## Thermodynamical Potentials of Classical and Quantum Systems

Ruikuan Liu, Tian Ma, Shouhong Wang, Jiayan Yang, Thermodynamical Potentials of Classical and Quantum Systems,  hal-01632278

The aim of the above paper is to systematically introduce thermodynamic potentials for thermodynamic systems and Hamiltonian energies for quantum systems of condensates.

I. In a recent paper [Ma-Wang, Dynamic Law of Physical Motion and Potential-Descending Principle, J. Math. Study, 50:3 (2017), 215-241; see also hal-01558752 and here], we postulated the potential-descending principle (PDP). We have shown the following conclusions:

• PDP leads to the first and second laws of thermodynamics,
• PDP provides the first principle for describing irreversibility, and
• leads to all three distributions: the Maxwell-Boltzmann distribution, the Fermi-Dirac distribution and the Bose-Einstein distribution in statistical physics.

In a nutshell, PDP is the first principle of statistical physics.

II. For a thermodynamic system with thermodynamic potential ${F(u, \lambda)}$, order parameters ${u}$ and control parameters ${\lambda}$, PDP gives rise to the following dynamic equation:

$\displaystyle \frac{du}{dt} = -\delta F(u, \lambda). \ \ \ \ \ (1)$

which offers a complete description of associated phase transitions and transformation of the system from non-equilibrium states to equilibrium states.

Consequently an important issue in statistical physics boils down to find a better and more accurate account of the thermodynamic potentials, which justifies the objectives of this paper.

III. The paper studies statistical systems in three categories:

1. conventional thermodynamic systems,
2.  thermodynamic systems of condensates, and
3. quantum systems of condensates.

The typical conventional thermodynamic systems include the physical-vapor transport (PVT) systems, the ${N}$-component systems, and the magnetic and dielectric systems.

There are two cases of condensates. The first is the case where the system is near the critical temperature ${T_c}$, the condensation is in its early stage and the condensed particle density is small. At this stage, the system is treated essentially as a thermodynamic system, and it is crucial then to find its thermodynamic potential. Such thermodynamic condensate systems belong to Category 2 above, and include thermodynamic systems of superconductors, superfluids, and the Bose-Einstein condensates.

The second is the case where away from the critical temperature, the system enters a deeper condensate state. In this case, the system is a quantum system and obeys the principle of Hamiltonian dynamics. We need to derive the related Hamiltonian energy. Such systems are quantum systems of condensates, and as classified as Category 3 above, which include superconducting systems, superfluid systems, and the Bose-Einstein condensates.

IV. Our study in this paper is based on

1.  SO(3) symmetry of thermodynamical potentials,
2.  theory of fundamental interaction of particles,
3. the statistical theory of heat developed by Ma-Wang,
4. quantum rules for condensates, and
5. the dynamical transition theory developed by Ma-Wang.

Of course, as mentioned earlier, the study presented in this paper certainly relies on the rich previous work done by pioneers in the related fields. It is worth mentioning that the potentials and Hamiltonians we shall introduce are based on first principles, and no mean-field theoretic expansions are used.

Ruikuan Liu, Tian Ma, Shouhong Wang & Jiayan Yang