Three Constituents of the Fundamental Theory for the Four Interactions

The aim of this blog is to clarify that the fundamental theory of four interactions must contain the following three basic constituents:

• the symmetries,
• the Lagrangian actions, also called functionals, and
• the field equations.

The relation between these three components is

$\displaystyle \text{\it symmetries} \Longrightarrow \text{\it {\bf unique} actions (functionals)} \Longrightarrow \text{\it field equations}$

Symmetries. The symmetry for gravity is the invariance under general coordinate transformations, which is precisely described by the Einstein principle of general relativity (PGR), and the symmetry for the electromagnetism, the weak and the strong interactions is the gauge symmetry, originally proposed by Herman Weyl.

Uniqueness of Actions. The symmetries determine uniquely the actions (functionals): the PGR uniquely determines the Einstein-Hilbert functional, and the gauge symmetry uniquely dictates the Yang-Mills action.

Of course, the uniqueness is derived under the principle that the law of nature must be simple; simplicity implies stability and beauty.

Field Equations by PID. The principle of interaction dynamics (PID) takes variation of the actions subject to generalized energy-momentum conservation constraints. It is the direct consequence of the presence of dark energy and dark matter, is the requirement of the presence of the Higgs field for the weak interaction, and is the consequence of the quark confinement phenomena for the strong interaction. Hence

PID is the principle for deriving the field equations of fundamental interaction.

Summary. The fundamental theory of four interaction is now complete. The symmetry for gravity is different from the gauge symmetry for the electromagnetism, the weak and the strong interactions, leading to different actions and field equations. In essence, the electromagnetism, the weak and the strong is unified by the gauge field theory. The symmetry for gravity was discovered by Einstein and the action by Einstein and Hilbert. For the electromagnetism, the weak and the strong interactions, the gauge symmetry was discovered by Weyl. The general ${SU(N)}$ action was introduced by Yang and Mills.

Tian Ma & Shouhong Wang

Dynamic Theory of Fluctuations And Critical Exponents of Thermodynamic Phase Transitions

Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang, Dynamic Theory of Fluctuations And Critical Exponents of Thermodynamic Phase Transitions, Hal-01674269

This paper is aimed to establish a dynamical law of fluctuations, and to derive the critical exponents based on the standard model with fluctuations, leading to correct critical exponents in agreement with experimental results.

1. For a thermodynamic system, the PDP proposed in

Tian Ma & Shouhong Wang, Dynamic Law of Physical Motion and Potential-Descending Principle, J. Math. Study, 50:3 (2017), pp. 215-241; see also HAL preprint: hal-01558752

provides the dynamic law for statistical physics:

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

which offers a complete description of associated phase transitions and transformation of the system from non-equilibrium states to equilibrium states. This dynamic law (1) also describes automatically irreversibility.

In view of (1), we developed a systematic theory in

Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang, Thermodynamical Potentials of Classical and Quantum Systems, 2017, hal-01632278

for deriving explicit expressions of thermodynamic potentials, based on first principles, rather than on the mean-field theoretic expansions.

The dynamic law (1) with expression formulas for the thermodynamic potentials and the dynamic transition theory developed in

Tian Ma & Shouhong Wang, Phase Transition Dynamics, Springer-Verlag, xxii, 555pp., 2013

provide a complete theoretical understanding of phase transitions and critical phenomena for thermodynamic systems. This is the basic theory of the standard model for thermodynamical systems.

2. There is, however, a discrepancy between the theoretical exponents and their experimental values, as in the case of mean-field theoretic approach. We demonstrate in this paper that in reality, there is a critical fluctuation effect, and we show that the discrepancy just mentioned is due entirely to the spontaneous fluctuation.

To have an accurate account of the fluctuations, we need to derive its governing fundamental law, which have to stem from the thermodynamic potential and the dynamic law (1).

In fact, for an equilibrium state ${u_0}$ of a thermodynamic system, the fluctuation of ${u}$ is the deviation from ${u_0}$:

$\displaystyle \text{fluctuation} \ w=u-u_0. \ \ \ \ \ (2)$

Then the needed dynamic law for fluctuations is given by

$\displaystyle \frac{\text{d}w}{\text{d}t}=- [\delta F(u_0+w)-\delta F(u_0)]+\widetilde{f}. \ \ \ \ \ (3)$

where ${\widetilde{f}}$ is the fluctuation of the external force. 3. We derive two basic theorems on critical exponents. The first theorem is based on the dynamic law (1).

First Theorem of Critical-Exponents.

For a second-order phase transition, near the critical point, using the dynamical law (1) without fluctuations, we derive the theoretical critical exponents ${\beta,\ \delta,\ \alpha, \ \gamma}$ as given by

$\displaystyle \beta=\frac{1}{2},\quad \delta=3,\quad\alpha=0,\quad \gamma=1. \ \ \ \ \ (4)$

The second theorem takes into consideration of fluctuations.

Second Theorem of Critical-Exponents.

For a second-order phase transition, near the critical point, using the dynamical law (3) with fluctuations, the fluctuation critical exponents ${\beta,\ \delta,\ \alpha,\ \gamma}$ are given by

$\displaystyle \beta=\frac{1}{3},\quad 3<\delta<6,\quad 0<\alpha<\frac{2}{3},\quad 1 \le\gamma<\frac{5}{3}. \ \ \ \ \ (5)$

We now list three groups of exponent data for different thermodynamic systems:

1) experimental exponents,

2) theoretical exponents without taking into consideration of fluctuations, and

3) theoretical exponents using the standard model with fluctuations.

This table shows clearly the strong agreement of the results using the standard model of thermodynamics with fluctuations.

 exponent magnetic system PVT system binary system without fluctuation with fluctuation ${\beta}$ 0.30-0.36 0.32-0.35 0.30-0.34 1/2 1/3 ${\delta}$ 4.2-4.8 4.6-5.0 4.0-5.0 3 3.0-6.0 ${\alpha}$ 0.0-0.2 0.1-0.2 0.05-0.15 0 0-2/3 ${\gamma}$ 1.2-1.4 1.2-1.3 1.2-1.4 1 1-5/3

4. We have shown that the theoretical values from the dynamic law (1) do reflect the nature under the ideal assumption that no fluctuations are present in the system. However, the fluctuations are inevitable, and are completely accounted for by the dynamic law of fluctuations (3). In a nutshell,

• the standard model (1), together with the dynamic law of fluctuation (3), offers correct information for critical exponents; and

• this in return validates the standard model of thermodynamics, which is derived based on first principles.

Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang

Topological Phase Transition V: Interior Separations and Cyclone Formation Theory

Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang, Topological Phase Transition V: Interior Separations and Cyclone Formation Theory, 2017, Hal-01673496

This is the last one in this series of papers on topological phase transitions (TPTs); see I, II, III, and IV for details.

1. Interior separation of fluid flows is a common phenomenon in fluid dynamics, especially in geophysical fluid dynamics, such as the formation of hurricanes, typhoons and tornados, and gyres of oceanic flows. In general, the interior separation refers to sudden appearance of a vortex from the interior of a fluid flow.

It is clear that fluid interior separation is a typical TPT problem, similar to the TPT associated with boundary-layer separations. Also, mathematical, the geometric theory of incompressible flows developed by two of the authors offers the needed mathematical foundation for understanding interior separations, as well as for the quantum phase transitions of the Bose-Einstein condensates, superfluidity and superconductivity. For this geometric theory, see

[MW05] T. Ma & S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, AMS Mathematical Surveys and Monographs Series, vol. 119, 2005, 234 pp.

2. At the kinematic level, a structural bifurcation theorem theorem was proved in [MW05]. Basically, let ${u(x,t)}$ be a one-parameter family of 2D divergence-free vector fields with the first-order Taylor expansion with respect to ${t}$ at ${t_0>0}$:

$\displaystyle u(x,t)=u^0(x)+(t-t_0)u^1(x)+o(|t-t_0|), \ \ \ \ \ (1)$

Let ${x_0\in\Omega}$ be a degenerate singular point of ${u^0}$, and the Jacobian ${Du^0(x_0)\neq0}$. Then, there are two orthogonal unit vectors ${e_1}$ and ${e_2}$ such that

$\displaystyle Du^0(x_0)e_1=0, \qquad Du^0(x_0)e_2=\alpha e_1 \quad \text{ with } \alpha\neq0. \ \ \ \ \ (2)$

If ${x_0\in \Omega}$ is an isolated singular point if ${u^0(x)}$ with ${Du^0(x_0)\neq0}$, and satisfies that

$\displaystyle \text{\rm ind}(u^0,x_0)=0, \qquad u^1(x_0)\cdot e_2\neq0, \ \ \ \ \ (3)$

then ${u(x,t)}$ has an interior separation from ${(x_0,t_0)}$.

3. The above result is of kinematic in nature. We need to derive a separation equation, which connects the topological structure of the fluid flows to the solutions of the Navier-Stokes equations, which govern the the fluid motion.

For geophysical fluid phenomena such as hurricanes, typhoons, and tornados, the typical interior separation phenomena are caused by external wind-driven forces and by the non-homogenous temperature distributions. Therefore, the crucial factors for the formation of interior separations in the atmospheric and oceanic flows are

• the initial velocity field,
• the external force, and
• the temperature.

Hence the dynamical fluid model for interior separations has to incorporate properly the heat effect.

The Boussinesq equations are mainly for convective flows, and are not suitable for studying interior separations, associated in particular with the such geophysical processes as hurricanes, typhoons, and tornados.

For this purpose, we use the horizontal heat-driven fluid dynamical equations by Yang and Liu [18], which couple the Navier-Stokes equation and the heat diffusion equation with the following equations of state:

$\displaystyle p= \beta_0\rho T \qquad \text{for gaseous systems}, \ \ \ \ \ (4)$

and

$\displaystyle p=\beta_1T-\beta_2\rho^{-1}+\beta_3 \qquad \mbox{for liquid systems}. \ \ \ \ \ (5)$

4. Consider the solution

$\displaystyle (u(x,t), T(x,t)) = (\varphi(x)+t\widetilde{u}(x,t), T_0(x)+t\widetilde{T}(x,t)),$

of the aforementioned fluid model with initial velocity ${\varphi=(\varphi_1, \varphi_2)}$ and initial temperature ${T_0}$. One main result we obtain is the following interior separation equations:

$\displaystyle u(x,t)=\varphi+v t, \ \ \ \ \ (6)$

$\displaystyle v(x)=\nu\Delta\varphi-(\varphi\cdot\nabla)\varphi-\alpha \nabla T_0 +f, \ \ \ \ \ (7)$

$\displaystyle \frac{\partial \varphi_1}{\partial x_2}\frac{\partial \varphi_2}{\partial x_1}+\bigg(\frac{\partial \varphi_1}{\partial x_1}\bigg)^2 +\frac{\alpha}{2}\Delta T_0-\frac{1}{2}\mbox{div}f\simeq 0. \ \ \ \ \ (8)$

These separations include all physical information about the interior separations of the solution ${u}$ for the system, in terms of the initial state ${(\varphi, T_0)}$ and the external force ${f}$.

5. Theoretical analysis and observations show that interior separation can only occur when

• one of you ${\varphi}$ or ${v}$ in (6) is U-shaped, which we call U-flow, and the other is either a U-flow or a flat flow; and
• ${\varphi}$ and ${v}$ have reversed orientations.

For the case where ${\varphi}$ is a U-flow and ${v}$ is a flat flow, the interior separation equations are expressed as:

$\displaystyle u_1= (a_0+a_1x_2+a_2x^2_2)-t(\alpha \tau_1+f^0_1)+\text{h.o.s.t}, \ \ \ \ \ (9)$

$\displaystyle u_2=b_0 x_1-t(\alpha \tau_2+a_0b_0+f^0_2)+\text{h.o.s.t.}, \ \ \ \ \ (10)$

with the parameters satisfy

$\displaystyle \alpha\tau_1+f^0_1>0, \quad 0<4a_0a_2-a^2_1\ll 4a_2(\alpha\tau_1+f^0_1). \ \ \ \ \ (11)$

Here h.o.s.t referes to higher-order terms with small cofficients. Also we show that an interior separation takes place from ${(x_0,t_0)}$:

$\displaystyle x_0=\bigg(\frac{t_0(\alpha\tau_2+a_0b_0+f^0_2)}{b_0},\ -\frac{a_1}{2a_2}\bigg), \qquad t_0=\frac{4a_0a_2-a^2_1}{4a_2(\alpha\tau_1+f^0_1)}. \ \ \ \ \ (12)$

6. A typical development of a hurricane consists of several stages including an early tropical disturbance, a tropical depression, a tropical storm, and finally a hurricane stage.

Using the U-flow theory as described in Section 5 above, we derive the formation mechanism of tornados and hurricanes, providing precise conditions for their formation and explicit formulas on the time and location where tornados and hurricanes form:

Basically, we demonstrate that the early stage of a hurricane is through the horizontal interior flow separations, and we identify the physical conditions for the formation of the U-flow, corresponding to the tropical disturbance, and the temperature-driven counteracting force needed as the source for tropical depression. Basically, we demonstrate that the early stage of a hurricane is through the horizontal interior flow separations, and we identify the physical conditions for the formation of the U-flow, corresponding to the tropical disturbance, and the temperature-driven counteracting force needed as the source for tropical depression.

Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang

Topological Phase Transitions IV: Dynamic Theory of Boundary-Layer Separations

Tian Ma & Shouhong Wang, Topological Phase Transitions IV: Dynamic Theory of Boundary-Layer Separations, Hal-01672759

This is the fourth of a series of papers on topological phase transitions (TPTs), including

1. A TPT refers to the change in the topological structure in the physical space of the solutions of the governing partial differential equation (PDE) models of the underlying physical problem.

Boundary-layer separation phenomenon is one of the most important processes in fluid flows, and there is a long history of studies which go back at least, if not earlier, to the pioneering work of L. Prandtl in 1904. Basically, in the boundary-layer, the shear flow can detach/separate from the boundary, generating vortices and leading to more complicated turbulent behavior. The fundamental level understanding of this challenge problem boils down to TPTs of fluid flows. Mathematically, the velocity field ${u(x, t)}$ of the fluid satisfies the Navier-Stokes equations (NSEs) or their variations, and ${u(x, t)}$ defines its own topological structure in the physical space ${x\in \Omega}$, where ${\Omega}$ is the physical domain that the fluid occupies. Then the TPT associated with the boundary-layer or interior separations studies transition of the topological structure of ${u(x, t)}$ as the system control parameter varies.

2. It is clear then the geometric theory of incompressible flows developed by the authors plays a crucial role for the study of TPTs of fluids; and the complete account of this geometric theory is given in the authors’ research monograph

[MW05] T. Ma & S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, AMS Mathematical Surveys and Monographs Series, vol. 119, 2005, 234 pp.

One component of this geometric theory is the necessary and sufficient conditions for structural stability of divergence-free vector fields. Another component of the theory crucial for the study in this paper is the theorems on structural bifurcations. These theorems form the kinematic theory for understanding the topological phase transitions associated with fluid flows.

3. The most difficult and important aspect of TPTs associated with fluid flows is to make connections between the solutions of the NSEs and their structure in the physical space. The first such connection is the separation equation [MW05, Theorem 5.4.1] for the NSEs with the rigid boundary condition:

$\displaystyle \frac{\partial u_\tau(x,t)}{\partial n} =\frac{\partial \varphi_\tau}{\partial n} +\int_{0}^{t}[\nu\nabla\times\Delta u+k\nu\Delta u\cdot \tau+\nabla\times f+kf_\tau]\mbox{d}t. \ \ \ \ \ (1)$

In this paper, we derive the following separation equation for the NSE with the free-slip boundary condition:

$\displaystyle u_\tau(x,t)=\varphi_\tau(x)+\int_{0}^{t} \bigg[\nu\bigg(\frac{\partial^2 u_\tau}{\partial \tau^2}+\frac{\partial^2 u_\tau}{\partial n^2}\bigg)-g_\tau(u)+F_\tau\bigg]\mbox{d}t, \ \ \ \ \ (2)$

where ${F}$ and ${g}$ are the divergence-free parts of the external forcing and the nonlinear term ${u\cdot \nabla u}$ as defined:

$\displaystyle f=F+\nabla \phi,\qquad \text{div}F=0,\qquad F_n|_{\partial \Omega}=0. \ \ \ \ \ (3)$

$\displaystyle (u\cdot\nabla)u=g(u)+\nabla\Phi(u), \qquad \text{div}g(u)=0,\qquad g_n|_{\partial\Omega}=0. \ \ \ \ \ (4)$

The separation equations (1) and (2) provide necessary and sufficient conditions for the flow separation at a boundary point:

• the complete information for boundary-layer separation is encoded in the separation equations (1) and (2), which are therefore crucial for all applications; and
• the separation equations (1) and (2) link precisely the separation point ${(x, t)}$, the external forcing and the initial velocity field ${\varphi}$.

By exploring the leading order terms of the forcing ${f}$ and the initial velocity field ${\varphi}$ (Taylor expansions), more detailed condition, called predicable condition, are derived in [3, 14] for the Dirichlet boundary condition case, and in this paper for the free boundary condition case.

The separation equations (1) and (2), as well as the predicable conditions determine precisely when, where, and how a boundary-layer separation occurs.

4. For example, using the separation equations (1) and (2), we derive

• precise criteria on critical curvature for generating vortices from boundary tip points,
• critical velocity formula for surface turbulence; and
• the mechanism of the formation of the subpolar gyre and the formation of the small scale wind-driven vortex oceanic flows, in the north Atlantic ocean.

In particular, for the wind-driven north Atlantic circulations, with careful analysis using the separation equations (1) and (2), we derive the following conclusions:

• If the mid-latitude seasonal wind strength ${\lambda}$ exceeds certain threshold ${\lambda_c}$,vortices near the north Atlantic coast will form. Moreover, the scale (radius) of the vortices is an increasing function of ${\lambda-\lambda_c}$;
•  the condition for the initial formation of the subpolar gyre is that the curvature ${k}$ of ${\partial\Omega}$ at the tip point on the east coast of Canada is sufficiently large, and the combined effect of the convexity of the tangential component of the Gulf stream shear flow and the strength of the tangential friction force is positive; and
•  the vortex separated from the boundary tip point is then amplified and maintained by the wind stress, the strong Gulf stream current and the Coriolis effect, leading to the big subpolar gyre that we observe.

Tian Ma, Shouhong Wang

Topological Phase Transition III: Solar Surface Eruptions and Sunspots

Tian Ma & Shouhong Wang, Topological Phase Transition III: Solar Surface Eruptions and Sunspots, 2017, hal-01672381

The physical and mathematical reasons for solar surface eruptions and sunspots are not satisfactorily understood. This paper is aimed to provide a new theory for the formation of the solar surface eruptions, sunspots and solar prominences, based on the recently developed statistical theory of heat by the authors, and on the theory and notion of topological phase transitions.

1. The most important ingredient of the study is the recently developed theory of heat [T. Ma & S. Wang, Statistical Theory of Heat, 2017, hal-01578634]. We derived the energy level temperature formula, showing that the temperature is essentially the average energy level of system particles. We also obtained the photon number entropy formula, demonstrating that the entropy is the number of photons in the gap between system particles, and the physical carrier of heat is the photons.

Another important component of the theory is the vibratory mechanism of photon absorption and radiation:

a particle can only absorb and radiate photons while experiencing vibratory motion. The higher the frequency of the vibration of the particle, the larger the absorbing and radiating energy. The vibration or irregular motion of particles in a system is caused by collisions between particles and by absorbing and radiating photons.

2. This above mechanism shows immediately that for particles in high speed vibration and irregular motion, the rate of photon emission and absorption increases, leading to the number density of photons to increase, and further causing the particle energy levels to elevate. Hence, the photon absorption and emission induce the concentration of temperature, which we call the anti-diffusive effect of heat:

Due to the higher rate of photon absorption and emission of the particles with higher energy levels, the photon flux will move toward to the higher temperature regions from the lower temperature regions.

By the Stefan-Boltzmann law, the reversed heat flux measuring the anti-diffusive effect is expressed as

$\displaystyle \bigg(\frac{\text{d}T}{\text{d}t}\bigg)_{ADE}=\beta_0T^4,$

where ${\beta_0}$ is the heat effect coefficient.

3. Then by the Fourier law, we derive the following law for heat transfer for the solar atmosphere:

$\displaystyle \frac{\partial T}{\partial t}+(u\cdot \nabla)T=\kappa \Delta T+\beta_0T^4+\beta_1(\pmb{E}^2+\pmb{H}^2). \ \ \ \ \ (1)$

Here on the right-hand side, the first term represents the usual diffusion of heat, the last term is the heat source due to the solar electromagnetic fields. Importantly the second term represents the anti-diffusive effect of heat, and it is this anti-diffusive effect that leads to the formation of sunspots, the solar flares and the prominences.

4. The full model governing Sun’s surface plasma fluid combines the fluid dynamical equations, the above new heat equation (1), and the Maxwell equations. They are given as follows:

$\displaystyle \rho\bigg[\frac{\partial u}{\partial t}+(u\cdot\nabla)u\bigg]= \mu\Delta u-\nabla p +\rho_e(\pmb{E}+ u\times \pmb{H})-g\pmb{k}\rho(1-\alpha T), \ \ \ \ \ (2)$

$\displaystyle \frac{\partial T}{\partial t}+(u\cdot \nabla)T=\kappa \Delta T+\beta_0T^4+\beta_1(\pmb{E}^2+\pmb{H}^2), \ \ \ \ \ (3)$

$\displaystyle \frac{\partial \pmb{H}}{\partial t}=-\frac{1}{\mu_0}\text{curl }\pmb{E}, \ \ \ \ \ (4)$

$\displaystyle \frac{\partial \pmb{E}}{\partial t}=\frac{1}{\varepsilon_0}\text{curl }\pmb{H}-\frac{1}{\varepsilon_0}\rho_eu, \ \ \ \ \ (5)$

$\displaystyle \mbox{div }\pmb{H}=0, \ \ \ \ \ (6)$

$\displaystyle \mbox{div } \pmb{E}= \rho_e, \ \ \ \ \ (7)$

$\displaystyle \frac{\partial \rho}{\partial t}=-\text{div}(\rho u). \ \ \ \ \ (8)$

5. As the anti-diffusive term in (3) counteracts with the diffusion term, we are able to prove a temperature blow-up theorem, which shows that there exist ${x_0\in\Omega}$ and ${t_0>0}$, such that the temperature ${T}$ blows up at ${(x_0,t_0)}$ with blow-up time estimated as

$\displaystyle t_0=\frac{|\Omega|^3 }{3 a^3\beta_0}, \qquad a=\int_\Omega T_0(x) dx, \ \ \ \ \ (9)$

where ${T_0}$ is the initial value of temperature, ${ \Omega=S^2\times(r_0,r_1)}$, ${r_0}$ is the solar radius, and ${r_1=r_0+h}$ with the thickness of solar atmosphere ${h}$.

6. The sunspots can now be clearly explained by the anti-diffusive effect of heat and the temperature blow-up that we just mentioned. We summarize this explanation as follows:

• Due to thermal fluctuations, the temperature in the solar atmosphere is nonhomogeneous, leading to elevated temperature in some local areas. The anti-diffusive effect of heat then shows that the higher temperature regions absorb more photons from their surrounding areas, leading to their temperature decreasing, and consequently generating sunspots;
• The anti-diffusive effect of heat makes the temperature around the sunspot areas increasing rapidly, generating the temperature blow-up, and leading to solar eruptions.

In fact, equation (2) dictates the behavior of mass ejections. When the temperature ${T}$ blows up at ${(x_0,t_0)}$, the maximal forces acting on the particles near ${x_0}$ are just ${\nabla p}$. Hence, in the neighborhood of ${(x_0,t_0)}$, (2) can be approximatively expressed as

$\displaystyle \frac{\text{d} u}{\text{d} t}=-\frac{1}{\rho}\nabla p. \ \ \ \ \ (10)$

By the gaseous equation of state: ${p=R\rho T/m}$, where ${R}$ is the gas constant and ${m}$ is the particle mass, the equation (10) is written as

$\displaystyle \frac{\text{d} u}{\text{d} t}=-\frac{R}{m}\nabla T. \ \ \ \ \ (11)$

The temperature blow-up shows that

$\displaystyle \lim\limits_{t\rightarrow t_0}|\nabla T(x_0,t)|=\infty.$

Therefore we deduce from (11) that

$\displaystyle \lim\limits_{t\rightarrow t_0}\bigg|\frac{\text{d} u(x_0,t)}{\text{d} t}\bigg|=\infty, \ \ \ \ \ (12)$

which represents the high speed gas explosion and particle ejections. The ejection direction is

$\displaystyle \vec{r}=-\lim\limits_{t\rightarrow t_0}\frac{\nabla T(x_0,t)}{|\nabla T(x_0,t)|}. \ \ \ \ \ (13)$

• It is clear that the temperature blow-up generates solar flares.
• By the Maxwell equations (4) and (5) can be written as

$\displaystyle \frac{\partial^2 \pmb{E}}{\partial t^2}+\frac{1}{\varepsilon_0\mu_0}\text{curl}^2 \pmb{E}=-\frac{\rho_e}{\varepsilon_0}\frac{\partial u}{\partial t}, \ \ \ \ \ (14)$

$\displaystyle \frac{\partial^2 \pmb{H}}{\partial t^2}+\frac{1}{\varepsilon_0\mu_0}\text{curl}^2 \pmb{H}= \frac{\rho_e}{\varepsilon_0}u. \ \ \ \ \ (15)$

In view of (12) and (13), the equations (14) and (15) generate very strong electromagnetic radiation in the ${\vec{r}}$ direction.

• The eruption (11) leads also to a huge current jet ${J=\rho_eu}$ in the ${\vec{r}}$ direction.Also, (5) leads to the approximately the Ampère law:

$\displaystyle \text{curl } \pmb{H}=\rho_e u, \ \ \ \ \ (16)$

which gives rise to violent magnetic loops, perpendicular to the direction ${\vec{r}}$, leading to the solar prominences.

• Astronomical observations show that sunspots and solar flares occur periodically in an 11-year cycle. The blow-up time (9) links the initial temperature ${T_0}$, the solar eruption period, and the anti-diffusive effect coefficient ${\beta_0}$. Such a link is applicable to all stars. For the Sun, we can easily estimate

$\displaystyle \beta_0=2.85\times 10^{-22}/(K^3\cdot s).$

This is part of the research program initiated recently by the authors on theory and applications of topological phase transitions, including

Tian Ma, Shouhong Wang

Topological Phase Transitions II: Spiral Structure of Galaxies

Tian Ma & Shouhong Wang, Topological Phase Transitions II: Spiral Structure of Galaxies, 2017, hal-01671178

The aim of this paper is to derive a new mechanism for the formation of the galactic spiral patterns.

1. There are three types of galactic structures: the spiral, the elliptical, and the irregular. The existing theory of the formation of the spiral galactic structure is the density wave theory by Chia-Ciao Lin and Frank Shu in 1964. They proposed that the spiral arms, being non material, are caused by the non-homogeneous velocity of stars and nebulae, similar to a traffic jam in a highway. However, the reasons behind the nonhomogeneous velocity of stars and nebulae are still not clear, and the density wave theory is not completely satisfactory.

In essence the physical origin of the density wave theory is not clearly understood. This is resolved in this paper using the recent development by the authors on gravitational field particle and gravitational radiation.

2. Due to the presence of dark matter and dark energy, the Einstein general theory of relativity can be uniquely modified using PID to take into account the effect of dark energy and dark matter phenomena, and to preserve the Einstein’s two fundamental principles: the principle of equivalence and the principle of general relativity. The new field equations are given by

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{8\pi G}{c^4}T_{\mu\nu}-\nabla_\mu\Phi_\nu, \ \ \ \ \ (1)$

where

•  ${\{g_{\mu\nu}\}}$ is the Riemannian metrics of 4-dimensional space-time, representing the gravitational potential, depicting the curved space-time;
•  ${\{\Phi_\nu\}}$ is the dual gravitational potential, representing the gravitational field particle and carrying the field energy, which is similar to the electromagnetic interaction field particle: the photon.

The gravitational field particle is described by the dual field ${\{\Phi_\mu\}}$, which is regarded as the graviton.

3. Motivated by the new statistical theory of heat by the authors, we have the following conclusions for the gravitational field particle:

the absorption and radiation of gravitons could generate a gravitational temperature field, representing the average energy level of massive matter, reminiscent of the photons yielding the temperature in thermodynamical systems.

Also, the gravitational temperature ${\mathcal{T}}$, which we call G-temperature, satisfies a diffusion equation given by

$\displaystyle \frac{\partial \mathcal{T}}{\partial t}+\frac{1}{\rho}(P\cdot\nabla)\mathcal{T}=\kappa \Delta \mathcal{T} +Q, \ \ \ \ \ (2)$

where ${Q}$ represents the gravitational source.

The new gravitational temperature field provides the needed key source for the formation of different galactic patterns, which was entirely missing in existing theories such as the pioneering work of Lin and Shu.

4. With the G-temparature equation at our disposal, we can derive the momentum form of the astrophysical fluid dynamical model coupling the diffusion equation of G-temperature field. This new astrophysical galactic dynamics model is a dissipative system. Hence the mechanism of the formation of different galactic structures is of characteristic for dissipative systems, in contrast with the density wave theory.

5. The spatial domain of the model is the ring domain in polar coordinates

$\displaystyle \Omega=\{(r,\theta)~|~0\leq \theta\leq 2\pi,~r_0

and we take the transformation

$\displaystyle x_1=r\theta,\quad x_2=r-r_0.$

The Rubin rotational curve suggests that the momentum density ${\overline{P}}$ in the steady state solution of the model is

$\displaystyle (\overline{P},\overline{\mathcal{T}},\overline{p})=\left((\rho_0v_0,0), \mathcal{T}_0-\beta x_2, -\rho_0g\int(1-\alpha\overline{\mathcal{T}})\text{d}x_2 \right) \ \ \ \ \ (3)$

where ${v_0}$ is the constant velocity in the Rubin rotational curve. Then the nondimensional deviation system from the basic solution is given by

$\displaystyle \frac{1}{\text{Pr}}\bigg[\frac{\partial P}{\partial t}+(P\cdot\nabla)P\bigg] =\Delta P -a\frac{\partial P}{\partial x_1}-\frac{1}{\text{Pr}}\nabla P+\sqrt{\text{Ra}}\mathcal{T}\vec{k}, \ \ \ \ \ (4)$

$\displaystyle \frac{\partial \mathcal{T}}{\partial t}+(P\cdot\nabla)\mathcal{T} = \Delta \mathcal{T}-a\text{Pr}\frac{\partial \mathcal {T}}{\partial x_1}+\sqrt{\text{Ra}}P_2, \ \ \ \ \ (5)$

$\displaystyle \text{div}P=0. \ \ \ \ \ (6)$

Here Pr is the Prandtl number, ${a}$ is the Rubin number, and ${\ell}$ is the ratio between the disk width and the halo radius. These are non-dimensional parameters given by

$\displaystyle \text{Ra}=\frac{g\alpha\rho_0\beta}{\kappa\mu}r^4_0, \qquad \beta=\frac{T_0-T_1}{r_1-r_0},\quad a=\frac{r_0 v_0}{\mu},\quad \ell=\frac{r_1-r_0}{r_0}. \ \ \ \ \ (7)$

For this galactic dynamics system, there are three crucial parameters for the formation of different galactic structures:

• Ra represents the G-temperature gradient,
• ${a}$ represents the average velocity of stars in the galaxy, and
• ${\ell}$ represents the relative ratio between the inner and outer radii of galactic disk.

5. Mathematically, using the dynamical transition theory and the geometric theory of incompressible flows, both developed by the authors, we can demonstrate that both topological phase transition and dynamic phase transition occur at the same critical control parameters. In particular we derive the following conclusions on the formation of galactic structures, dictated by the parameters Ra, ${a, \ell}$:

• if ${\text{Ra} or ${\ell}$ is small, then the galaxy is elliptic; and
• if ${\text{Ra}>R_c(a)}$ and ${\ell}$ is relatively large, then the galaxy is spiral.

This paper is part of the research program initiated recently by the authors on theory and applications of topological phase transitions, including

• quantum phase transitions,
• electromagnetic eruptions on solar surface,
• boundary-layer separation of fluid flows, and
• interior separation of fluid flows.

Tian Ma, Shouhong Wang

Topological Phase Transitions I: Quantum Phase Transitions

Tian Ma & Shouhong Wang, Topological Phase Transitions I: Quantum Phase Transitions, hal-01651908

The aim of the above paper is to provide a systematic theoretical study on quantum phase transitions associated with the Bose-Einstein condensates, the superfluidity and the superconductivity.

I. Based on current developments, we now know that there are only two types of phase transitions:

• dynamical phase transitions and
• topological phase transitions (TPTs).

The authors have developed the dynamic transition theory for dissipative systems; see [Ma & Wang, Phase Transition Dynamics, Springer-Verlag, 2013, 555pp.] and the references therein. This is a new notion of phase transitions, applicable to all dissipative systems, including nonlinear dissipative systems in statistical physics, fluid dynamics, atmospheric and oceanic sciences, biological and chemical systems etc.

A TPT refers to the transition in its topological structure in the physical space of the system, and quantum phase transitions (QPTs) belong to the category of TPTs. The notion of TPTs is originated from the pioneering work by J. Michael Kosterlitz and David J. Thouless (1972) where they identified a completely new type of phase transitions in two-dimensional systems where topological defects play a crucial role. With this work, together with F. Duncan M. Haldane, they received 2016 Nobel prize in physics.

There have been many attempts, but the basic theoretical understanding of TPTs is still largely open.

II. QPTs are TPTs for condensate systems, including the gaseous Bose-Einstein condensates, superconductivity, and superfluidity. The basic physical characteristics of a QPT are as follows:

1.  A QPT is a transition between quantum states, and the state quantities or the order parameters of the quantum system describing the transition should be the wave functions ${\psi}$ of the quantum states;
2. the control parameters are non-thermal; and
3. a QPT is a TPT of a condensate system, rather than a dynamical phase transition. Consequently, the state functions describing a QPT are ${(\zeta,\varphi)}$ in the wave function ${\psi=\zeta e^{i \varphi}}$.

III. The field equations governing the condensation are determined by the principle of Hamiltonian dynamics (PHD) or equivalently by the principle of Lagrangian dynamics (PLD):

$\displaystyle i\hbar\frac{\partial\Psi}{\partial t}=\frac{\delta}{\delta \Psi^*}\mathcal{H}(\Psi,\lambda), \ \ \ \ \ (1)$

where ${\lambda}$ is the control parameter. The associated phase transition equations are then the following topological structure equations:

• for particle number conserved systems,

$\displaystyle \frac{\delta}{\delta \psi^*}\mathcal{H}(\psi,\lambda)= \mu\psi, \ \ \ \ \ (2)$

where ${\Psi=e^{-i\mu t/\hbar}\psi(x)}$, ${\psi=\zeta e^{i \varphi}}$, and ${\mu \in\mathbb{R}}$ represents the chemical potential;

• for non-conserved systems,

$\displaystyle \frac{\delta}{\delta \psi^*}\mathcal{H}(\psi,\lambda)=0. \ \ \ \ \ (3)$

IV. Based on the above three characteristics and the topological structure equations (2), the wave function of the condensate is a function of the control parameter ${\lambda}$, i.e. ${\psi(\lambda)=\zeta e^{i\varphi}}$. Then mathematically we say that the system undergoes a QPT at a critical ${\lambda_c}$ if the topological structure of ${(\zeta(\lambda_c-\varepsilon),\varphi(\lambda_c-\varepsilon))}$ is different from that of ${(\zeta(\lambda_c+\varepsilon),\varphi(\lambda_c+\varepsilon))}$.

V. We deduce from the topological structure equations (2) that the supercurrents of superconductivity and superfluidity are as follows:

• for gaseous Bose-Einstein systems and for liquid helium systems,

$\displaystyle J= \frac{\hbar}{m}\zeta^2\nabla\varphi, \ \ \ \ \ (4)$

• for superconductivity,

$\displaystyle \frac{e_s}{m_sc}\zeta^2\bigg(\hbar\nabla\varphi-\frac{e_s}{c}\textbf{A}\bigg). \ \ \ \ \ (5)$

VI. The supercurrents enjoy the divergence–free condition (incompressibility):

$\displaystyle \text{div}J=0. \ \ \ \ \ (6)$

The authors have developed a geometric theory for incompressible flows to study the structure and its stability and transitions of incompressible fluid flows in the physical spaces. The complete account of this geometric theory is given in the authors’ research monograph [Ma & Wang, Geometric Theory of Incompressible Flows and Applications to Fluid Dynamics, AMS, 2005]. This geometric theory can then be directly applied to study the transitions of topological structure associated with the quantum phase transitions of BEC, superfluidity and superconductivity. We derive in particular the basic theory for different TPTs, leading to transparent physical pictures of various condensates.

VII. After careful examination of the formation of Cooper pairs in superconductivity, we derive the following microscopic mechanism for the Meissner effect:

•  Below the critical temperature ${T_c}$, an applied magnetic field ${\textbf{H}_a}$ induces spin ${J=1}$ Cooper pairs;
• only the Cooper pairs reversely parallel to the applied ${\textbf{H}_a}$ are stable, leading to their physical formation; and
• the total magnetic moment of all Cooper pairs with ${s=-1}$, together with the surface supercurrents, can cancel out the magnetism induced by the applied field ${\textbf{H}_a}$ in the superconductor, and resists the applied field ${\textbf{H}_a}$ from entering its body.

Also, this paper is the first one in the series of papers by the authors on TPTs, including the forthcoming papers for the following science problems:

1. galactic spiral structures,
2. electromagnetic eruptions on solar surface,
3. boundary-layer separation of fluid flows, and
4. interior separation of fluid flows.

Tian Ma, Shouhong Wang