Linear and Nonlinear Responses and Universality of Critical Exponents in Hamiltonian Systems with Long-Range Interaction

We investigate the linear and the nonlinear responses to the external field in the mean-field interaction systems whose dynamics is described by the Vlasov equation in the limit of large population. A nonlinear response formula is derived by the asymptotic-transient decomposition, which is developed for analyzing the nonlinear Landau damping and plasma waves. The formula gives a self-consistent equation for the asymptotic state, which is in general not in thermal equilibrium but in a long-lasting nonequilibrium quasistationary state due to the long-range nature of the interaction. This formula is useful to compute the critical exponent analytically for the nonlinear response at the critical point, denoted by δ. Further advantage of this formula is that it includes the linear response formula obtained from the naively linearized equation, and unifies the nonlinear and the linear responses, where the latter induces another critical exponent γ for the zero-field susceptibility. Interestingly, these critical exponents for the quasistationary states differ from ones in thermal equilibria, but the Widom’s scaling relation holds true even for the non-classical critical exponents. We discuss universality of the critical exponents and the scaling relation for spatially periodic one-dimensional systems.