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Author(s):
Alexei Ivanov
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Title:
Scaling of the Distribution Function and the Critical Exponents near the Point of a Marginal Stability under the Vlasov-Poisson Equations
Date of publication:
Aug. 2000
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Key words:
Fluctuation phenomena-Phase transitions: general studies-Critical point phenomena
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Abstract:
A model system, described by the consistent Vlassov-Poisson equations under periodical boundary conditions, has been studied numerically near the point of a marginal stability. The power laws, typical for a system, undergoing a second-order phase transition, hold in a vicinity of the critical point: (i) A propto -theta^beta, beta = 1.907 pm 0.006 for theta leq 0, where A is the saturated amplitude of the marginally-stable mode; (ii) chi propto theta^-gamma as theta rightarrow 0, gamma = gamma- = 1.020 pm 0.008 for theta < 0, and gamma = gamma+ = 0.995 pm 0.020 for theta > 0, where chi = partial A/partial F_1 at F_1 rightarrow 0 is the susceptibility to external drive of the strain F1; (iii) at theta = 0 the system responds to external drive as A pm F_1^1/delta, and delta = 1.544 pm 0.002. theta = (<upsilon^2 > - <upsilon_cr^2 >)/<upsilon_cr^2> is the dimensionless reduced velocity dispersion. Within the error of computation these critical exponents satisfy to equality gamma = beta(delta - 1), known in thermodynamics as the Widom equality, which is direct consequence of scaling invariance of the Fourier components fm of the distribution function f at |theta| ll 1, i.e. fm(lambda^at t, lambda^a upsilon upsilon, lambda^a theta theta, lambda^aA 0 A_0, lambda^aF F_1) = lambda f_m (t, upsilon, theta, A_0, F_1) at theta approx 0. On the contrary to thermodynamics these critical indices indicate to a very wide critical area. In turn, it means that critical phenomena may determine macroscopic dynamics of a large fraction of systems.
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