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We analyse the Jackiw-Teitelboim model of 2D gravity coupled to $N$ masslessfree scalar fields in the semi-classical limit. Two systems are studied whichessentially differ in the boundary conditions that are imposed. We find thatthe thermodynamics has interesting differences. We also analyse the response toadditional infalling matter which satisfies the null energy condition. Thesecond law is shown to be valid in both systems for the generalised entropywhich takes into account the entanglement across the event horizon due to thematter fields. Similarly we find that the generalised entropy increases alongfuture Q-screens in both systems.

Jackiw-Teitelboim Model Coupled to Conformal Matter in the Semi-Classical Limit