# American Institute of Mathematical Sciences

April  2019, 24(4): 1411-1448. doi: 10.3934/dcdsb.2018214

## Thermodynamical potentials of classical and quantum systems

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA 3 School of Medical Informatics and Engineering, Southwest Medical University, Luzhou, Sichuan 646000, China

* Corresponding author: Shouhong Wang

Received  November 2017 Published  April 2019 Early access  June 2018

Fund Project: The work was supported in part by the US National Science Foundation (NSF), the Office of Naval Research (ONR) and by the Chinese National Science Foundation (11771306).

The aim of the paper is to systematically introduce thermodynamic potentials for thermodynamic systems and Hamiltonian energies for quantum systems of condensates. The study is based on the rich previous work done by pioneers in the related fields. The main ingredients of the study consist of 1) SO(3) symmetry of thermodynamical potentials, 2) theory of fundamental interaction of particles, 3) the statistical theory of heat developed recently [23], 4) quantum rules for condensates that we postulate in Quantum Rule 4.1, and 5) the dynamical transition theory developed by Ma and Wang [20]. The statistical and quantum systems we study in this paper include conventional thermodynamic systems, thermodynamic systems of condensates, as well as quantum condensate systems. The potentials and Hamiltonian energies that we derive are based on first principles, and no mean-field theoretic expansions are used.

Citation: Ruikuan Liu, Tian Ma, Shouhong Wang, Jiayan Yang. Thermodynamical potentials of classical and quantum systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1411-1448. doi: 10.3934/dcdsb.2018214
##### References:

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##### References:
An electron rotating around a direction $n$ with velocity $v$ induces a magnetic moment $m = ev{\bf s}$, where ${\bf s}$ is the area vector enclosed by the electron orbit
The coexistence curve of $^3$He without an applied magnetic field
$PT$-phase diagram of $^3$He in a magnetic field
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