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We generalize the second law of thermodynamics in its maximum work formulation for a nonequilibrium initial distribution. It is found that in an isothermal process, the Boltzmann relative entropy (H-function) is not just a Lyapunov function but also tells us the maximum work that may be gained from a nonequilibrium initial state. The generalized second law also gives a fundamental relation between work and information. It is valid even for a small Hamiltonian system not in contact with a heat reservoir but with an effective temperature determined by the isentropic condition. Our relation can be tested in the Szilard engine, which will be realized in the laboratory. The maximum work formulation of the second law of thermodynamics relates the work needed to move a system from one equilibrium state to another to the free energy difference between those states. It tells us that the work must be greater than or equal to the difference in free energies. In recent refinements of the second law, such as the fluctuation theorem[1] and the Jarzynski equality [2][3], the maximum work formulation in the sense of the average work is rigorously shown for an initial canonical distribution [2].In this letter we generalize the maximum work formulation of the second law to transitions between nonequilibrium states. Our generalization allows one to find the maximum work that can be gained from such a transition and the processes that realize it. We are able to consider isolated systems as well as those coupled to a heat reservoir and both adiabatic and isothermal transitions.The derivation is based on the Jarzynski equality modified for a nonequilibrium initial distribution. This leads to a relation between the work and the Boltzmann relative entropy with an effective temperature. The Boltzmann relative entropy, also known as the KullbackLeibler divergence, is always positive and gives a "distance" between the nonequilibrium initial distribution and the canonical distribution [4].For a finite Hamiltonian system without a heat reservoir, the effective temperature is determined by an isentropic condition. The maximum work is realized in two successive processes: an instantaneous stabilization of the nonequilibrium initial distribution and an isentropic process.When the system is coupled to a large heat reservoir, the effective temperature is the temperature of the heat reservoir. From the generalized second law, the max-

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phenomena and complex systems, which is one of the most fascinating fields of science today, to publish books that cover the essential concepts in this area, as well as the latest developments. As the number of scientists involved in the subject increases continually, so does the number of new questions and results. Nonlinear effects are essential to understand the behaviour of nature, and the methods and ideas introduced to treat them are increasingly used in new applications to a variety of problems ranging from physics to human sciences. Most of the books in this series will be about physical and mathematical aspects of nonlinear science, since A C.I.P. CataIogue record for this book is available from the Library of Congress.