We apply a finite temperature path integral method to study strongly anharmonic thermal factors in extended X-ray absorption fine structure (EXAFS), electron energy-loss spectroscopy (EELS) and angle resolved X-ray photoelectron spectroscopy (ARXPS). Two examples, symmetric and asymmetric double-well potentials are considered for the anharmonic potentials, which are used to describe structural phase transition and rattling mechanism. We discuss the quantum tunneling effects in the thermal

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... s. The validity of the classical approximation and the cumulant expansion are also studied for various asymmetric potentials. Complex plane expression is introduced to explain the specific features of the EXAFS thermal damping function. Degree of asymmetry of the potential can be estimated from the complex plane expression. In the temperature dependence of the higher order cumulant (third and fourth), minimum point (third order) and maximum point (fourth order) appear at the characteristic temperature, which depends on the potential asymmetry.