Stefans Law and the greek letters (#201)

Author: kloimhardt

src/mentat_collective/emmy/fdg_ch01_ys.clj

@@ -7,7 +7,7 @@
                               :date     "2025-11-19"
                               :image    "fdg_ch01_ys.jpg"
                               :category :libs
-                              :tags     [:emmy :infix :yamlscript]}}}
+                              :tags     [:emmy :infix :yamlscript :notation]}}}
 
 (ns mentat-collective.emmy.fdg-ch01-ys
     (:refer-clojure :exclude [+ - * / zero? compare divide numerator denominator
@@ -78,30 +78,28 @@
 
 (ys "
 defn LFree(mass):
-  fn([_ _ v]): mass * 1/2 * square(v)
+  fn([_ _ v]): 1/2 * mass * square(v)
 ")
 
 ;; ## Another one
 
-;; I proceed to the next infix snippet
+(kind/scittle
+  '(defn sphere->R3 [R]
+     (fn [[_ [theta phi]]]
+       (up (* R (sin theta) (cos phi))
+           (* R (sin theta) (sin phi))
+           (* R (cos theta))))))
 
 (ys "
 defn sphere-to-R3(R):
   fn([_ [theta phi]]):
     up:
-     =>: R * sin(theta) * cos(phi)
-     =>: R * sin(theta) * sin(phi)
-     =>: R * cos(theta)
+     R *: sin(theta) * cos(phi)
+     R *: sin(theta) * sin(phi)
+     R *: cos(theta)
 ")
 
-;; which is the following in Clojure
-
-(kind/scittle
-  '(defn sphere->R3 [R]
-     (fn [[_ [theta phi]]]
-       (up (* R (sin theta) (cos phi))
-           (* R (sin theta) (sin phi))
-           (* R (cos theta))))))
+;; The `*:` means nothing but normal multiplication.
 
 ;; ## Higher order functions
 
@@ -120,10 +118,12 @@ defn F-to-C(F):
     up:
      time: state
      F: state
-     =>: D(0).of(F).at(state) + ( D(1).of(F).at(state) * velocity(state) )
+     D(0).of(F).at(state) +:
+       D(1).of(F).at(state) *
+       velocity(state)
 ")
 
-;; With `of` and `at` like that, the above `D(0).of(F).at(state)` (which means "take the zeroth derivative of the function F at point state") translates into what are higher order functions in the Clojure version.
+;; Again, the above `+:` is normal addition. With `of` and `at` like that, the above `D(0).of(F).at(state)` (which means "take the zeroth derivative of the function F at point state") translates into what are higher order functions in the Clojure version.
 
 (kind/scittle
   '(defn F->C [F]
@@ -145,13 +145,15 @@ defn Lsphere(m R):
 
 (kind/scittle
   '(defn Lsphere [m R]
-     (compose (Lfree m) (F->C (sphere->R3 R)))))
+     (compose (Lfree m)
+              (F->C (sphere->R3 R)))))
 
 ;; ## The proof is in the pudding
 
-;; In order to quote symbols, I introduce a macro called `q`
+;; Emmy is a symbolic algebra system, `m:q` produces a symbol named `m`.
 
-(defmacro q [f] (list 'quote f))
+^:kindly/hide-code
+(defmacro q [s] (list 'quote s))
 
 (ys "
 simplify:
@@ -166,7 +168,9 @@ simplify:
   (vector :tt
     '(simplify
        ((Lsphere 'm 'R)
-        (up 't (up 'theta 'phi) (up 'thetadot 'phidot))))))
+        (up 't
+            (up 'theta 'phi)
+            (up 'thetadot 'phidot))))))
 
 ;; Indeed the results are the same which proves the infix technically works.
 
@@ -192,12 +196,18 @@ yamlscript/core {:git/url \"https://github.com/yaml/yamlscript\"
 ;; ## 1
 (ys "
 defn L2(mass metric):
-  fn(place velocity): mass * 1/2 * metric(velocity velocity).at(place)
+  fn(place velocity):
+    1/2 *:
+      mass *
+      metric(velocity velocity).at(place)
 ")
+
 (kind/scittle
   '(defn L2 [mass metric]
      (fn [place velocity]
-       (* 1/2 mass ((metric velocity velocity) place)))))
+       (* 1/2
+          mass
+          ((metric velocity velocity) place)))))
 
 
 ;; ## 2
@@ -209,9 +219,11 @@ defn L2(mass metric):
 defn Lc(mass metric coordsys):
   e =: coordinate-system-to-vector-basis(coordsys)
   fn([_ x v]):
-    L2(mass metric): point(coordsys).at(x) (e * v)
+    L2(mass metric): point(coordsys).at(x), (e * v)
 ")
 
+;; The `,` above is a matter of taste and can also be omitted
+
 (kind/scittle
   '(defn Lc [mass metric coordsys]
      (let [e (coordinate-system->vector-basis coordsys)]
@@ -251,4 +263,6 @@ simplify:
 (kind/reagent
   (vector :tt
     '(simplify
-       (L (up 't (up 'x 'y) (up 'vx 'vy))))))
+       (L (up 't
+              (up 'x 'y)
+              (up 'vx 'vy))))))

src/mentat_collective/emmy/josefstefan.clj

@@ -0,0 +1,154 @@
+^{:kindly/hide-code true
+  :clay             {:title  "How to pronounce greek letters"
+                     :quarto {:author      :kloimhardt
+                              :type        :post
+                              :description "Josef Stefan's fourth-power law, written in both greek and english."
+                              :date        "2025-11-27"
+                              :image       "josefstefan.png"
+                              :category    :libs
+                              :tags        [:emmy :physics :notation]}}}
+(ns mentat-collective.emmy.josefstefan
+  (:refer-clojure :exclude [+ - * / = abs compare zero? ref partial
+                            times numerator denominator infinite?])
+  (:require [emmy.env :as e :refer :all :exclude [F->C times]]
+            [mentat-collective.emmy.spell :refer :all]
+            [scicloj.kindly.v4.api :as kindly]
+            [scicloj.kindly.v4.kind :as kind]))
+
+;; The greek letters $\xi$ ("xi"), $\eta$ ("eta") and $\zeta$ ("zeta") are often used in textbooks as counterparts to the latin x,y and z. This needs practice. All the more so that there also are the two greeks $\chi$ ("chi") and $\omega$ ("omega").
+
+;; I'd like to make a table for this:
+
+^:kindly/hide-code
+(kind/table
+  [["xi" (kind/md "$\\xi$")]
+   ["eta" (kind/md "$\\eta$")]
+   ["zeta" (kind/md "$\\zeta$")]
+   ["chi" (kind/md "$\\chi$")]
+   ["omega" (kind/md "$\\omega$")]]
+  )
+
+;; For practice, I start with a widely forgotten formula by Messieurs Dulong and Petit
+
+^:kindly/hide-code
+(def tex (comp kind/tex emmy.expression.render/->TeX))
+
+^:kindly/hide-code
+(defn dp-formel2 [Celsius constants]
+  (calcbox [((((eta power xi) minus 1) times zeta)
+             with
+             [xi in Celsius])
+            [and [[eta zeta] being constants]]]))
+
+^:kindly/hide-code
+(def greek-alphabet {:eta 'eta :zeta 'zeta :chi 'chi :omega 'omega})
+
+^:kindly/hide-code
+(tex (:calc (dp-formel2 'xi greek-alphabet)))
+
+;; How on earth do you pronounce that? Here we go
+
+^:kindly/hide-code
+(kind/hiccup [:blockquote (:hiccup (dp-formel2 0 greek-alphabet))])
+
+;; Included is the hint that the xi-$\xi$ is a temperature measured in degree Celsius. But what are the constants eta-$\eta$ and zeta-$\zeta$?
+
+;; ## The formula of Dulong&Petit
+
+;; I define a function which includes the sought-for numbers
+
+(defn dp-formula [Celsius]
+  (calcbox [((((eta power xi) minus 1) times zeta)
+             with
+             [xi in Celsius])
+            [and [(1 comma 0 0 77) for eta]]
+            [and [(2 comma 0 2) for zeta]]]))
+
+;; This reads as
+
+(kind/hiccup [:blockquote (:hiccup (dp-formula 0))])
+
+;; The formula prints like this
+
+(tex (:calc (dp-formula 'xi)))
+
+;; I can now also calculate a number, e.g. for 100 °C
+
+(:calc (dp-formula 100))
+
+;; But what does this number mean? It is degrees per minute. In 1817, Dulong and Petit measured the rate of change with time of the indicated temperature on a previously heated mercury-in-glass thermometer with a spherical bulb placed centrally in a spherical enclosure held at zero degrees Celsius.
+
+;; These were their measured values
+
+^:kindly/hide-code
+(def temper [80 100 120 140 160 180 200 220 240])
+
+^:kindly/hide-code
+(def dp-meas [1.74 2.30 3.02 3.88 4.89 6.10 7.40 8.81 10.69])
+
+^:kindly/hide-code
+(kind/table
+  {"°C"       temper
+   "°C / min" dp-meas})
+
+;; So, when the thermometer was at 100°C, within the first minute it lost 2.3 degrees due to radiation. In that case, their formula was pretty accurate. But a man named Pouillet used the formula to estimate the temperature of the sun, got a value of some 1700 degrees, and that seemed pretty low to a certain Josef Stefan. He re-published the above data and proposed his famous fourth-power law on the relationship between heat-radiation and temperature on pages 391-428 of the "Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Classe, Neunundsiebzigster Band, Wien, 1879".
+
+;; ## The law of Stefan
+
+(defn stefan-law [Celsius constants]
+  (calcbox [((((chi plus xi) power 4 )
+              minus
+              (chi power 4))
+             times
+             omega)
+            [with [xi in Celsius]]
+            [and [[chi omega] being constants]]]))
+
+^:kindly/hide-code
+(tex (:calc (stefan-law 'xi greek-alphabet)))
+
+^:kindly/hide-code
+(kind/hiccup [:blockquote (:hiccup (stefan-law 0 greek-alphabet))])
+
+;; We need to set omega-$\omega$ to one six billionth. The other constant is given by the absolute zero temperature, chi-$\chi$ = 273.
+
+;; As an exposition, we calculate the fourth-power of 273.
+;; The result is a pretty big number.
+
+^:kindly/hide-code
+(def pow_273_4 (calcbox [(((chi times chi) times chi) times chi) [with [chi equals 273]]]))
+
+^:kindly/hide-code
+(kind/hiccup [:blockquote (:hiccup pow_273_4)])
+
+^:kindly/hide-code
+(:calc pow_273_4)
+
+;; I can imagine that in the 19th century, without having computers, to fit some data it took considerable guts to take on a fourth power law.
+
+(defn stefan-law-numbers [Celsius]
+  (calcbox [((((chi plus xi) power 4 )
+              minus
+              (chi power 4))
+             times
+             omega)
+            [with [xi in Celsius]]
+            [and [(one (6 billion) th) for omega]]
+            [and [273 for chi]]]))
+
+^:kindly/hide-code
+(kind/hiccup [:blockquote (:hiccup (stefan-law-numbers 0))])
+
+^:kindly/hide-code
+(tex (:calc (stefan-law-numbers 'xi)))
+
+;; Stefan's Law passes the first test in fitting the data as well as the old model.
+
+^:kindly/hide-code
+(kind/table
+  {"°C"          temper
+   "°C / min"    dp-meas
+   "D&P-formula" (map #(round (:calc (dp-formula %)) 2) temper)
+   "Stefan law"  (map #(round (:calc (stefan-law-numbers %)) 2) temper)})
+
+;; With his new formula, Josef Stefan estimated the lower bound of the temperature of the sun to be around 5600 °C which means he was pretty much bang-on within some 100 degrees.