User:Tohline/Appendix/Ramblings/StrongNuclearForce - Revision history

Tohline: /* Radial Dependance of the Strong Nuclear Force */

2021-07-22T00:00:25Z

Radial Dependance of the Strong Nuclear Force

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	__FORCETOC__		__FORCETOC__
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Radial ~~Dependance~~ of the Strong Nuclear Force=		=Radial Dependence of the Strong Nuclear Force=

	{{LSU_HBook_header}}		{{LSU_HBook_header}}

Tohline: /* Insert Dependence on (Energy) Density */

2021-07-21T22:14:22Z

Insert Dependence on (Energy) Density

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	The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential. The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime. Generally speaking, a QGP is achieved in a very high energy-density environment.		The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential. The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime. Generally speaking, a QGP is achieved in a very high energy-density environment.

	We can mimic this behavior in our modified cosmology by assuming that the coefficient on the ~~potential's logarithmic~~ term varies with the energy-density of the fluid. (More simply, let's have it vary with the ''mass''-dentiy.) We want to "kill off" the ~~logarithmic~~ term when the density climbs above some threshold, <math>\rho_H</math>. Let's try …		We can mimic this behavior in our modified cosmology by assuming that the coefficient on the <math>1/r</math> term in the gravitational acceleration varies with the energy-density of the fluid. (More simply, let's have it vary with the ''mass''-dentiy.) We want to kill off the <math>1/r</math> term when the density climbs above some threshold, <math>\rho_H</math>. Let's try …
	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">
	<tr>		<tr>
	<td align="right"><math>\~~Phi~~</math></td>		<td align="right"><math>\ddot{R}</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	- \frac{~~GM_r~~}{r}		<math>- \frac{GM}{R^2} \biggl\{ 1 + \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \frac{R}{a_T} \biggr\} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			Note that the <math>~R-</math>dependent potential from which this expression for the acceleration is derived is,
			<table border="0" align="center" cellpadding="5">
			<tr>
			<td align="right"><math>~\Phi(R)</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>+ \frac{GM}{R} - \frac{GM_r}{a_T} \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \ln \biggl(\frac{R}{a_T}\biggr) \, .</math>
			</td>
			</tr>
			</table>

			This expression for the gravitational acceleration has the desired properties:
			<ul>
			<li>
			In the early universe, when <math>\rho/\rho_H \gg 1</math>, the density-dependent coefficient of the second (confining) term goes to zero; we have an ''asymptotically free regime'' in which a Coulomb-like potential dominates throughout the universe.
			</li>
			<li>
			As the universe expands, the density will steadily drop. For <math>\rho/\rho_H \ll 1</math>, the density-dependent coefficient of the confining term approaches unity and we retrieve our originally proposed, modified cosmology; that is, the potential is dominated by a logarithmic term for all distances greater than <math>~a_T</math>.
			</li>
			</ul>

	=Potentially Useful References=		=Potentially Useful References=

Tohline: /* Cosmologies */

2021-07-21T17:20:47Z

Cosmologies

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	</table>		</table>
	</div>		</div>

			==Insert Dependence on (Energy) Density==

			The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential. The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime. Generally speaking, a QGP is achieved in a very high energy-density environment.

			We can mimic this behavior in our modified cosmology by assuming that the coefficient on the potential's logarithmic term varies with the energy-density of the fluid. (More simply, let's have it vary with the ''mass''-dentiy.) We want to "kill off" the logarithmic term when the density climbs above some threshold, <math>\rho_H</math>. Let's try …
			<table border="0" align="center" cellpadding="5">
			<tr>
			<td align="right"><math>\Phi</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			- \frac{GM_r}{r}
			</td>
			</tr>
			</table>

	=Potentially Useful References=		=Potentially Useful References=

Tohline: /* Tidbits */

2021-07-20T15:37:09Z

Tidbits

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	Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.		Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
			</li>
			</ul>

			From the arXiv preprint of a review article by [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force A. Deur, S. J. Brodsky, & G. F. de Téramond (2020)] titled, ''The QCD Running Coupling'':
			<ul>
			<li>
			(middle of p. 10) "We illustrate this behavior" — that is, "… the scale dependence of the coupling" — "for the coupling that arises in the static case of heavy sources and which provides a simple physical picture. Historically, and in the case of linear theories with massless force carriers, a force coupling constant is a universal coefficient that links the force to the 'charges' of two bodies (''e.g.,'' the electric charge for electricity or the mass for gravity) divided by the distance dependence <math>1/r^2</math>."
			</li>
			<li>
			(middle of p. 10, continued) "In QFT (quantum field theory) … for weak enough forces, the first Born approximation dominates higher order contributions and the <math>1/q^2</math> propagator in momentum yields the familiar <math>1/r^2</math> factor in coordinate space. However, higher orders do contribute and deviations from the <math>1/r^2</math> law thus occur. This extra r-dependence is folded in the coupling which then acquires a scale dependence."
	</li>		</li>
	</ul>		</ul>

Tohline: /* Wikipedia as a Resource */

2021-07-20T15:18:30Z

Wikipedia as a Resource

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	[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:		[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:
	<ul>		<ul>
			<li>
			"QGP (quark-gluon plasma) is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities."
			</li>
	<li>		<li>
	"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."		"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."

Tohline: /* Wikipedia as a Resource */

2021-07-19T15:14:13Z

Wikipedia as a Resource

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	"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."		"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
	</li>		</li>
			</ul>

			[https://en.wikipedia.org/wiki/Strong_interaction Strong interaction]:
			<ul>
			<li>
			"Unlike all other forces … the strong force does not diminish in strength with increasing distance between pairs of quarks. After a limiting distance (about the size of a hadron) has been reached, it remains at a strength of about 10,000 newtons, no matter how much farther the distance between the quarks." Hence, the effective potential has a term that is linear in r.
			</li>
	</ul>		</ul>

Tohline: /* Wikipedia as a Resource */

2021-07-19T15:03:18Z

Wikipedia as a Resource

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	<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>		<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
	</ul>		</ul>
			</li>
			</ul>

			[https://en.wikipedia.org/wiki/Color_confinement Color confinement]:
			<ul>
			<li>
			"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
	</li>		</li>
	</ul>		</ul>

Tohline: /* Wikipedia as a Resource */

2021-07-19T14:47:31Z

Wikipedia as a Resource

← Older revision		Revision as of 14:47, 19 July 2021
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	==Wikipedia as a Resource==		==Wikipedia as a Resource==
	[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]		[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:
			<ul>
			<li>
			"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
			</li>
			<li>
			"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
			<ul>
			<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
			<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
			</ul>
			</li>
			</ul>

	==Tidbits==		==Tidbits==

Tohline: /* Radial Dependance of the Strong Nuclear Force */

2021-07-19T14:29:30Z

Radial Dependance of the Strong Nuclear Force

← Older revision		Revision as of 14:29, 19 July 2021
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	{{LSU_HBook_header}}		{{LSU_HBook_header}}

			==Wikipedia as a Resource==
			[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]

	==Tidbits==		==Tidbits==

Tohline: /* ASTR4422 Class Notes */

2018-03-07T00:30:01Z

ASTR4422 Class Notes

← Older revision		Revision as of 00:30, 7 March 2018
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{2G}{~~r_0~~} \biggl[ \frac{4}{3} \pi \~~rho_0 r_0~~^3 \biggr] \, .</math>		<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{8}{3}\pi G \rho - \frac{2G}{~~r_0~~} \biggl[ \frac{4}{3} \pi \~~rho_0 r_0~~^3 \biggr] \frac{1}{R^2}</math>		<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{8}{3}\pi G \~~rho_0~~ \biggl[ \frac{\rho}{\~~rho_0~~} - \frac{~~r_0~~^2}{R^2} \biggr] </math>		<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{8}{3}\pi G \~~rho_0~~ \biggl[ \biggl(\frac{~~r_0~~}{R}\biggr)^3 - \biggl(\frac{~~r_0~~}{R}\biggr)^2 \biggr] </math>		<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{8}{3}\pi G \~~rho_0~~ \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>		<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{8}{3}\pi G \~~rho_0~~ \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>		<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{\~~rho_0~~}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>		<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	Hence, if in the present epoch, <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,		Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{8}{3}\pi G \~~rho_0~~ \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>		<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{8}{3}\pi G \~~rho_0~~ \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \~~rho_0~~ </math>		<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\~~rho_0~~}\biggr] \frac{32}{3}\pi G \~~rho_0~~ = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>		<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

	~~Quite~~ generally,		As our [[User:Tohline/SSC/FreeFall#Parametric\|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~tH_0~~</math>		<math>~t</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{~~t}{~~\tau_\mathrm{ff}} \~~cdot H_0~~</math>		<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~t \biggl[ \frac{~~32 G~~ \~~rho_0~~}{3\pi} \biggr]^{1 / 2} \~~cdot H_0~~</math>		<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
	</td>		</td>
	</tr>		</tr>