Mathematics – Probability
Scientific paper
Sep 1994
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994a%26a...289..325v&link_type=abstract
Astronomy and Astrophysics (ISSN 0004-6361), vol. 289, no. 2, p. 325-356
Mathematics
Probability
122
Burger Equation, Cosmology, Functions (Mathematics), Mass Distribution, Mathematical Models, Probability Theory, Universe, Adhesion, Brownian Movements, Dimensions, Fractals, Lagrange Coordinates, Legendre Functions, Power Spectra, Simulation
Scientific paper
In the simple one-dimensional case, with ordinary Brownian motion as initial condition, numerically supported conjectures by She et al. have led to a proof by Sinai of the following result: there is a Devil's staricase of dimension 1/2 in the Lagranian map for the solution of the Burgers equation in the limit of vanishing viscosity. The main goal of this paper is to give a comprehensive introduction to these recent theoretical develpments and the extend them to initial power-law spectra with a wide range of exponents and to more than one dimension. The necessary geometric and probablistic tools, due mostly to Sinai, are here presented in a detailed but rather elementary way, intended for a readership of general physicists. The extensions of Sinai's theory presented here offer included heuristic elements. Most of the predictions are however tested by accurate numerical experiments in one and two dimensions, using a new 'Fast Legendre Transform' algorithm which exploits a monotonicity property and has very low storage requirements. Predictions of the present theory for the mass function are compared to those of Press and Schechter (1974). Their expression for the mass function is found to agree with the adhesion model at large masses in any dimension. At small masses, there are discrpancies in dimension. At small masses, there are discrpancies in dimensions higher than one. In one dimension the scaling behavior at small masses is correctly given by the Press-Schechter theory. Sinai's theory, recast in the language of gravitational collapse, tells us that the scaling originates not from the condition of collapse of a given region of small size, but from a condition of non-collapse of an extended halo around the region.
Dubrulle Bérangère
Frisch Uriel
Noullez Alain
Vergassola Massimo
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