Theory of the Self-Organized Critical State in Nonequilibrium 4He

Journal of Low Temperature Physics

Volumn 119, Issue 1-2, 2000, Pages 155-179

  Weichman, P B ^a   Miller, J ^b  

^a Blackhawk Golden   (United States)

^b NEC LABORATORIES AMERICA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EQUATIONS OF STATE OF GASES; MATHEMATICAL MODELS; PHASE TRANSITIONS; TEMPERATURE; THERMAL CONDUCTIVITY OF GASES;

HEAT CURRENT; NONEQUILIBRIUM HELIUM; SELF ORGANIZED CRITICAL STATE; SUPERFLUID TRANSITION TEMPERATURE;

SUPERFLUID HELIUM;

EID: 0033879335     PISSN: 00222291     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1004668820870     Document Type: Article

References (37)

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15. 9, Q > 0 this state has a finite coherence length (though, like all superfluid states, it has an infinite thermal conductivity), and is therefore noncritical, hence our preferred notation.
16. ∞(Q) is well defined.
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22. - malization group theory of the SOC state. This theory improves on our mean field results in one important respect: it accounts approximately for renormalization of universal properties due to critical fluctuations. However, since vortices and phase slips exist only for n = 1, the dynamics of the large-n SOC state will be very different from the true dynamics (in particular, there is no persistent superflow for n > 1), and is therefore missing important physics that the mean field approximation treats correctly. At present the two approaches should be viewed as complementary, each containing a distinct subset of the key ingredients required for a full theory.
23. An anonymous referee has pointed out that a more general class of periodic pattern forming systems with an Eckhaus instability [see, e.g., P. C. Hohenberg and M. C. Cross, Rev. Mod. Phys. 65, 851 (1993) for a review] might also show self-organization of this same kind.
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25. David Goodstein, private communication.
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27. An equivalent condition was obtained by Onuki in Ref. 17.
28. If the pressure were not static, one would have to include here also thermal expansion effects (Talso Chui, private communication). Thus, to obtain a closed system of equations one also requires separate hydrodynamic equations for the pressure (or density) and for the local fluid velocity. In addition to the heat diffusion mode, these equations would also contain the ordinary (first) sound mode. When temperature varies on time scales long compared to relevant first sound frequencies, the isobaric assumption is correct and the single closed equation (11) for the temperature emerges.
29. p. This is presumably tied to the existence in this case of solutions with an oscillating component.
30. 0 = 0, the first of equations (30) is non-dissipative, and in fact identical in form to the Gross-Pitaevski equation for the order parameter field 〈Ψ〈 at zero temperature, the linearized solution to which contains a sound mode with a Bogoliubov-type spectrum. However, because the normal fluid density vanishes at T = 0, this mode corresponds to first sound, not second sound. The Gross-Pitaevski equation therefore has a very different interpretation from the Model F equation, and the latter should not be thought of as some finite temperature correction to the former.
31. s remains nonzero.
32. ∞ will presumably be metastable to an interface state, so that in principle the interface should grow continuously out of the boundary as the superfluid is heated in a quasistatic fashion (consistent with the theory presented in Ref. 18). However in any real experiment where the heating takes place at a finite rate, hysteresis effects should occur with a finite "superheating" of the superfluid state. The magnitude of the superheating will be very sensitive to boundary nucleation effects, and it remains to be seen whether it is experimentally observable.
33. See also R. Haussmann, J. Low Temp. Phys. 88, 249 (1992).
34. The experimental observability of this effect is limited by the same metastability issues described in footnote 28 above.
35. This is completely analogous to the behavior of a spin system in a magnetic field gradient: different spins will precess at different rates.
36. R. V. Duncan, private communication.
37. 1) slightly smaller than unity.