Thermodynamical potentials of classical and quantum systems

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.

Origin of the Electron–Phonon Interaction of Topological Semimetal Surfaces Measured with Helium Atom Scattering

“…This is clearly not true for the diffraction from a surface CDW which forms below T c from a Fermi surface instability and has the temperature-dependent population of electron states near the Fermi level according to Fermi statistics. In this case, I 0 has an implicit dependence on T, which generally is negligible with respect to that of W(T), except near T c : here its square root I 0 works as an order parameter 25,26 and vanishes for increasing T → T c as (1 − T/T c ) β , where β is the orderparameter critical exponent (typically β = 1/3, 12,27−29 while T c ≈ 280 K in the present case 12 ). As a good 1DEG example, it is shown that a CDW diffraction peak also may be used to extract λ HAS away from the critical region.…”

Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates

“…Also, the Hamiltonian energy for such a BEC system with J = 1 is derived by T.-L. Ho [14], T. Ohmi and K. Machida [31]. These two quantum systems are receiving an increasing attention from many researchers in the past several decades, see [2,5,6,8,10,11,16,17,19,20,21,22,33,34,35,38,39]. More recently, Luckins and Van Gorder [19] studied stationary and quasi-stationary solutions for the cubic-quintic Gross-Pitaevskii equation modeling Bose-Einstein condensates in one, two, and three spatial dimensions under the assumption of radial symmetry with the BEC dynamics influenced by a confining potential.…”

“…In order to better understand QPT related to spinor Bose-Einstein condensates(sBEC), what we are concerned in this paper is to study bifurcation solutions for the Bose-Einstein condensate system with the spinor J = 1. First, we establish a pattern formation equation related to sBEC with the Hamilton energy [14,17,31] basing on two fundamental physical principles, which are Principle of Hamilton Dynamics (PHD) and Principle of Lagrangian Dynamics (PLD) in [29]. Furthermore, we get three kinds of critical conditions related to eigenvalues basing on the spectrum analysis and three kinds of confining potentials.…”

“…where ψ is defined as (9), µ ∈ R is the chemical potential, which is a constant and the more physical properties of chemical potential refer to [17,22]. By (10) and 14, we can derive that…”

“…where Ψ * is the conjugate of Ψ, g 1 and g 2 are shown as (17). Owing to the quantum system is energy conservation, the wave function should take the following form basing on the equivalent form (14) in subsection 3.1 Ψ = e −iµt/ Φ = e −iµt/ (ϕ…”