A Characterization of the Entropy--Gibbs Transformations

Let h be a finite dimensional complex Hilbert space, b(h)+ be the set of all positive semi-definite operators on h and Phi is a (not necessarily linear) unital map of B(H) + preserving the Entropy-Gibbs transformation. Then there exists either a unitary or an anti-unitary operator U on H such that Phi(A) = UAU* for any B(H) +. Thermodynamics, a branch of physics that is concerned with the study of heat (thermo) and power (dynamics), might at first seem more important for engineers trying to in- vent a new engine than for biochemists trying to understand the mechanisms of life. However, since chemical reactions involve atoms and molecules that are bound by the laws of physics, studying thermodynamics should be a priority for every aspiring biochemist. There are two laws of thermodynamics that are important to the study of biochemistry. These two laws have to do with energy and order both essential for life as we know it. It is easy to understand that our bodies need energy to function from the visible muscle movement that gets us where we want to go, to the microscopic cellular processes that keep our brains thinking and our organs functioning. Order is also important. Our bodies represent a high degree of order: atoms and molecules are meticulously organized into a complex system ranging in scale from the microscopic to the macroscopic.