Math of convergent assembly - Revision history

Apm: Category:Far term target

2026-03-29T18:21:19Z

← Older revision		Revision as of 20:21, 29 March 2026
Line 188:		Line 188:

	[[Category:Pages with math]]		[[Category:Pages with math]]
			[[Category:Far term target]]

Apm: /* In terms of scaling of volumetric throughput density */ added link to Continuity of throughput

2022-09-05T07:37:36Z

In terms of scaling of volumetric throughput density: added link to Continuity of throughput

← Older revision		Revision as of 09:37, 5 September 2022
Line 36:		Line 36:
	<math> V(1) = (s/B^1)^3 (B^2)^1 = s^3 / B^1 </math> <br>		<math> V(1) = (s/B^1)^3 (B^2)^1 = s^3 / B^1 </math> <br>
	<math> V_C(1) = (s F/B^1)^3 (B^2)^1 = (s F)^3 / B^1 </math> <br>		<math> V_C(1) = (s F/B^1)^3 (B^2)^1 = (s F)^3 / B^1 </math> <br>
	'''Note: The absolute throughput matches up Q(1) = Q(0) = Q. Just as it needs to.'''<br>		'''Note: The absolute throughput matches up Q(1) = Q(0) = Q. Just as it needs to.''' See: [[Continuity of throughput]]<br>
	<math> Q(1) = V(1)/t(1) = (s^2 v) / (B^3 C F) = Q(0) = Q </math> <br>		<math> Q(1) = V(1)/t(1) = (s^2 v) / (B^3 C F) = Q(0) = Q </math> <br>
	<math> D(1) = Q(1)/V_C(1) = (v B^1) / (s B^3 C F^4) </math> <br>		<math> D(1) = Q(1)/V_C(1) = (v B^1) / (s B^3 C F^4) </math> <br>

Apm at 07:26, 5 September 2022

2022-09-05T07:26:31Z

← Older revision		Revision as of 09:26, 5 September 2022
Line 57:		Line 57:
	<math> D(L) = v / (L B^3 C F^3) </math> <br>		<math> D(L) = v / (L B^3 C F^3) </math> <br>
	Units are correct: 1/s = (m³/s)/m³ <br>		Units are correct: 1/s = (m³/s)/m³ <br>

			We see that: <br>
	<math> D \propto L^{-1} </math> <br>		<math> D \propto L^{-1} </math> <br>
	'''Halving the size of nanomachinery doubles the volumetric throughput density.''' <br>		'''Halving the size of nanomachinery doubles the volumetric throughput density.''' <br>
	This is the scaling law we wanted to derive and ~~the whole~~ point of the exercise. <br>		This is the scaling law we wanted to derive and a main point of the exercise <br>
			(beside showing continuity of throughput).<br>
	See also: [[Higher throughput of smaller machinery]]		See also: [[Higher throughput of smaller machinery]]

Apm: added F reducing its final power in the denominator from 4 to 3

2022-09-05T07:22:43Z

added F reducing its final power in the denominator from 4 to 3

← Older revision		Revision as of 09:22, 5 September 2022
Line 52:		Line 52:
	But i is discrete. We rather want a continuous scaling law for a given continuous sizescale L. <br>		But i is discrete. We rather want a continuous scaling law for a given continuous sizescale L. <br>
	We can do that by calculating L from i like so: <br>		We can do that by calculating L from i like so: <br>
	<math> L = s/B^i</math> <br>		<math> L = sF/B^i</math> <br>
	Now to get a continuous scaling law for size scales L we can substitute B^i for s/L giving: <br>		Now to get a continuous scaling law for size scales L we can substitute B^i for sF/L giving: <br>
	'''And finally we arrive at:''' <br>		'''And finally we arrive at:''' <br>
	<math> D(L) = v / (L B^3 C F^4) </math> <br>		<math> D(L) = v / (L B^3 C F^3) </math> <br>
	Units are correct: 1/s = (m³/s)/m³ <br>		Units are correct: 1/s = (m³/s)/m³ <br>
	<math> D \propto L^{-1} </math> <br>		<math> D \propto L^{-1} </math> <br>

Apm: carries => carry

2022-09-05T07:19:37Z

carries => carry

← Older revision		Revision as of 09:19, 5 September 2022
Line 49:		Line 49:
	Be careful! While s is a size it is a constant. <br>		Be careful! While s is a size it is a constant. <br>
	The size s does not carry the information about the scale that we want to consider. <br>		The size s does not carry the information about the scale that we want to consider. <br>
	What does ~~carries~~ the information about the scale we want to consider is the index i. <br>		What does carriy the information about the scale we want to consider is the index i. <br>
	But i is discrete. We rather want a continuous scaling law for a given continuous sizescale L. <br>		But i is discrete. We rather want a continuous scaling law for a given continuous sizescale L. <br>
	We can do that by calculating L from i like so: <br>		We can do that by calculating L from i like so: <br>

Apm: improved on the explanation on the transition from discrete to continuous

2022-09-05T07:19:00Z

improved on the explanation on the transition from discrete to continuous

← Older revision		Revision as of 09:19, 5 September 2022
Line 47:		Line 47:
	<math> D(i) = Q(i)/V_C(i) = (v B^i) / (s B^3 C F^4) </math> <br>		<math> D(i) = Q(i)/V_C(i) = (v B^i) / (s B^3 C F^4) </math> <br>
	----		----
	~~To get~~ a continuous scaling law for ~~size scales~~ L we can ~~substitute for~~ i <br>		Be careful! While s is a size it is a constant. <br>
	<math> i = ~~- \log_B(L~~/s) </math> <br>		The size s does not carry the information about the scale that we want to consider. <br>
			What does carries the information about the scale we want to consider is the index i. <br>
			But i is discrete. We rather want a continuous scaling law for a given continuous sizescale L. <br>
			We can do that by calculating L from i like so: <br>
			<math> L = s/B^i</math> <br>
			Now to get a continuous scaling law for size scales L we can substitute B^i for s/L giving: <br>
	'''And finally we arrive at:''' <br>		'''And finally we arrive at:''' <br>
	<math> D(L) = v / (L B^3 C F^4) </math> <br>		<math> D(L) = v / (L B^3 C F^4) </math> <br>

Apm: /* Related */ added link to yet unwritten page: Mesoscale friction

2022-08-30T14:53:44Z

Related: added link to yet unwritten page: Mesoscale friction

← Older revision		Revision as of 16:53, 30 August 2022
Line 169:		Line 169:
	* [[Level throughput balancing]]		* [[Level throughput balancing]]
	* [[Continuity of throughput]]		* [[Continuity of throughput]]
			* [[Mesoscale friction]]

	= External links =		= External links =

Apm: /* Related / added Increasing bearing area to decrease friction & sub-layer

2022-08-30T14:52:49Z

← Older revision		Revision as of 16:52, 30 August 2022
Line 165:		Line 165:
	* '''[[Higher throughput of smaller machinery]]'''		* '''[[Higher throughput of smaller machinery]]'''
	* [[Deliberate slowdown at the lower assembly levels]]		* [[Deliberate slowdown at the lower assembly levels]]
	* [[Optimal sublayernumber for minimal friction]]		* [[Increasing bearing area to decrease friction]]
			* [[Optimal sublayernumber for minimal friction]] & [[sub-layer]]
	* [[Level throughput balancing]]		* [[Level throughput balancing]]
	* [[Continuity of throughput]]		* [[Continuity of throughput]]

Apm: /* Related */ added links to related pages

2022-08-30T14:50:49Z

Related: added links to related pages

← Older revision		Revision as of 16:50, 30 August 2022
Line 164:		Line 164:

	* '''[[Higher throughput of smaller machinery]]'''		* '''[[Higher throughput of smaller machinery]]'''
			* [[Deliberate slowdown at the lower assembly levels]]
			* [[Optimal sublayernumber for minimal friction]]
			* [[Level throughput balancing]]
			* [[Continuity of throughput]]

	= External links =		= External links =

Apm: /* In terms of replication time */ poer => power

2022-08-30T14:43:07Z

In terms of replication time: poer => power

← Older revision		Revision as of 16:43, 30 August 2022
Line 76:		Line 76:
	'''Artificial systems:''' "... a one third power scaling law of replication time to mass – i.e. replication time is proportional to the cube root of the mass." <br>		'''Artificial systems:''' "... a one third power scaling law of replication time to mass – i.e. replication time is proportional to the cube root of the mass." <br>
	<math> t_{repl} \propto \sqrt[3]{m_{device}} \propto L^1 </math> <br>		<math> t_{repl} \propto \sqrt[3]{m_{device}} \propto L^1 </math> <br>
	'''Natural systems:''' " ... like animals and plants, have empirically a well-documented one-''fourth'' ~~poer~~ scaling law" <br>		'''Natural systems:''' " ... like animals and plants, have empirically a well-documented one-''fourth'' power scaling law" <br>
	<math> t_{repl} \propto \sqrt[4]{m_{device}} </math> <br>		<math> t_{repl} \propto \sqrt[4]{m_{device}} </math> <br>