  For thermal transport in one-dimensional (1D) systems, recent studies have
suggested that employing different theoretical models and different numerical
simulations under different system's parameter regimes might lead to different
universality classes of the scaling exponents. In order to well understand the
universality class(es), here we perform a direct dynamics simulation for two
archetype 1D oscillator systems with quite different phonon dispersions under
various system's parameters and find that there is a crossover between the
different universality classes. We show that by varying anharmonicity and
temperatures, the space-time scaling exponents for the systems with different
dispersions can be feasibly tuned in different ways. The underlying picture is
suggested to be understood by phonons performing various kinds of
continuous-time random walks (in most cases, be the L\'{e}vy walks but not
always), probably induced by the peculiar phonon dispersions along with
nonlinearity. The results and suggested mechanisms may provide insights into
controlling the transport of heat in some 1D materials.

Statistical Mechanics