×

Statistical properties of the Burgers equation with Brownian initial velocity. (English) Zbl 1162.82017

Summary: We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin \(x=0\)). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its \(n\)-point correlations. In the same limit, we derive the \(n\)-point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
60J65 Brownian motion
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1970) · Zbl 0171.38503
[2] Aurell, E., Gurbatov, S.N., Wertgeim, I.I.: Self-preservation of large-scale structures in Burgers’ turbulence. Phys. Lett. A 182, 109–113 (1993) · doi:10.1016/0375-9601(93)90062-5
[3] Aurell, E., Frisch, U., Noullez, A., Blank, M.: Bifractality of the devil’s staircase appearing in the Burgers equation with Brownian initial velocity. J. Stat. Phys. 88, 1151–1164 (1997) · Zbl 0924.60044 · doi:10.1007/BF02732429
[4] Balian, R., Schaeffer, R.: Scale-invariant matter distribution in the universe. I–Counts in cells. Astron. Astrophys. 220, 1–29 (1989)
[5] Balian, R., Schaeffer, R.: Scale-invariant matter distribution in the universe. II–Bifractal behaviour. Astron. Astrophys. 226, 373–414 (1989)
[6] Bec, J., Khanin, K.: Burgers turbulence. Phys. Rep. 447, 1–66 (2007) · doi:10.1016/j.physrep.2007.04.002
[7] Bernardeau, F.: The effects of smoothing on the statistical properties of large-scale cosmic fields. Astron. Astrophys. 291, 697–712 (1994)
[8] Bernardeau, F., Colombi, S., Gaztaaga, E., Scoccimarro, R.: Large-scale structure of the universe and cosmological perturbation theory. Phys. Rep. 367, 1–248 (2002) · Zbl 0996.85005 · doi:10.1016/S0370-1573(02)00135-7
[9] Bertoin, J.: The inviscid Burgers equation with Brownian initial velocity. Commun. Math. Phys. 193, 397–406 (1998) · Zbl 0917.60063 · doi:10.1007/s002200050334
[10] Burgers, J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974) · Zbl 0302.60048
[11] Burkhardt, T.W.: Semiflexible polymer in the half plane and statistics of the integral of a Brownian curve. J. Phys. A 26, L1157–L1162 (1993) · doi:10.1088/0305-4470/26/22/005
[12] Carraro, L., Duchon, J.: Equation de Burgers avec conditions initiales a accroissements independants et homogenes. Ann. Inst. Henri Poincare 15, 431–458 (1998) · Zbl 0912.35163 · doi:10.1016/S0294-1449(98)80030-9
[13] Cole, J.D.: On a quasi-linear parabolic equation occuring in aerodynamics. Q. Appl. Math. 9, 225–236 (1951) · Zbl 0043.09902
[14] Colombi, S., Bouchet, F.R., Schaeffer, R.: Large scale structure statistics: Finite volume effects. Astron. Astrophys. 281, 301–313 (1994)
[15] Davis, M., Peebles, P.J.E.: On the integration of the bbgky equations for the development of strongly nonlinear clustering in an expanding universe. Astrophys. J. Suppl. S. 34, 425–450 (1977) · doi:10.1086/190456
[16] Fournier, J.-D., Frisch, U.: L’equation de Burgers deterministe et statistique. J. Mec. Theor. Appl. 2, 699–750 (1983) · Zbl 0573.76058
[17] Frachebourg, L., Martin, Ph.A.: Exact statistical properties of the Burgers equation. J. Fluid Mech. 417, 323–349 (2000) · Zbl 0961.76016 · doi:10.1017/S0022112000001142
[18] Frisch, U., Bec, J.: ”Burgulence”. In: Lesieur, M., Yaglom, A., David, F. (eds.) Les Houches 2000: New trends in turbulence. Springer, Berlin (2001)
[19] Frisch, U., Bec, J., Aurell, E.: ”Locally homogeneous turbulence” is it an inconsistent framework? Phys. Fluids 17, 081706 (2005) · Zbl 1187.76166 · doi:10.1063/1.2008994
[20] Fry, J.N.: Galaxy n-point correlation functions–theoretical amplitudes for arbitrary n. Astrophys. J. 277, L5–L8 (1984) · doi:10.1086/184189
[21] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, New York (1965) · Zbl 0918.65002
[22] Gurbatov, S., Malakhov, A., Saichev, A.: Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles. Manchester University Press, Manchester (1991) · Zbl 0860.76002
[23] Gurbatov, S.N., Pasmanik, G.V.: Self-preservation of large-scale structures in a nonlinear viscous medium described by the Burgers equation. Sov. Phys. JETP 88, 309–319 (1999) · doi:10.1134/1.558798
[24] Gurbatov, S.N., Saichev, A.I.: Degeneracy of one-dimensional acoustic turbulence at large Reynolds numbers. Sov. Phys. JETP 53, 347–354 (1981) · Zbl 0483.76062
[25] Gurbatov, S.N., Saichev, A.I., Shandarin, S.F.: The large-scale structure of the universe in the frame of the model equation of non-linear diffusion. Mon. Not. R. Astron. Soc. 236, 385–402 (1989) · Zbl 0662.76062
[26] Gurbatov, S.N., Simdyankin, S.I., Aurell, E., Frisch, U., Toth, G.: On the decay of Burgers turbulence. J. Fluid Mech. 344, 339–374 (1997) · Zbl 0941.76040 · doi:10.1017/S0022112097006241
[27] Hagan, P.S., Doering, C.R., Levermore, C.D.: The distribution of exit times for weakly colored noise. J. Stat. Phys. 54, 1321–1352 (1989) · Zbl 0714.60072 · doi:10.1007/BF01044718
[28] Hopf, E.: The partial differential equation u t +uu x =u xx . Commun. Pure Appl. Mech. 3, 201–230 (1950) · Zbl 0039.10403 · doi:10.1002/cpa.3160030302
[29] Kida, S.: Asymptotic properties of Burgers turbulence. J. Fluid Mech. 93, 337–377 (1979) · Zbl 0436.76031 · doi:10.1017/S0022112079001932
[30] Le Doussal, P.: Exact results and open questions in first principle functional rg. arXiv:0809.1192 (2008) · Zbl 1187.82059
[31] Marshall, T.W., Watson, E.J.: A drop of ink falls from my pen... it comes to earth, I know not when. J. Phys. A, Math. Gen. 18, 3531–3559 (1985) · Zbl 0592.60073 · doi:10.1088/0305-4470/18/18/016
[32] Marshall, T.W., Watson, E.J.: The analytic solutions of some boundary layer problems in the theory of Brownian motion. J. Phys. A, Math. Gen. 20, 1345–1354 (1987) · Zbl 0634.35075 · doi:10.1088/0305-4470/20/6/018
[33] Melott, A.L., Shandarin, S.F., Weinberg, D.H.: A test of the adhesion approximation for gravitational clustering. Astrophys. J. 428, 28–34 (1994) · doi:10.1086/174216
[34] Molchan, G.M.: Burgers equation with self-similar Gaussian initial data: tail probabilities. J. Stat. Phys. 88, 1139–1150 (1997) · Zbl 0944.60073 · doi:10.1007/BF02732428
[35] Molchanov, S.A., Surgailis, D., Woyczynski, W.A.: Hyperbolic asymptotics in Burgers’ turbulence and extremal processes. Commun. Math. Phys. 168, 209–226 (1995) · Zbl 0818.60046 · doi:10.1007/BF02099589
[36] Noullez, A., Gurbatov, S.N., Aurell, E., Simdyankin, S.I.: Global picture of self-similar and non-self-similar decay in Burgers turbulence. Phys. Rev. E 71, 056305 (2005) · doi:10.1103/PhysRevE.71.056305
[37] Peebles, P.J.E.: The Large Scale Structure of the Universe. Princeton University Press, Princeton (1980)
[38] Press, W., Schechter, P.: Formation of galaxies and clusters of galaxies by self-similar gravitational condensation. Astrophys. J. 187, 425–438 (1974) · doi:10.1086/152650
[39] Prodinger, H., Urbanek, F.J.: On monotone functions of tree structures. Discrete Appl. Math. 5, 223–239 (1983) · Zbl 0508.05042 · doi:10.1016/0166-218X(83)90043-4
[40] Schaeffer, R.: The probability generating function for galaxy clustering. Astron. Astrophys. 144, L1–L4 (1985)
[41] She, Z.-S., Aurell, E., Frisch, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–641 (1992) · Zbl 0755.60104 · doi:10.1007/BF02096551
[42] Sheth, R.K., Tormen, G.: Large-scale bias and the peak background split. Mon. Not. R. Astron. Soc. 308, 119–126 (1999) · doi:10.1046/j.1365-8711.1999.02692.x
[43] Sinai, Ya.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148, 601–621 (1992) · Zbl 0755.60105 · doi:10.1007/BF02096550
[44] Valageas, P.: Non-linear gravitational clustering: smooth halos, substructures and scaling exponents. Astron. Astrophys. 347, 757–768 (1999)
[45] Valageas, P.: Dynamics of gravitational clustering. II. Steepest-descent method for the quasi-linear regime. Astron. Astrophys. 382, 412–430 (2002) · Zbl 1032.83056 · doi:10.1051/0004-6361:20011663
[46] Valageas, P.: Dynamics of gravitational clustering. IV. The probability distribution of rare events. Astron. Astrophys. 382, 450–476 (2002) · Zbl 1032.83057 · doi:10.1051/0004-6361:20011673
[47] Valageas, P.: Using the Zeldovich dynamics to test expansion schemes. Astron. Astrophys. 476, 31–58 (2007) · Zbl 1130.85341 · doi:10.1051/0004-6361:20078065
[48] Valageas, P.: Ballistic aggregation for one-sided Brownian initial velocity. Physica A 388, 1031–1045 (2009). arXiv:0809.1192 · doi:10.1016/j.physa.2008.12.033
[49] Valageas, P., Munshi, D.: Evolution of the cosmological density distribution function: a new analytical model. Mon. Not. R. Astron. Soc. 354, 1146–1158 (2004) · doi:10.1111/j.1365-2966.2004.08325.x
[50] Vallée, O., Soares, M.: Les Fonctions d’Airy pour la Physique. Diderot, Paris (1998)
[51] Vergassola, M., Dubrulle, B., Frisch, U., Noullez, A.: Burgers’equation, devil’s staircases and the mass distribution for large-scale structures. Astron. Astrophys. 289, 325–356 (1994)
[52] Zeldovich, Y.B.: Gravitational instability: An approximate theory for large density perturbations. Astron. Astrophys. 5, 84–89 (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.