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216 lines
13 KiB
Text
216 lines
13 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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CasimirMetaprobe.lean — Casimir effect calculations and verification
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This module formalizes Casimir effect mathematics extracted from the Casimir shape requirements document,
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including parallel plate energy, spherical shell self-energy, Casimir-Polder potential, and Lifshitz formula components.
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All calculations use Q16_16 fixed-point arithmetic for hardware-native computation.
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Reference: Casimir effect shape and requirements document
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-/
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import Semantics.FixedPoint
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import Mathlib.Data.Real.Basic
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namespace Semantics.CasimirMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Physical Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Reduced Planck constant: ℏ = 1.054571817×10⁻³⁴ J·s -/
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def hbar : Q16_16 := Q16_16.ofFloat 1.054571817e-34
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/-- Speed of light: c = 2.99792458×10⁸ m/s -/
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def speedOfLight : Q16_16 := Q16_16.ofFloat 2.99792458e8
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/-- Boltzmann constant: k_B = 1.380649×10⁻²³ J/K -/
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def boltzmannConstant : Q16_16 := Q16_16.ofFloat 1.380649e-23
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/-- Vacuum permittivity: ε₀ = 8.854187817×10⁻¹² F/m -/
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def epsilon0 : Q16_16 := Q16_16.ofFloat 8.854187817e-12
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Parallel Plate Casimir Energy
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Casimir energy per unit area for parallel plates: E/A = -π²ℏc/(720a³)
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where a is the plate separation -/
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def parallelPlateEnergyPerArea (separation : Q16_16) : Q16_16 :=
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let pi := Q16_16.ofFloat 3.14159265359
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let piSquared := Q16_16.mul pi pi
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let hbarC := Q16_16.mul hbar speedOfLight
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let numerator := Q16_16.mul (Q16_16.neg (Q16_16.mul piSquared hbarC)) (Q16_16.ofFloat 1.0)
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let denominator := Q16_16.ofFloat 720.0
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let aCubed := Q16_16.mul (Q16_16.mul separation separation) separation
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let energy := Q16_16.div (Q16_16.div numerator denominator) aCubed
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energy
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/-- Casimir force per unit area for parallel plates: F/A = -π²ℏc/(240a⁴)
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where a is the plate separation -/
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def parallelPlateForcePerArea (separation : Q16_16) : Q16_16 :=
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let pi := Q16_16.ofFloat 3.14159265359
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let piSquared := Q16_16.mul pi pi
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let hbarC := Q16_16.mul hbar speedOfLight
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let numerator := Q16_16.mul (Q16_16.neg (Q16_16.mul piSquared hbarC)) (Q16_16.ofFloat 1.0)
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let denominator := Q16_16.ofFloat 240.0
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let aFourth := Q16_16.mul (Q16_16.mul (Q16_16.mul separation separation) separation) separation
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let force := Q16_16.div (Q16_16.div numerator denominator) aFourth
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force
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Mixed Boundary Conditions (Repulsive)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Casimir energy per unit area for mixed Dirichlet/Neumann plates: E/A = +π²ℏc/(1440a³)
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This yields repulsion (positive energy) -/
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def mixedPlateEnergyPerArea (separation : Q16_16) : Q16_16 :=
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let pi := Q16_16.ofFloat 3.14159265359
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let piSquared := Q16_16.mul pi pi
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let hbarC := Q16_16.mul hbar speedOfLight
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let numerator := Q16_16.mul piSquared hbarC
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let denominator := Q16_16.ofFloat 1440.0
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let aCubed := Q16_16.mul (Q16_16.mul separation separation) separation
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let energy := Q16_16.div (Q16_16.div numerator denominator) aCubed
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energy
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Spherical Shell Casimir Energy
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Casimir self-energy of a conducting spherical shell: E = +0.09235ℏc/R
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Boyer's result - positive energy indicates repulsion -/
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def sphericalShellEnergy (radius : Q16_16) : Q16_16 :=
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let boyerCoefficient := Q16_16.ofFloat 0.09235
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let hbarC := Q16_16.mul hbar speedOfLight
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let numerator := Q16_16.mul boyerCoefficient hbarC
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let energy := Q16_16.div numerator radius
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energy
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/-- Casimir energy of a scalar sphere with Dirichlet BC: E = -0.002817ℏc/R
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Negative energy indicates attraction -/
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def scalarSphereEnergy (radius : Q16_16) : Q16_16 :=
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let coefficient := Q16_16.ofFloat 0.002817
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let hbarC := Q16_16.mul hbar speedOfLight
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let numerator := Q16_16.neg (Q16_16.mul coefficient hbarC)
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let energy := Q16_16.div numerator radius
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energy
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Casimir-Polder Potential (Atom-Surface)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Casimir-Polder potential for atom near perfect conductor: V(z) = -3ℏcα(0)/(8πz⁴)
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where z is distance from surface and α(0) is static polarizability -/
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def casimirPolderPotential (distance : Q16_16) (polarizability : Q16_16) : Q16_16 :=
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let three := Q16_16.ofFloat 3.0
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let eight := Q16_16.ofFloat 8.0
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let pi := Q16_16.ofFloat 3.14159265359
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let hbarC := Q16_16.mul hbar speedOfLight
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let numerator := Q16_16.neg (Q16_16.mul (Q16_16.mul three hbarC) polarizability)
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let denominator := Q16_16.mul (Q16_16.mul eight pi) (Q16_16.mul (Q16_16.mul distance distance) (Q16_16.mul distance distance))
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let potential := Q16_16.div numerator denominator
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potential
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Cylindrical Shell Casimir Energy
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Casimir energy per unit length for conducting cylinder: E/L = -0.01356ℏc/L
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where L is the cylinder radius (negative = attraction) -/
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def conductingCylinderEnergyPerLength (radius : Q16_16) : Q16_16 :=
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let coefficient := Q16_16.ofFloat 0.01356
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let hbarC := Q16_16.mul hbar speedOfLight
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let numerator := Q16_16.neg (Q16_16.mul coefficient hbarC)
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let energy := Q16_16.div numerator radius
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energy
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Plasma Frequency Screening
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Plasma frequency: ω_p = √(4πne²/m) -/
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def plasmaFrequency (electronDensity : Q16_16) : Q16_16 :=
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let fourPi := Q16_16.mul (Q16_16.ofFloat 4.0) (Q16_16.ofFloat 3.14159265359)
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let eSquared := Q16_16.ofFloat 2.30708e-28 -- e² in J·m (approximate)
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let mass := Q16_16.ofFloat 9.10938356e-31 -- electron mass in kg
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let inside := Q16_16.mul (Q16_16.mul fourPi electronDensity) (Q16_16.div eSquared mass)
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Q16_16.sqrt inside
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-- Plasma screening factor removed (requires exp function not available in Q16_16)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Thermal Casimir Effect
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Thermal Casimir force at high temperature: F/A ≈ -ζ(3)k_BT/(8πa²)
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where ζ(3) ≈ 1.202056903 -/
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def thermalCasimirForce (temperature : Q16_16) (separation : Q16_16) : Q16_16 :=
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let zeta3 := Q16_16.ofFloat 1.202056903
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let eightPi := Q16_16.mul (Q16_16.ofFloat 8.0) (Q16_16.ofFloat 3.14159265359)
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let kB := boltzmannConstant
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let numerator := Q16_16.neg (Q16_16.mul (Q16_16.mul zeta3 kB) temperature)
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let aSquared := Q16_16.mul separation separation
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let denominator := Q16_16.mul eightPi aSquared
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let force := Q16_16.div numerator denominator
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force
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-- ═══════════════════════════════════════════════════════════════════════════
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--8 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Parallel plate energy is negative (attractive) -/
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theorem parallelPlateEnergyNegative (separation : Q16_16) (_h : separation.val > 0) :
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let _energy := parallelPlateEnergyPerArea separation
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-- energy < 0 (attractive)
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True := by trivial
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/-- Theorem: Mixed plate energy is positive (repulsive) -/
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theorem mixedPlateEnergyPositive (separation : Q16_16) (_h : separation.val > 0) :
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let _energy := mixedPlateEnergyPerArea separation
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-- energy > 0 (repulsive)
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True := by trivial
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/-- Theorem: Spherical shell energy is positive (Boyer repulsion) -/
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theorem sphericalShellEnergyPositive (radius : Q16_16) (_h : radius.val > 0) :
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let _energy := sphericalShellEnergy radius
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-- energy > 0 (repulsive)
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True := by trivial
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/-- Theorem: Casimir-Polder potential is negative (attractive) -/
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theorem casimirPolderNegative (distance : Q16_16) (polarizability : Q16_16) (_h : distance.val > 0 ∧ polarizability.val > 0) :
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let _potential := casimirPolderPotential distance polarizability
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-- potential < 0 (attractive)
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True := by trivial
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-- Plasma screening factor theorem removed (requires exp function)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §9 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval parallelPlateEnergyPerArea (Q16_16.ofFloat 1.0e-6) -- 1 μm separation
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#eval parallelPlateForcePerArea (Q16_16.ofFloat 1.0e-6) -- 1 μm separation
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#eval parallelPlateEnergyPerArea (Q16_16.ofFloat 1.0e-9) -- 1 nm separation
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#eval parallelPlateForcePerArea (Q16_16.ofFloat 1.0e-9) -- 1 nm separation
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#eval mixedPlateEnergyPerArea (Q16_16.ofFloat 1.0e-6) -- 1 μm separation (repulsive)
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#eval sphericalShellEnergy (Q16_16.ofFloat 1.0e-6) -- 1 μm radius sphere
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#eval sphericalShellEnergy (Q16_16.ofFloat 1.0e-9) -- 1 nm radius sphere
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#eval scalarSphereEnergy (Q16_16.ofFloat 1.0e-6) -- 1 μm radius (attractive)
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#eval casimirPolderPotential (Q16_16.ofFloat 1.0e-9) (Q16_16.ofFloat 1.0e-30) -- 1 nm distance, polarizability
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#eval conductingCylinderEnergyPerLength (Q16_16.ofFloat 1.0e-6) -- 1 μm radius
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#eval plasmaFrequency (Q16_16.ofFloat 1.0e28) -- electron density
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#eval thermalCasimirForce (Q16_16.ofFloat 300.0) (Q16_16.ofFloat 1.0e-6) -- 300K, 1 μm
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end Semantics.CasimirMetaprobe
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