Chemical Engg. Seminar Series : Dr. Ramandeep Johal

Equilibrium statistical mechanics via microscopic Legendre transform

Legendre transform between thermodynamic quantities such as the Helmholtz free energy and entropy plays a key role in the formulation of the canonical ensemble. The transform helps to exchange the independent variable from the system’s internal energy to its conjugate variable—the inverse temperature of the reservoir. In this talk, we formulate the Legendre transform between the free energy and Shannon entropy, denoted as the microscopic Legendre transform, where the conjugate variables are the microstate probabilities and the scaled dimensionless energies. Based upon the above definitions, we derive central relations within the canonical ensemble. Thermodynamics of a system in contact with a heat reservoir is now discussed based on Shannon entropy rather than Boltzmann entropy. Jaynes’ maximum entropy principle under the constraint of a specified mean energy is compared with the present approach. Applications of the microscopic transform to other problems such as dynamical systems, generalized entropies and other ensembles will be highlighted.