Critical exponents for a model Vlasov-Poisson system at marginal stability

Abstract

The remarkable property of critical phenomena is the universal scaling appearing in vast variety of systems, e.g., magnets and gases follow simple power laws for the order parameter, specific heat capacity, susceptibility, compressibility, etc. In thermodynamic systems, phase transitions take place at a critical temperature when the coefficients, characterizing the linear response of the system to external perturbations, diverge and long-range order appears, causing a transition to a new phase due to collective behavior of an entire system. Thus, irrespective of the amplitude of perturbations, the system behaves nonlinearly at the critical point. Nonlinearity, inherent to criticality, leads to scalings that are universal for the thermodynamic systems as well as for plasmas and gravitating systems. However, analysis based on the partition function (as in thermodynamics) is generally not applicable to the kinetic systems governed by coupled Vlasov and Poisson equations which, under certain conditions, lose their stability in a manner analogous to a second-order phase transition. Unlike second-order phase transitions, studied as nonlinear phenomena from the very beginning, analysis of Vlasov-Poisson systems is largely based on dispersion relations obtained by linearization of initially nonlinear equations. This allows one to predict criticality, but, similar to linear thermodynamics, cannot describe all properties of the critical state. Thus nonlinear analysis is important for plasmas where critical states are common because of the wealth of instabilities. Another example can be found far from plasmas: the dynamics of a normal brain are revealed to be close to marginal stability, apparently a vital condition for successful adaptability. Here, a model Vlasov-Poisson gravitating system is simulated near the point of marginal stability and the critical exponents are calculated. These exponents obey the Widom, Rushbrooke, and Griffith laws at the formal dimensionality d=5 and are not the Landau-Weiss exponents contrary to the expectation for the equivalent mean field model.