Books

PDE/Applied Math Seminar

Institute Seminar

Fall 14 course: S311

Shouhong Wang

## New Wordpress Blog

Interplay between Mathematics and Physics

## Field Theory and Particle Physics (Ma & Wang)

• Two Fundamental Principles: We have postulated two basic principles, the principle of interaction dynamics (PID) and the principle of representation invariance (PRI), unifying Nature's four fundamental forces/interactions. Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint, and PRI requires that physical laws be independent of representations of the gauge groups.
• Unified Field Model: A unified field model is then derived using these two principles. One important outcome of this unified field model is a natural duality between the interacting fields corresponding to graviton, photon, intermediate vector bosons and gluons, and the adjoint bosonic type fields. The unified field model can be naturally decoupled to study individual interactions.
• Theory for Dark Energy & Dark Matter: For gravity, we derive modified Einstein equations, giving rise to a unified theory for dark matter and dark energy: $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R =- \frac{8\pi G}{c^4} T_{\mu\nu} -\color{red}{ D_\mu D_\nu \varphi}, \qquad D^\mu \left( \color{red}{D_\mu D_\nu \varphi }+ \frac{8\pi G}{c^4}T_{\mu\nu}\right)=0$$
• Duality of Strong Interaction: We derive three levels of strong interaction potentials for quark, nucleon/hadron, and atom respectively. These potential/force formulas offer a clear mechanism for quark confinement, short-range nature of strong interaction and asymptotic freedom.
• New Mass Generation Mchanism and Duality Theory of Weak Interaction: A completely different, but much simpler electroweak theory is derived using PID and PRI. This theory reveals a much deeper level of understanding of electroweak interaction, including e.g. a new mass generation mechanism, the introduction of weak and strong charges, and weak interaction potential/force formulas.
• Weakton Model of Elementary Particles and Decay Mechanism: Sub-atomic decays and electron radiations indicate interior structure for charged leptons, quarks and mediators. The weakton model postulates that all matter particles (leptons, quarks) and mediators are made up of massless weaktons. The weakton model offers a perfect explanation for all sub-atomic decays and all generation/annihilation precesses of matter-antimatter: the precise constituents of particles involved in all decays both before and after the reaction can now be precisely derived. In addition, the bremsstrahlung phenomena can be understood using the weakton model. Also, the weakton model offers an explanation to the baryon asymmetry problem.

## Statistical Physics and Phase Transition Dynamics

Ma and Wang have developed a new mathematical theory for dynamic transitions. The key philosophy of the dynamic transition theory is to search for the full set of transition states, giving a complete characterization on stability and transition. One important part of the theory is to establish a general principle that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic, and random. Their theory has been applied to a wide range of problems in nonlinear sciences, leading to precise physical understanding and predictions of the underlying problems.

It is well-known that the gas-liquid coexistence curve terminates at a critical point, also called the Andrews critical point. It is a longstanding open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. For the first time, we show that 1) the gas-liquid co-existence curve can be extended beyond the Andrews critical point, and 2) the transition is first order before the critical point, second-order at the critical point, and third order beyond the Andrews critical point.

We have derived new Ginzburg-Landau type of models for liquid helium-3 [78], helium-4 [77] and their mixture [82], leading to various physical predictions, such as the existence of a new phase $C$ for helium-3. Although these predictions need yet to be verified experimentally, they certainly offer new insights to both theoretical and experimental studies for a better understanding of the underlying physical problems.

## Geophysical Fluid Dynamics and Climate Dynamics

In the early 90s, Jacques-Louis Lions and Roger Temam and Shouhong Wang initiated a mathematical study on general circulation models in climate and geophysical fluid dynamics. The main focus was on the primitive equations (PEs) of atmosphere-only, ocean-only, and the coupled ocean-atmosphere systems.

Another direction of research is on specific geophysical phenomena. This part of the work is carried out in collaboration with Roger Samelson, Henk Dijkstra, Michael Ghil, and Tian Ma.

• New Mechanism of El Nino Southern Oscillation (ENSO) [91, 86]

We discovered a new mechanism of the ENSO, as a self-organizing and self-excitation system, with two highly coupled oscillatory processes: 1) the oscillation between the two metastable warm (El Nino phase) and cold events (La Nina phase), and 2) the spatiotemporal oscillation of the sea surface temperature (SST) field. The interplay between these two processes gives rises the climate variability associated with the ENSO, leads to both the random and deterministic features of the ENSO, and defines a new natural feedback mechanism, which drives the sporadic oscillation of the ENSO. The randomness is closely related to the uncertainty/fluctuations of the initial data between the narrow basins of attractions of the corresponding metastable events, and the deterministic feature is represented by a deterministic coupled atmospheric and oceanic model predicting the basins of attraction and the sea-surface temperature (SST).

• Thermohaline Circulation [88]

Oceanic circulation is one of key sources of internal climate variability. One important source of such variability is the thermohaline circulation (THC). Physically speaking, the buoyancy fluxes at the ocean surface give rise to gradients in temperature and salinity, which produce, in turn, density gradients. These gradients are, overall, sharper in the vertical than in the horizontal and are associated therefore with an overturning or THC. A mathematical theory associated with the thermohaline circulations (THC) is derived in [88], using the Boussinesq system, governing the motion and states of the large-scale ocean circulation. First, it is shown that the first transition is either to multiple steady states or to oscillations (periodic solutions), determined by the sign of a non-dimensional parameter $K$, depending on the geometry of the physical domain and the thermal and saline Rayleigh numbers. Second, for both the multiple equilibria and periodic solutions transitions, both Type-I (continuous) and Type-II (jump) transitions can occur, and precise criteria are derived in terms of two computable nondimensional parameters $b_1$ and $b_2$. Associated with Type-II transitions is the hysteresis phenomena, and the physical reality is represented by either metastable states or by a local attractor away from the basic solution, showing more complex dynamical behavior. Third, a convection scale law is introduced, leading to an introduction of proper friction terms in the model in order to derive the correct circulation length scale. In particular, the dynamic transitions of the model with the derived friction terms suggest that the THC favors the continuous transitions to stable multiple equilibria.

 Last updated on July 25, 2014