Abstract
We consider two stochastic equations that describe the turbulent transfer of a passive scalar field θ(x) ≡ θ(t,x) and generalize the known Obukhov–Kraichnan model to the case of a possible compressibility and large-scale anisotropy. The pair correlation function of the field θ(x) is characterized by an infinite collection of anomalous indices, which have previously been found exactly using the zero-mode method. In the quantum field formulation, these indices are identified with the critical dimensions of an infinite family of tensor composite operators that are quadratic in the field θ(x), which allows obtaining exact values for the latter (the values not restricted to the ε-expansion) and then using them to find the corresponding renormalization constants. The identification of the correlation function indices with the composite-operator dimensions itself is supported by a direct calculation of the critical dimensions in the one-loop approximation.
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REFERENCES
G. Falkovich, K. Gawedzki, and M. Vergassola, Rev. Modern Phys., 73, 913 (2001).
D. Bernard, K. Gawedzki, and A. Kupiainen, Phys. Rev. E, 54, 2564 (1996).
M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Phys.Rev.E, 52, 4924 (1995).
L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil'ev, Phys. Rev. E, 58, 1823 (1998); Theor. Math. Phys., 120, 1074 (1999).
L. Ts. Adzhemyan and N. V. Antonov, Phys.Rev.E, 58, 7381 (1998).
L. Ts. Adzhemyan et al., Phys. Rev. E, 64, 056306 (2001).
M. E. Fisher, The Nature ofCriticalPoints, Univ. of Colorado Press, Boulder, Colo. (1965); H. Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon, Oxford (1971); F. J. Dyson, “Stability of matter,” in: Statistical Physics, Phase Transitions, and Superfluidity (M. Chrötiln et al., eds.), Vol. 1, Gordon and Breach, New York (1968), p. 179; M. Kac, “Mathematical mechanism of phase transitions,” in: Statistical Physics, Phase Transitions, and Superfluidity (M. Chrötiln et al., eds.), Vol. 1, Gordon and Breach, New York (1968), p. 241; E. W. Montroll, “Lectures on the Ising model of phase transitions,” in: Statistical Physics, Phase Transitions, and Superfluidity (M. Chr´etiln et al., eds.), Gordon and Breach, New York (1968), p. 195; M. E. Fisher, “The theory of critical point singularities,” in: Enrico Fermi Summer School of “Critical Phenomena,” (Villa Monastero, Varenna sub Lago di Como, Italy, 1970, M. B. Green, ed.), Acad. Press, New York (1971).
N. V. Antonov and J. Honkonen, Phys.Rev.E, 63, 036302 (2001).
N. V. Antonov and J. Honkonen, Vestn. S.-Petersb. Univ. Ser. Fiz. Khim., No. 1 (in print 2004).
W. Heisenberg, Z. Phys., 124, 628 (1948).
L. Ts. Adzhemyan and N. V. Antonov, Theor. Math. Phys., 115, 562 (1998).
D. V. Shirkov, Sov. Phys. Dokl., 27, 197 (1982); Theor. Math. Phys., 60, 778 (1985).
T. D. Lee, Phys. Rev., 95, 1329 (1954).
W. Thirring, Ann. Phys., 3, 91 (1958).
K. G. Wilson, Phys. Rev. D, 2, 1438 (1970).
A. N. Vasilöv, Quantum Field Renormlization Group in the Theory of Critical Behavior and Stochastic Dynamics [in Russian], Petersburg Inst. for Nuclear Phys. Publ., St. Petersburg (1998); English transl.: The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics, Chapman and Hall/CRC, Boca Raton, Flor. (2004).
U. Frisch, A. Mazzino, A. Noullez, and M. Vergassola, Phys. Fluids, 11, 2178 (1999); A. Mazzino and P. Muratore Ginanneschi, Phys. Rev. E, 63, 015302(R) (2001).
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Antonov, N.V., Gol'din, P.B. Exact Anomalous Dimensions of Composite Operators in the Obukhov–Kraichnan Model. Theoretical and Mathematical Physics 141, 1725–1736 (2004). https://doi.org/10.1023/B:TAMP.0000049764.37693.6d
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DOI: https://doi.org/10.1023/B:TAMP.0000049764.37693.6d