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The linear response is investigated in a long-range Hamiltonian system fromthe view point of dynamics, which is described by the Vlasov equation in thelimit of large population. Due to existence of the Casimir invariants of theVlasov dynamics, an external field does not drive the system to the forcedthermal equilibrium in general, and the linear response is suppressed. With theaid of a linear response theory based on the Vlasov dynamics, we compute thesuppressed linear response in a system having two order parameters, whichintroduce the conjugate two external fields and the susceptibility matrix ofsize two accordingly. Moreover, the two order parameters bring three phases andthe three types of second-order phase transitions between two of them. For eachtype of the phase transitions, all the critical exponents for elements of thesusceptibility matrix are computed. The critical exponents reveal that someelements of the matrices do not diverge even at critical points, while themean-field theory predicts divergences. The linear response theory alsosuggests appearance of negative off-diagonal elements, in other words, anapplied external field decreases the value of an order parameter. Thesetheoretical predictions are confirmed by direct numerical simulations of theVlasov equation.

Nondivergent and negative susceptibilities around critical points of a long-range Hamiltonian system with two order parameters