Laboratoire de Physique Théorique
Toulouse - UMR 5152

Systems with long-range interaction are trapped in long-lasting nonequilibrium steady states, called quasistationary states, before they equilibrate [1]. Temporal evolution of such systems is approximately described by a single-body distribution function defined on the single-particle phase space, which is a solution to the Vlasov equation (also called the collisionless Boltzmann equation). The quasistationary states are associated with stable stationary solutions to the Vlasov equation, and duration of them diverges with system size [1]. The topics of this talk are the time asymptotic behaviour of linear and nonlinear response to external force and critical phenomena of one dimensional systems in quasi-stationary states.

The linear response theory based on the Vlasov equation is reviewed quickly, and a nonclassical critical exponent gamma characterising the divergence of isolated susceptibility [2] is introduced. Since the Vlasov equation has infinitely many invariants, the critical exponent gamma is different from the one obtained in the equilibrium statistical mechanics. Is this difference shown for other kind of critical exponents ? Do scaling relations among several exponents hold true in the isolated systems in quasi-stationary states, as equilibrium state ? Do they have some universality ? The nonlinear response theory answers them partially. To compute the nonlinear response, we use the transient-linearisation method which was developed for analysing the nonlinear Landau damping and plasma oscillation [3], instead of the naive perturbation technique. An asymptotic state is unknown and is assumed to be stationary. We linearise the Vlasov equation around this unknown asymptotic state. Solving this equation implicitly and taking the long-time limit, we derive a self-consistent equation which determines the asymptotic state. By analysing this equation, it is shown that this nonlinear response formula includes the linear response at the first order, and we can compute the nonlinear effect whose order is less than 2 [4,5]. By use of this formula, the nonlinear response on the critical point can be computed, and we can get the non-classical critical exponent delta characterising how the order parameter responds to the external force on the critical point. The critical exponents gamma and delta are different from the ones obtained in the equilibrium state. However, it is shown that these exponents hold Widom’s scaling relation inevitably. Further the result exhibited in this talk is independent of the detail of the system and the one-parameter family of single-body distributions associated with quasistationary states [5].

[1] A. Campa, T. Dauxois, and S. Ruffo, Phys. Rep. 480, 57 (2009).
[2] SO, A. Patelli, and Y. Y. Yamaguchi, Phys. Rev. E 89, 032131 (2014).
[3] C. Lancellotti and J. J. Dorning, Trans. Th. Stat. Phys. 38, 1 (2009).
[4] SO and Y. Y. Yamaguchi, Phys. Rev. E 89, 052114 (2014).
[5] SO and Y. Y. Yamaguchi, Phys. Rev. E 91, 062108 (2015).