Based on the definition of the apparent horizon in a general two-dimensional dilaton gravity theory, we analyze the tunnelling phenomenon of the apparent horizon by using Hamilton-Jacobi method. In this theory the definition of the horizon is very different from those in higher-dimensional gravity theories. The spectrum of the radiation is obtained and the temperature of the radiation is read out from this spectrum and it satisfies the usual relationship with the surface gravity. Besides, the calculation with Parikhs null geodesic method for a simple example conforms to our result in general stationary cases.
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