boltzmann.lisp

;;; The Bolzmann formula for entropy as a measure of molecular disorder.
;;;
;;; Consider a container partitioned into two equal volumes. The number of ways,
;;; P, in which N molecules can be divided into two groups, N1 and N2, is given by
;;; the simple combinatorial formula
;;;
;;;                     P = N! / N1! N2!
;;;
;;; in which N! = N(N-1)(N-2)...3*2*1. The quantity P is called the "number of
;;; complections" (see Landau and Lifschitz, 1968) [p. 10]
;;;
;;; Here, the difference between N1 and N2 is described by the "ratio" parameter,
;;; which is expected to be a decimal.

(defun factorial (n &optional (acc 1))
  "Naive factorial implementation"
  (if (<= n 1)
      acc
      (factorial (- n 1) (* acc n))))

(defun boltzmann (n ratio)
  "Prigogine 1980, p.10"
  (/ (factorial n)
     (* (factorial (* n ratio))
	(factorial (* n (- 1 ratio))))))

(defun boltzmann-entropy (w)
  "Derived version of Boltzmann's formula. k is Boltzmann's constant, or
  1.38062*10^-23"
  )

(defun random-walk-1 (m n)
  "'An idealized but nevertheless successful model for Brownian motion.'
       - Prigogine 1980, p.11"
  (* .5
     (/ (factorial n)
	(* (factorial (* .5 (+ n m)))
	   (factorial (* .5 (- n m)))))))

(defun random-walk-2 (m n)
  "Shouldn't random-walk-1 and random-walk-2 be equal? Because they aren't...
  I think I need to revist this."
  (* (expt .5 (/ 2 (* pi n)))
     (exp (* (/ (- 1 (expt m 2))
		2)
	     n))))