Abstract
Results from the theory ofU-statistics are used to characterize the microcanonical partition function of theN-vortex system in a rectangular region for largeN, under various boundary conditions, and for neutral, asymptotically neutral, and nonneutral systems. Numerical estimates show that the limiting distribution is well matched in the region of major probability forN larger than 20. Implications for the thermodynamic limit are discussed. Vortex clustering is quantitatively studied via the average interaction energy between negative and positive vortices. Vortex states for which clustering is generic (in a statistical sense) are shown to result from two modeling processes: the approximation of a continuous inviscid fluid by point vortex configurations; and the modeling of the evolution of a continuous fluid at high Reynolds number by point vortex configurations, with the viscosity represented by the annihilation of close positive-negative vortex pairs. This last process, with the vortex dynamics replaced by a random walk, reproduces quite well the late-time features seen in spectral integration of the 2d Navier-Stokes equation.
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References
L. Onsager, Statistical hydrodynamics,Nuovo Cimento (Suppl.) 6:279–287 (1949).
J. Fröhlich and D. Ruelle, Statistical mechanics of vortices in an inviscid two-dimensional fluid,Commun. Math. Phys. 87:1–36 (1982).
E. H. Hauge and P. C. Hemmer, The two-dimensional Coulomb gas,Phys. Norveg. 5:209–217 (1971).
Y. B. Pointin and T. S. Lundgren, Equation of state of a vortex fluid,Phys. Rev. A 13:1274–1275 (1976).
R. H. Kraichnan and D. Montgomery, Two-dimensional turbulence,Rep. Prog. Phys. 43:547–619 (1980).
D. Montgomery, Two-dimensional vortex motion and negative temperatures,Phys. Lett. 39A:7–8 (1972).
D. Montgomery and G. Joyce, Statistical mechanics of ‘negative temperature’ states,Phys. Fluids 17:1139–1145 (1974).
S. F. Edwardsand J. B. Taylor, Negative temperature states of two-dimensional plasmas and vortex fluids,Proc. R. Soc. Lond. A 336:257–271 (1974).
C. E. Seyler, Jr., Thermodynamics of two-dimensional plasmas or discrete line vortex fluids,Phys. Fluids 19:1336–1341 (1976).
Y. B. Pointin and T. S. Lundgren, Statistical mechanics of two-dimensional vortices in a bounded container,Phys. Fluids 19:1459–1470 (1976).
T. S. Lundgren and Y. B. Pointin, Non-Gaussian probability distributions for a vortex fluid,Phys. Fluids 20:356–363 (1977).
T. S. Lundgren and Y. B. Pointin, Statistical mechanics of two-dimensional vortices,J. Stat. Phys. 17:323–355 (1977).
J. L. Lebowitz and E. H. Lieb, Existence of thermodynamics for real matter,Phys. Rev. Lett. 22:631–634 (1969); E. H. Lieb and J. L. Lebowitz, The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei,Adv. Math. 9:316–398 (1972).
K. A. O'Neil, The energy of a vortex lattice configuration, inMathematical Aspects of Vortex Dynamics, R. E. Caflisch, ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1988), pp. 217–220.
L. J. Campbell, M. M. Doria, and J. B. Kadtke, Energy of infinite vortex lattices,Phys. Rev. A 39:5436–5439 (1989); L. J. Campbell, Vortex lattices in theory and practice, inMathematical Aspects of Vortex Dynamics, R. E. Caflisch, ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1988), pp. 195–204.
K. A. O'Neil and R. A. Redner, On the limiting distribution of pair-summable potential functions in many particle systems,J. Stat. Phys. 62 (1991), to be published.
R. L. Serfling,Approximation Theorems of Mathematical Statistics (Wiley, New York, 1980), esp. Chapter V.
V. Penna, Dynamics and spectral properties of a quantum 2d bounded vortex system,Physica A 152:400–419 (1988).
A. C. Ting, H. H. Chen, and Y. C. Lee, Exact vortex solutions of two-dimensional guiding-center plasmas,Phys. Rev. Lett. 53:1348–1351 (1984).
J. Goodman, T. Y. Hou, and J. Lowengrub, Convergence of the point vortex method for the 2-D Euler equations,Commun. Pure Appl. Math. 43:415–430 (1990).
J. Miller, Statistical mechanics of Euler equations in two dimensions,Phys. Rev. Lett. 65:2137–2140 (1990).
G. F. Carnevale and G. K. Vallis, Pseudo-advective relaxation to stable states of inviscid two-dimensional fluids,J. Fluid Mech. 213:549–571 (1990).
A. J. Chorin, Numerical study of slightly viscous flow,J. Fluid Mech. 57:785–796 (1973).
J. B. Weiss and J. C. McWilliams, Non-ergodicity of point vortices,Phys. Fluids A 3:835–844 (1991).
J. C. McWilliams, The emergence of isolated coherent vortices in turbulent flow,J. Fluid Mech. 146:21–43 (1984).
W. H. Matthaeus, W. T. Stribling, D. Martinez, S. Oughton, and D. Montgomery, Selective decay and coherent vortices in two-dimensional incompressible turbulence,Phys. Rev. Lett. 66:2731–2734 (1991).
J. B. Taylor, Negative temperatures in two dimensional vortex motion,Phys. Lett. 40A:1–2 (1972).
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Campbell, L.J., O'Neil, K. Statistics of two-dimensional point vortices and high-energy vortex states. J Stat Phys 65, 495–529 (1991). https://doi.org/10.1007/BF01053742
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DOI: https://doi.org/10.1007/BF01053742