arXiv subjects

Steven D Miller

Publications and source records attributed to Steven D Miller.

12 recordsLinked to original sources

arXiv2025

Kinetic Theory and the Mechanics of Isothermal Gas Spheres: Derivation of the Classical Emden--Chandrasekhar Equation via the Vlasov--Poisson Formalism

We present a derivation of the mechanics of isothermal gas spheres directly from the Vlasov--Poisson equation. By extremising the Boltzmann entropy, we obtain the Maxwell--Boltzmann distribution for a self-gravitating isothermal Newtonian gas, which is a stationary solution of the Vlasov--Poisson system. From this distribution, the corresponding Poisson--Boltzmann equation for the gravitational potential is deduced. The second variation of entropy reproduces the classical Antonov instability criterion: the critical energy is $E_c \simeq -0.335\,\frac{G M^2}{R}$, below which no local entropy maximum exists and the configuration becomes unstable (the so-called "gravothermal catastrophe"). In this work, we assume $E>E_c$, so all equilibria lie on the stable branch, and the Antonov instability does not affect the analysis. Specializing to spherical symmetry, we recover the classical equation for the isothermal gas sphere, as originally studied by Chandrasekhar, which has applications to the formation of red giant stars. We also derive the fundamental equation of hydrostatic equilibrium, the energy integral and the virial theorem directly from the stationary Vlasov--Poisson solution, demonstrating also that an isothermal gas exhibits negative specific heat. Furthermore, we show that an isothermal gas sphere of strictly constant density is an impossibility. This exposition emphasizes some of the deep connections and self-consistency between kinetic theory, statistical mechanics, and stellar structure, while highlighting some formal aspects of classical astrophysical models.

astro-ph.GA↗

arXiv2025

The Kelly Criterion And Utility Function Optimisation For Stochastic Binary Games: Submartingale And Supermartingale Regimes

A reformulation of the Kelly Criterion is presented. Let $\mathfrak{G}$ be a generic stochastic Bernoulli binary game with outcomes $\mathscr{Z}(I)\in\lbrace -1,1\rbrace$ of N trials for $I=1...N$. The binomial probabilities are $\mathsf{P}(\mathscr{Z}(I)=1)=p$ and ${\mathsf{P}}(\mathscr{Z}(I)=-1)=q$ with $p+q=1$. For a fair game $p=q=\tfrac{1}{2}$ and for a biased game $p>q$. If $\mathscr{W}(0)$ is the initial wealth then at the $I^{th}$ trial one bets a fraction $\mathcal{F}$ so that the bet is $B(I)=\mathcal{F}\mathscr{W}(I-1)$. If one wagers $B(I)$ and wins one recovers the original wager plus $B(I)$ if $\mathscr{Z}(I)=+1$, or a loss of $B(I)$ if $\mathscr{Z}(I)=-1$. The wealth at the $N^{th}$ trial/bet for large $N$ is the random walk $\mathscr{W}(N)=\mathscr{W} (0)+\sum_{I=1}^{N}B(I)\mathscr{Z}(I)=\mathscr{W}(0)\prod_{I=1}^{N}(1+\mathcal{F}\mathscr{Z}(I))$ with expectation $\mathsf{E}[\mathscr{W}(N)]$. Defining a 'utility function' $\mathsf{U}(\mathcal{F},p)=\mathsf{E}[\log(\mathscr{W}(N)/\mathscr{W}(0))^{1/N}]$ then $\mathsf{U}(\mathcal{F},p)$ is optimised by the Kelly fraction $\mathcal{F}=\mathcal{F}_{K}=p-q=2p-1$, which is essentially a critical point of $\mathsf{U}(\mathcal{F},p)$. Also $\mathsf{U}(\mathcal{F}_{K},p)$ can be related to the Shannon entropy. If $[0,1]=[0,\mathcal{F}_{*})\bigcup [\mathcal{F}_{*}]\bigcup (\mathcal{F}_{*},1]$ with $\mathsf{U}(\mathcal{F}_{*},p)=0$ then $\mathsf{U}(\mathcal{F},p)>0, \forall\mathcal{F}\in[0,\mathcal{F}_{*})$ and $\mathscr{W}(N)$ is a submartingale for $p>1/2$; also $\mathsf{U}(\mathcal{F},p)<0,\forall \mathcal{F}\in(\mathcal{F}_{*},1]$, and $\mathscr{W}(\mathcal{F},p)$ is a supermartingale. Estimates are derived for variance and volatility $\mathsf{VAR}(\mathscr{W}(N))$ and $\sigma(\mathscr{W}(N))=\sqrt{\mathsf{VAR}(\mathscr{W}(N)})$. For large $N$ and $\mathcal{F}=\mathcal{F}_{K}$, $\mathsf{E}[\mathscr{W}(N)]$ grows exponentially.

math.PR↗

arXiv2024

A Spectral Representation of a Weighted Random Vectorial Field: Potential Applications to Turbulence and the Problem of Anomalous Dissipation in the Inviscid Limit

Let ${\mathfrak{G}}\subset\mathbb{R}^{3}$ with $vol(\mathfrak{G})\sim L^{3}$. Let ${\mathscr{T}}(x)$ be a Gaussian random field $\forall~x\in\mathfrak{G}$ with expectation $\mathbf{E}[{\mathscr{T}}(x)]=0$ and correlation $\mathbf{E}[{\mathscr{T}}(x)\otimes{\mathscr{T}}(y)]=K(x,y;\lambda)$, an isotropic and regulated kernel with correlation length $\lambda$. The field has a Karhunen-Loeve spectral representation ${\mathscr{T}}(x)=\sum_{I=1}^{\infty}\mathrm{Z}^{1/2}_{I}f_{I}(x)\otimes\mathscr{Z}_{I}$, with eigenvalues $\lbrace\mathrm{Z}_{I}\rbrace$, eigenfunctions $\lbrace f_{I}(x)\rbrace $ and Gaussian random variables $\mathscr{Z}_{I}$ with $\mathbf{E}[\mathscr{Z}_{I}]=0$ and $\mathbf{E}[\mathscr{Z}_{I}\otimes\mathscr{Z}_{J}]=\delta_{IJ}$. If $\mathfrak{G}$ contains incompressible fluid of viscosity $\nu$ with velocity $u_{a}(x,t)$ that evolves via the Navier-Stokes equations with a high 'Reynolds function' $\mathsf{RE}(x,t)=\tfrac{\|u_{a}(x,t)\|L}{\nu} $ then aspects of a turbulent flow with $\mathsf{RE}(x,t)\gg \mathsf{RE}_{*}$, a critical Reynolds number, might be represented by the 'weighted' random field $\mathscr{U}_{a}(x,t)= u_{a}(x,t)+\mathrm{A}u_{a}(x,t)\big(\mathsf{RE}(x,t)-\mathsf{RE}_{*}\big)^{\beta}\sum_{I=1}^{\infty} \mathrm{Z}^{1/2}_{I}f_{I}(x)\otimes\mathscr{Z}_{I}$ where random fluctuations and amplitude scale nonlinearly with $\mathsf{RE}(x,t)$, with mean $\mathbf{E}[{\mathscr{U}}_{a}(x,t)] =u_{a}(x,t)$. In the inviscid limit one can prove an anomalous dissipation-type law \begin{align} \lim_{\nu\rightarrow 0}\bigg(\lim_{u_{a}(x,t)\rightarrow {u}_{a}}\sup~\nu \int_{\mathfrak{G}}\int_{0}^{T}{\mathbf{E}}\bigg[\bigg|{\nabla}_{a}{\mathscr{U}}_{a}(x,s)\bigg|^{2}\bigg]d\mathcal{V}(x) ds\bigg)>0 \end{align} iff $\beta=\tfrac{1}{2}$ and $\sum_{I=1}^{\infty}\mathrm{Z}_{I}\int_{{\mathfrak{G}}}{\nabla}_{a}f_{I}(x){\nabla}^{a}f_{I}(x)d\mathcal{V}(x)>0$.

math.PR↗

arXiv2023

Second And Third-Order Structure Functions Of An 'Engineered' Random Field And Emergence Of The Kolmogorov 4/5 And 2/3-Scaling Laws Of Turbulence

The 4/5 and 2/3 laws of turbulence can emerge from a theory of 'engineered' random vector fields $\mathcal{X}_{i}(x,t) =X_{i}(x,t)+\tfracθ{\sqrt{d(d+2)}} X_{i}(x,t)ψ(x)$ existing within $\mathbf{D}\subset\mathbf{R}^{d}$. Here, $X_{i}(x,t)$ is a smooth deterministic vector field obeying a nonlinear PDE for all $(x,t)\in\mathbf{D}\times\mathbf{R}^{+}$, and $θ$ is a small parameter. The field $ψ(x)$ is a regulated and differentiable Gaussian random field with expectation $\mathbb{E}[ψ(x)]=0$, but having an antisymmetric covariance kernel $\mathscr{K}(x,y)=\mathbb{E}[ψ(x)ψ(y)]=f(x,y)K(\|x-y\|;λ)$ with $f(x,y)=-f(y,x)=1,f(x,x)=f(y,y)=0$ and with $K(\|x-y\|;λ)$ a standard stationary symmetric kernel. For $0\le\ell\le λ<L$ with $X_{i}(x,t)=X_{i}=(0,0,X)$ and $θ=1$ then for $d=3$, the third-order structure function is \begin{align} S_{3}[\ell]=\mathbb{E}\left[|\mathcal{X}_{i}(x+\ell,t)-\mathcal{X}(x,t)|^{3}\right]=-\frac{4}{5}\|X_{i}\|^{3}=-\frac{4}{5}X^{3}\nonumber \end{align} and $S_{2}[\ell]=CX^{2}$. The classical 4/5 and 2/3-scaling laws then emerge if one identifies the random field $\mathcal{X}_{i}(x,t)$ with a turbulent fluid flow $\mathcal{U}_{i}(x,t)$ or velocity, with mean flow $\mathbb{E}[\mathcal{U}_{i}(x,t)]=U_{i}(x,t)=U_{i}$ being a trivial solution of Burger's equation. Assuming constant dissipation rate $ε$, small constant viscosity $ν$, corresponding to high Reynolds number, and the standard energy balance law, then for a range $η\le\ell\ll λ<L$ \begin{align} S_{3}[\ell]=\mathbb{E}\left[|\mathcal{U}_{i}(x+\ell,t)-\mathcal{U}(x,t)|^{3}\right]=-\frac{4}{5}ε\ell\nonumber \end{align} where $η=(ν^{3/4}ε)^{-1/4}$. For the second-order structure function, the 2/3-law emerges as $S_{2}[\ell]=Cε^{2/3}\ell^{2/3}$.

physics.flu-dyn↗

arXiv2023

A Cluster Expansion Proof That The Stochastic Exponential Of A Brownian Motion Is A Martingale

Let $ψ:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth and continuous real function and $ψ\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let ${B}(t)$ be a standard Brownian motion defined with respect to a probability space $(Ω,\mathscr{F},{\mathsf{P}})$ and where $d{B}(t)=ξ(t)dt$ and $t\in\mathbb{R}^{+}$. The process $ξ(t)$ is a Gaussian white noise with expectation $\mathbf{\mathsf{E}}~ξ(t)=0$ and with covariance ${\mathsf{E}}~ξ(t)ξ(s)=δ(t-s)$. The Dolean-Dades stochastic exponential ${Z}(t)$ is the solution to the linear stochastic differential equation describing a geometric Brownian motion such that $d{Z}(t)=ψ(t){Z}(t)d{B}(t)=ψ(t){Z}(t)ξ(t)dt$. Using a cluster expansion method, and the moment and cumulant generating functions for $ξ(t)$, it is shown that ${Z}(t)$ is a martingale. The original Novikov criteria for ${Z}(t)$ being a true martingale are reproduced and exactly satisfied, namely that \begin{align} {\mathsf{E}}\mathrm{Z}(t)={\mathsf{E}}\exp\left(\int_{o}^{t}ψ(u)d{B}(u) -\frac{1}{2}\int_{0}^{t}|ψ(u)|^{2}du\right)=1\nonumber \end{align} provided that $\exp\big(\int_{0}^{t}|ψ(u)|^{2}du\big)<\infty$ for all $t>0$. However, ${\mathsf{E}}\big[|{Z}(t)|^{p}\big] =\exp(\tfrac{1}{2}p(p-1)ϕ(t))$, if $ϕ(t)=\int_{0}^{t}|ψ(u)|^{2}du$ is monotone increasing and is a submartingale for all $p>1$.

math.PR↗

arXiv2023

A Stochastic Differential Equation For Laser Propagation In Medias With Random Gaussian Absorption Coefficients: A Modified Beer's Law Solution Via A Van Kampen Cluster Expansion

Let $\mathbb{I\!D}=[0,\mathrm{L}]\subset\mathbb{R}^{+}$ be a slab geometry with boundaries $z=0$ and $z=\mathrm{L}$. A laser beam with a flat incident intensity $ψ_{o}$ enters the slab along the z-axis or unit vector $\widehat{\mathbf{e}}_{3}$ at $z=0$. The slab contains matter with an absorption coefficient of $\mathsf{A}$ with respect to the wavelength. If $\mathsf{A}$ is constant and homogenous then the beam decays as Beer's law $ψ(z,\widehat{\mathbf{e}}_{3})=ψ_{o}\exp(-\mathsf{A} z)$. If the absorption coefficient is randomly fluctuating in space as $\mathbf{A}(z)=\mathsf{A}(1+α\mathbf{G}(z))$--where $α>0$ determines the magnitude of the fluctuations, and the Gaussian random function has expectation $\mathbb{E}\lbrace \mathbf{G}(z)\rbrace =0$ and a binary correlation $\mathbb{E}\lbrace\mathbf{G}(z_{1})\otimes\mathbf{G}(z_{2})\rbrace=ϕ(z_{1},z_{2};ξ)={\mathsf{C}}\exp(-|z_{1}-z_{2}|^{2}ξ^{-2})$ for all $(z_{1},z_{2})\in\mathbb{I\!D}$ with correlation length $ξ$--then the beam propagation and attentuation within the medium is described by the stochastic differential equation \begin{equation} d\widehat{ψ(z,\widehat{\mathbf{e}}_{3})}=-\mathsf{A}\widehat{ψ(z,\mathbf{e}_{3})}dz-α\mathsf{A}\widehat{ψ(z,\mathbf{e}_{3})}\mathbf{G}(z)dz \end{equation} The stochastically averaged solution is derived via a Van Kampen-type cluster expansion, truncated at 2nd order for Gaussianality, giving a modified Beer's law \begin{equation} \mathbb{I}(z,\widehat{\mathbf{e}}_{3})=\mathbb{E}\big\lbrace\widehat{ψ(z,\widehat{\mathbf{e}}_{3})}\big\rbrace=ψ_{o}\exp(-\mathsf{A}z)\exp\bigg(\frac{1}{4}α^{2}\mathsf{A}^{2}{\mathsf{C}}ξ\bigg[\exp(-z^{2}/ξ^{2})\bigg(\sqrtπz Erf(\tfrac{z}ξ)\exp\bigg(\frac{z^{2}}{ξ^{2}}\bigg)+ξ\bigg)-ξ\bigg]\bigg) \end{equation} The deterministic Beer's law is recovered as $α\rightarrow 0 $.

math-ph↗

arXiv2022

A Turbulent Fluid Mechanics Via Nonlinear Mixing Of Smooth Velocity Flows With Reynolds-Weighted Random Fields

We consider a finite-volume domain $\mathfrak{D}\subset\mathbb{R}^{3}$ of size $\mathrm{Vol}(\mathfrak{D})\sim \mathrm{L}^{3}$ containing a viscous fluid of kinematic viscosity $\nu$ with velocity field $U_{a}(x,t)$ satisfying the Navier--Stokes equations with prescribed boundary data. We introduce a zero-centred homogeneous-isotropic Gaussian field $\mathscr{B}(x)$ on $\mathfrak{D}$ with Bargmann--Fock correlation $\mathbb{E}\langle\mathscr{B}(x)\otimes\mathscr{B}(y)\rangle=\mathsf{C}\exp(-|x-y|^{2}\lambda^{-2})$, where $\lambda\le \mathrm{L}$. For the volume-averaged Reynolds number $\mathbf{Re}(\mathfrak{D},t)=(|\mathrm{Vol}(\mathfrak{D})|^{-1}\int_{\mathfrak{D}}|U_{a}(x,t)|d\mu(x))\mathrm{L}/\nu$, let $\mathbf{Re}_{c}(\mathfrak{D})$ denote the critical threshold for turbulence. We propose a Reynolds-weighted mixing ansatz for a turbulent velocity field \[\mathscr{U}_{a}(x,t)=U_{a}(x,t)+\alpha U_{a}(x,t)\psi(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)\mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)]\mathscr{B}(x)\] with $\alpha\ge 1$, $\psi$ monotone increasing, and $\mathbb{I}_{\mathcal{S}}$ active only for $\mathbf{Re}>\mathbf{Re}_{c}$. The construction preserves the mean flow, $\mathbb{E}\langle\mathscr{U}_{a}(x,t)\rangle=U_{a}(x,t)$, while allowing turbulence intensity to grow with the control parameter $\mathbf{Re}$. This provides a tentative stochastic closure for Navier--Stokes, enabling Reynolds-type correlations $\mathsf{T}_{ab}(x,y;t)=\mathbb{E}\langle\mathscr{U}_{a}(x,t)\otimes\mathscr{U}_{b}(y,t)\rangle$ and higher moments. For test functions $f$ and curves $\Im\subset\mathfrak{D}$ we define a Hopf-like functional \[\mathbb{H}[\mathscr{U}_{a},t]=\mathbb{E}\bigg\langle\exp\bigg(i\int_{\Im}f(x,t)\mathscr{U}_{a}(x,t)dx^{a}\bigg)\bigg\rangle\] encoding circulation statistics generated by the mixing ansatz.

math-ph↗

arXiv2021

The Fundamental Equilibrium Equation For Gaseous Stars And The Tolman-Oppenheimer-Volkoff Equation -- Derivations And Applications With Emphasis On Optimisational-Variational Methods

Stars are essentially gravitationally stabilised thermonuclear reactors in hydrostatic equilibrium. The fundamental differential equation for all Newtonian gaseous stars in equilibrium is \begin{align} \frac{dp(r)}{dr}=-\frac{\mathscr{G}\mathcal{M}(r)ρ(r)}{r^{2}}\nonumber \end{align} where $p(r),ρ(r)$ are the pressure, density at radius $r$ and $\mathcal{M}(r)$ is the mass contained within a shell of radius $r$ given by $\mathcal{M}(r)=\int_{0}^{r}4π\overline{r}^{2} ρ(\overline{r})d\overline{r}$, and $\mathscr{G}$ is Newton's constant. This simple but crucial differential equation for the pressure gradient within any star, underpins much of astrophysical theory and it can derived by various methods:via a simple heuristic argument; via the Euler-Poisson equations for a self-gravitating fluid/gas; via a variational method by taking the 1st variation of the sum of the thermal and gravitational energies of the star; via the 2nd variation of the Massiue thermodynamic functional for a self-gravitating isothermal perfect-gas sphere; from conservation of the virial tensor; as the non-relativistic limit of the Tolman-Oppenheimer-Volkoff equation (TOVE). The TOVE for equilibrium of relativistic stars in general relativity can in turn be derived by various methods: from the energy-momentum conservation constraint on the Einstein equations applied to a spherically symmetric perfect fluid/gas; via a constrained optimization method on the mass and nucleon number; via a maximum entropy variational method for a sphere of self-gravitating perfect fluid/gas or radiation. An overview is given of all derivations with emphasis on variational methods. Many important applications and astrophysical consequences of the Newtonian equilibrium equation are also reviewed.

gr-qc↗

arXiv2021

Stochastic Cauchy Initial Value Formulation Of The Heat Equation For Random Field Initial Data: Smoothing, Harnack-Type Bounds And p-Moments

The following stochastic Cauchy initial-value problem is studied for the parabolic heat equation on a domain $ \mathbf{Q}\subset{\mathbf{R}}^{n}$ with random field initial data. \begin{align} &{\square}\widehat{u(x,t)} \equiv \bigg(\frac{\partial}{\partial t}-Δ_{x}\bigg)\widehat{u(x,t)}=0,~x\in\mathbf{Q},t> 0 \end{align} \begin{align} \widehat{u(x,0)}=ϕ(x)+\mathscr{J}(x),~x\in\mathbf{Q},t=0 \end{align} where $ϕ(x)\in C^{\infty}({\mathbf{Q}})$, and $\mathscr{J}(x)$ is a classical Gaussian random scalar field with expectation $\mathbb{E}[\![\mathscr{J}(x)]\!]=0 $ and with a regulated covariance $\mathbb{E}[\![ \mathscr{J}(x)\otimes\mathscr{J}(y)]\!]=ζJ(x,y;\ell)$, correlation length $\ell$ and $\mathbb{E}[\![ \mathscr{J}(x)\otimes\mathscr{J}(x)]\!]=ζ<\infty$. The randomly perturbed solution $\widehat{u(x,t)}$ is a stochastic convolution integral. This leads to stochastic extensions and versions of some classical results for the heat equation; in particular, a Li-Yau differential Harnack inequality \begin{align} \mathbb{E}\left[\!\!\left[\frac{|\nabla\widehat{u(x,t)}|^{2}}{|\widehat{u(x,t)}|^{2}} \right]\!\!\right]-\mathbb{E}\left[\!\!\left[\frac{\tfrac{\partial}{\partial t}\widehat{u(x,t)}}{\widehat{u(x,t)}} \right]\!\!\right]\le \frac{1}{2}n\frac{1}{t} \end{align} and a parabolic Harnack inequality. Decay estimates and bounds for the volatility $\mathbb{E}[\![|\widehat{u(x,t)}|^{2}]\!]$ and p-moments $\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!] $ are derived. Since $\lim_{t\uparrow \infty}\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!]=0 $, the Cauchy evolution of the randomly perturbed solution is stable since the heat equation smooths out or dissipates volatility induced by initial data randomness as $t\rightarrow\infty$.

math.PR↗

arXiv2021

Harmonic Functions And Linear Elliptic Dirichlet Problems With Random Boundary Values--Stochastic Extensions Of Some Classical Theorems And Estimates

Let $ψ:{\mathcal{D}}\rightarrow{\mathbf{R}}$ be a harmonic function such that $Δψ(x)=0$ for all $x\in\mathcal{D}\subset{\mathbf{R}}^{n}$. There are then many well-established classical results:the Dirichlet problem and Poisson formula, Harnack inequality, the Maximum Principle, the Mean Value Property etc. Here, a 'noisy' or random domain is one for which there also exists a classical scalar Gaussian random field (GRF) ${\mathscr{J}(x)}$ defined for all $x\in{\mathcal{D}}$ or $x\in\partial {\mathcal{D}}$ with respect to a probability space $[Ω,\mathcal{F},{\mathrm{I\!P}}]$. The GRF has vanishing mean value $\mathbf{E}[\![\mathscr{J}(x)]\!] = 0$ and a regulated covariance ${\mathbf{E}}[\![{\mathscr{J}(x)} \otimes {\mathscr{J}(y)}]\!] = αJ(x,y;ξ)$ for all $(x,y)\in{\mathcal{D}}$ and/or $(x,y)\in{\partial\mathcal{D}}$, with correlation length $ξ$ and ${\mathbf{E}}[\![{\mathscr{J}(x)} \otimes {\mathscr{J}}(x)]\!] = α<\infty$. The gradient $\nabla{\mathscr{J}(x)}$ and integral $\int_{\mathcal{D}}{\mathscr{J}}(x) dμ(x)$ also exist on ${\mathcal{D}}\bigcup\partial\mathcal{D}$. Harmonic functions and potentials can become randomly perturbed GRFs of the form $\overline{ψ(x)}=ψ(x)+λ{\mathscr{J}}(x)$. Physically, this scenario arises from noisy sources or random fluctuations in mass/charge density, noisy or random boundary/surface data; and introducing turbulence/randomness into smooth fluid flows, steady state diffusions or heat flow. This leads to stochastic modifications of classical theorems for randomly perturbed harmonic functions and Riesz and Newtonian potentials; and to stability estimates and bounds for the growth and decay of their volatility and moments.

math-ph↗

arXiv2021

Stochastic Control of Tolman-Oppenheimer-Snyder Collapse of Zero-Pressure Stars to Black Holes: Rigorous Criteria for Density Bounds and Singularity Smoothing

The Tolman-Oppenheimer-Snyder description gives exact analytical solutions for an Einstein-matter system describing total gravitational collapse of a zero-pressure perfect-fluid sphere, representing a massive star which has exhausted its nuclear fuel. The star collapses to a point of infinite density within a finite comoving proper time interval $[0,t_{*}]$, and the exterior metric matches the Schwarzchild black hole metric. The description is re-expressed in terms of a 'density function' $u(t)=(ρ(t)/ρ_{o}))^{1/3}=R^{-1}(t)$ for initial density $u_{0}=R^{-1}(0)=1$ and radius $R(0)$, whereby the general-relativistic formulation reduces to an autonomous nonlinear ODE for $u(t)$. The solution blows up or is singular at $t=t_{*}=π/2(8πG/3ρ_{o})^{1/2}$. The blowup interval $[0,t_{*}]$ is partitioned into domains $[0,t_ε]\bigcup[t_ε,t_{*}]$,with $t_{*}=t_ε+|ε|$ and $|ε|\ll 1$, so that $t_ε$ can be infinitesimally close to $t_{*}$. Randomness or 'stochastic control' is introduced via the 'switching on' of specific (white-noise) perturbations at $t=t_ε$. Hybrid nonlinear ODES-SDES are then 'engineered' over the partition. Within the Ito interpretation, the resulting density function diffusion $\overline{u(t)}$ is proved to be a martingale whose supremum, volatility and higher-order moments are finite, bounded and singularity free for all finite $t>t_ε$. The collapse is (comovingly) eternal but never becomes singular. Extensive and rigorous boundedness and no-blowup criteria are established via various methods, and blowup probability is always zero. The density singularity is therefore smoothed or 'noise-suppressed'. Within the Stratanovitch interpretation, the singularity formation probability is unity; however, null recurrence ensures the expected comoving time for this to occur is now infinite.

math-ph↗

arXiv2020

Stochastic Averaging of The Einstein Vacuum Equations on a Toroidal Manifold with Randomly Perturbed Radial Moduli: Stability Criteria and Induced 'Cosmological Constant' Terms

The Einstein vacuum equations on an (n+1)-dimensional toroidal manifold $\mathbb{M}^{n+1}=\mathbb{T}^{n}\times\mathbb{R}^{+}$ reduce to a system of n-dimensional nonlinear ODEs in terms of the set of toroidal radii $(a_{i})_{i=1}^{n}$ or the radial moduli fields $(ψ_{i})_{i=1}^{n}=(\log(a_{i}(t))_{i=1}^{n}$ of the n-torus $\mathbb{T}^{n}$. This geometry is also the basis of Kasner-Bianchi-type cosmologies. The equations are trivially satisfied for static solutions $ψ_{i}^{E}=ψ^{E}$ or radii $a_{i}^{E}=a^{E}$, describing an initially static toroidal 'micro-universe' or 'vacuum bubble'. It is Lyapunov stable to short-pulse deterministic perturbations, which have a sharp Gaussian profile: the perturbed radii rapidly converge to new attractors and therefore to new stable equilibria. These perturbations induce transitions between stable states. Introducing classical random fluctuations or perturbations, with a regulated covariance, the radial moduli become Gaussian random fields paramatrizing a 'toroidal random geometry'. The randomly perturbed Einstein equations are then interpreted as a stochastic n-dimensional nonlinear dynamical system. Non-vanishing 'cosmological constant' terms are retained within the averaged equations since they are nonlinear. This is analogous to averaging the Navier-Stokes equations in statistical turbulence theory, which yields an additional non-vanishing Reynolds term, since like the Einstein equations they are also of nonlinear hyperbolic type. The expectations of the randomly perturbed toric radii can be estimated from a cumulant expansion method. The initially static toroidal vacuum bubble undergoes eternal 'noise-induced' stochastic exponential growth or inflation. Random radial moduli fields within this scenario therefore act like a 'dark energy'. Finally, a class of random perturbations is considered for which this Einstein system is stable.

math-ph↗