disks, whereas the normal fluid between the disks is dragged into rotation, we see that the moment of inertia must be proportional to Pn/P' The proportionality constant is determined by the fact that at the transition tempera-.

(13.6)

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FIGURE 5.17 The PSAR indicator is very easy to use: be long if the price is above the PSAR and short if the price is below it.

Penrose and Onsager* have given an estimate of the fraction of particles in liquid He 4 having zero momentum, by making use of (12.92) for a translationally invariant liquid. An equivalent expression for the one-particle density matrix in terms of the wave functions of the system is (cf. (A.53

PI(X - y) == (lP(X)1J;(y))

=NL:f d 3rz ... d3rN'I',,(y,fz, ,fN)e-,B..If''I',,*(X,fz, ,fN) (13.7)

PI(f) =Nf d 3rz ... d3rN'I'0(f,fz, ,fN)'I'O(0,fz, ,fN)

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where '1'0 is the ground state wave function. It is real, positive, and unique, as shown in the Appendix. Intuitively we expect '1'0 to be approximately constant except when two He atoms" touch." Then it is essentially zero. The main effect of the attractive part of the potential is to give the whole system a selfdetermined equilibrium density. It is plausible that this effect is not important for our considerations, as long as we assume the liquid to have the observed density of liquid He 4 We assume that

(13.8)

FN(fl, ,f N )=,

stop price every day based on the PSAR value, if you were using this indicator devoutly. Take a look at the chart of symbol HANS in Figure 5.18. The activity for most of April had prices moving higher, and the PSAR was below the prices. The last two price bars on the graph, however, have the PSAR above instead of below the prices. The reason is that the second-tolast price bar touched the PSAR dot, ipping this dot s location from beneath the price bar to above it. Until such time as the prices touch a PSAR dot, the PSAR will remain in this bearish position.

i - f) ::s; a, any i . (otherwIse)

* j)

(13.9)

= 2.56 A

The constant ZN/N! is numerically equal to the configuration integral of a classical hard-sphere gas. We note that

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