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We extend Jarzynski's work relation and the second law of thermodynamics to aheat conducting system which is operated by an external agent. These extensionscontain a new non equilibrium contribution expressed as the violation of the(linear) response relation caused by the operation. We find that a naturalextension of the minimum work principle involves information about thetime-reversed operation, and is far from straightforward. Our work relation maybe tested experimentally especially when the temperature gradient is small.

Work Relation and the Second Law of Thermodynamics in Nonequilibrium Steady States