Thermodynamical potentials of classical and quantum systems

Ruikuan Liu; Tian Ma; Shouhong Wang; Jiayan Yang

doi:10.3934/dcdsb.2018214

Article Contents

2019, Volume 24, Issue 4: 1411-1448. Doi: 10.3934/dcdsb.2018214

This issue Previous Article Next Article On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces

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
* Corresponding author: Shouhong Wang

Received: October 31, 2017

Early access: June 2018

Published: January 2019

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).

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Abstract

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.

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Mathematics Subject Classification: Primary: 74A15, 82B30; Secondary: 80A17.

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Figure 3.1. 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

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Figure 4.1. The coexistence curve of $^3$He without an applied magnetic field

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Figure 4.2. $PT$-phase diagram of $^3$He in a magnetic field

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