| dc.description.abstract | In spite of their simplicity, sandpile models have proved to be very useful for studying (a) SOC in model systems and (b) nonequilibrium steady states in driven systems. As we have mentioned in the brief overview given above, case (a) has been studied much more than case (b). In Chapter 2 of this thesis, we concentrate on one problem in case (a) where we analyze the possible large?scale behaviours of a sandpile model in one dimension via an exact decimation procedure and determine their universality classes. In Chapter 3 we present what we believe is the first study of phase transitions between nonequilibrium steady states in driven sandpile models where we show that an apparently simple automaton gives rise to a rich phase diagram involving a series of first?order and continuous phase transitions, thus making sandpile models a good laboratory for studying nonequilibrium critical phenomena.
In closing, let us first take up the question of the dynamical exponent. There is no way of solving the full dynamical problem exactly. However, since the continuum limit of our models is of the form
?h/?t = ??? ?H/?h + f ,?(2.43)
with
H = ? dx (?h/?x)? ,?(2.44)
we can make some conjectures: All nonlinear terms in Eq. 2.43 are proportional to a positive power of the wavenumber, as a result of the form of H (Eq. 2.44). Thus, if we try to obtain their effect perturbatively, they should not generate corrections at the lowest order in wavenumbers, so ?? should not get renormalized. Unlike in model A of Hohenberg and Halperin (1977), the Hamiltonian contains no terms depending purely on h (e.g., h²). Thus we expect that the relaxation time should simply be proportional to the susceptibility. If we interpret Eq. 2.42 in terms of a wavenumber?dependent susceptibility ?(q), then
?(q) ~ q?? ,?(2.45)
to ensure that ?h_q h_?q? ~ 1/q?. So that for a system of N sites,
?(q) ~ q?(1+2/n) , for 1 < n < 2.?(2.46)
In other words, the dynamic exponent z for this critical system is 1 + 2/n. Note that this means the n = 2 model (for which Eq. 2.46 is readily shown to be exact) is diffusive, whereas models with 1 < n < 2 are subdiffusive.
Note that, in the continuum limit, all the fixed?point models can be written in the form of Eq. 2.44. Power?counting would suggest (Grinstein, Lee and Sachdev 1990) that the higher powers of (?h/?x) are irrelevant, so that (?h/?x)², i.e., n = 2 is the only possible fixed point. That is true for models which differ only weakly from the n = 2 model. The fixed points we find are not accessible to perturbation theory about n = 2 in the conventional sense. Only models in which J decreases for large |?| may be expected to converge to the non?trivial fixed points. If, instead, we start with J ~ ?tanh ?, which saturates for large ?, we find, upon iteration, that it approaches n = 2 behaviour.
It is illuminating to look at the equation governing the slope ? alone. This is obtained in the continuum limit by differentiating the equation of motion (Eq. 2.43), which gives
??/?t = ?/?x (?? ?/?x ?H/?? + ?) ,?(2.48)
where the spatial Fourier-transform of the noise ? satisfies
??(k,t) ?(?k,t?)? ? k² ?(t ? t?).?(2.49)
Equation 2.48 describes a conserved variable ? governed by the Hamiltonian H and driven by a conserving noise ?. The current for ? is
j = ??? ?/?x (?H/??).?(2.50)
Specifically, for n = 1, j ? 0.
In other words, the effective slope?diffusivity at ? > 1 is negative. It is this instability which gives the 1 < n < 2 models their singular behaviours.
Note that our model is not in the universality class of the KPZ equation (Kardar, Parisi and Zhang 1986) for the height fluctuation of a growing one?dimensional interface. The latter contains the nonlinear term (?xh)², which is not the divergence of a current, and is hence qualitatively different from the form Eq. 2.4 that we consider.
Finally, there may well be higher?dimensional, possibly anisotropic, extensions of these models which yield to exact treatments. We hope to find some examples in the near future.
We have shown that our driven sandpile model displays a variety of steady states and many transitions between them. This richness, coupled with its simplicity, makes our model a promising one for the study of nonequilibrium phenomena in driven systems. It displays transitions similar to those in other driven systems; e.g., our onset transitions are like depinning transitions in sliding CDWs or in pinned flux?lattice systems as mentioned above. It would be interesting to study whether this similarity is merely superficial. For instance, there are some obvious ways in which the CDW models are different from ours: (1) They exhibit pinning because of quenched randomness but have no external noise; and (2) no current flows in their pinned states. Also, our sandpile models do not belong to the same universality class as the CDW systems since the critical exponents differ. However, the importance of these differences remains to be elucidated.
The discussion above shows that reduced models, in particular, the GOY shell model, provide an interesting way of studying intermittency in turbulent flows. Compared to phenomenological models, the GOY shell model is closer to the NS equations, but is at the same time numerically much more tractable than the NS equations. This is why the occurrence of intermittency in these models has received a lot of attention over the past few years. However, a theoretical understanding of the emergence of intermittency in these models is probably in its infancy.
It is debatable whether any real?space quantity can be obtained reliably from the GOY model, which is defined in a logarithmically discretised k?space. Recently Bohr et al. (1996) have proposed a procedure that can generate an artificial velocity field v(r,t) in three?dimensional real space from the shell?model velocity fields. The first step is to introduce a three?dimensional wavevector k = k? e?, where k? is the wavenumber of the n?th shell and e? are vectors of norm unity. Next the j?th component [j = x,y,z] of the real?space velocity field is constructed through some kind of an “inverse Fourier transform”:
v?(r,t) = ?? C?? u?(t) e^{i k??r} + C.C. ,?(4.38)
in which the coefficients C?? are random numbers of O(1). It can be checked that the v?’s calculated from Eq. 4.38 do obey the incompressibility criterion ?·v = 0, since the coefficients satisfy ?? C?? = 0 for all n. However, it remains to be checked explicitly whether the velocity fields calculated through such a prescription conform with those seen experimentally.
The studies mentioned above suggest that, if S is sufficiently large and ? = 1/(1 ? S), the GOY shell model yields velocity fields whose statistical properties, such as the scaling of structure functions in the inertial range, are remarkably close to those observed in experiments. In our studies in the next Chapter we will show that there is a marked similarity in the dissipation?range behaviour also. It is tempting to say, therefore, that the GOY model captures all the universal scaling properties of fluid turbulence because all known conservation laws have been built in by a suitable choice of parameters (see above). However, we must temper this optimistic view, for there are many features which seem important for fluid turbulence that are not contained in or poorly represented by the GOY model. As noted by Kadanoff et al. (1995), there is no analogue of sweeping effects in the GOY model. Furthermore, since the GOY model has only scalar velocities and k’s which are logarithmically spaced, it cannot represent high?vorticity filamentary structures, which are believed to be important in fully developed turbulence, very well. (The arguments that lead to the She–Leveque formula rely on the filamentary nature of these structures; yet, strangely enough, the GOY?model exponents ?? agree reasonably well (see Chapter 5) with this formula.) Are all these apparently important effects, which are not contained in the GOY shell model, irrelevant in some way? Unfortunately, there is no clear answer to this question at this moment. Further studies are needed to elucidate the similarities and differences between the solutions | |
