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We show that an exact solution of two-dimensional dilaton gravity with matterdiscovered previously exhibits an irreversible temporal flow towards flat spacewith a vanishing cosmological constant. This time flow is induced by the backreaction of matter on the space-time geometry. We demonstrate that the systemis not in equilibrium if the cosmological constant is non-zero, whereas thesolution with zero cosmological constant is stable. The flow of the systemtowards this stable end-point is derived from the renormalization-group flow ofthe Zamolodchikov function. This behaviour is interpreted in terms ofnon-critical Liouville string, with the Liouville field identified as thetarget time.Comment: 20 pages LATEX, three figures incorporate

IRREVERSIBLE TIME FLOW IN A TWO-DIMENSIONAL DILATON BLACK HOLE WITH MATTER