elastic materials and in muscle contraction. Other forms of work which must sometimes be considered are stress-strain, magnetic, as in adiabatic demagnetization used in the approach to absolute zero, and work due to electric polarization. These are described by tensors. In most cases of interest there are internal degrees of freedom and processes, such as chemical reactions and phase transitions, which create entropy. Even for homogeneous "bulk" materials, the free energy functions depend on the (often suppressed) composition, as do all proper thermodynamic potentials (extensive functions), including the internal energy. is the number of molecules (alternatively, moles) of type in the system. If these quantities do not appear, it is impossible to describe compositional changes. The differentials for processes at uniform pressure and temperature are (assuming only work): where μi is the chemical potential for the ith component in the system. The second relation is especially useful at constant and , conditions which are easy to achieve experimentally, and which approximately characterize living creatures. Under these conditions, it simplifies to Any decrease in the Gibbs function of a system is the upper limit for any isothermal, isobaric work that can be captured in the surroundings, or it may simply be dissipated, appearing as times a corresponding increase in the entropy of the system and/or its surrounding. An example is surface free energy, the amount of increase of free energy when the area of surface increases by every unit area. The path integral Monte Carlo method is a numerical approach for determining the values of free energies, based on quantum dynamical principles. Work and free energy change For a reversible isothermal process, ΔS = qrev/T and therefore the definition of A results in (at constant temperature) This tells us that the change in free energy equals the reversible or maximum work for a process performed at constant temperature. Under other conditions, free-energy change is not equal to work; for instance, for a reversible adiabatic expansion of an ideal gas, Importantly, for a heat engine, including the Carnot cycle, the free-energy change after a full cycle is zero, while the engine produces nonzero work. It is important to note that for heat engines and other thermal systems, the free energies do not offer convenient characterizations; internal energy and enthalpy are the preferred potentials for characterizing thermal systems. Free energy change and spontaneous processes According to the second law of thermodynamics, for any