The inverse problem for a linear vector field in thermodynamics

The inverse problem in thermodynamics is considered: For a linear vector field which generates an evolution on thermodynamic phase space, the generating scalar function has been found. The form of acceptable evolution for a linear vector field is determined and it is shown that such fields form a Lie subalgebra. Some examples and possible physical interpretations are given. The sufficient condition for a linear vector field to generate a thermodynamic process on a submanifold of ideal gas is derived. A simple generalization of the inverse problem for nonlinear fields is introduced.