We study the most suitable procedure to measure the effective temperature inoff-equilibrium systems. We analyze the stationary current established betweenan off-equilibrium system and a thermometer and the necessary conditions forthat current to vanish. We find that the thermometer must have a shortcharacteristic time-scale compared to the typical decorrelation time of theglassy system to correctly measure the effective temperature. This generalconclusion is confirmed analyzing an ensemble of harmonic oscillators withMonte Carlo dynamics as an illustrative example of a solvable model of a glass.We also find that the current defined allows to extend Fourier's law to theoff-equilibrium regime by consistently defining effective transportcoefficients. Our results for the oscillator model explain why thermalconductivities between thermalized and frozen degrees of freedom in structuralglasses are extremely small.
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