Numerical Ensemble of Harmonic Oscillators
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From: Han de Bruijn on 22 Jun 2010 05:22 Hypothesis: ---------- | ALL LAWS OF NATURE ARE ONLY APPROXIMATELY TRUE Just take it or leave it. I'm not going to defend this as a new kind of dogma. Instead, we are going to explore some consequences of the proposition, in a very specific mathematical sense. And that's it. As an application of the above Hypothesis, consider the classical _ideal_ harmonic oscillator, which is described analytically by the following differential equation and boundary conditions: d^2x/dt^2 + oo^2.x = 0 ; x(0) = 0 ; dx/dt(0) = oo ; x = x(t) x = space coordinate, t = time, oo = (angular) frequency. The "exact", but I would rather say ANALYTICAL, solution of the above equation is well known. And we call it the _ideal_ solution: x(t) = sin(oo.t) But according to our Hypothesis, the _true_ equations are not "exact"; they are _approximations_ of the above ideal differential equation: x(t + dt) = x(t) + dt.x'(t) + dt^2/2.x''(t) + dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (1) x(t - dt) = x(t) - dt.x'(t) + dt^2/2.x''(t) - dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (2) - 2.x(t) = - 2.x(t) (3) ------------------------------------------------------------------ + x(t + dt) - 2.x(t) + x(t - dt) = dt^2.x''(t) + dt^4/12.x''''(t) .. So the TRUE equation of motion is, with dt = finite and uncertain: x(t + dt) - 2.x(t) + x(t - dt) ------------------------------ + oo^2.x(t) = dt^2 = h.o.t. + dt^2/12.d^4x(t)/dx^4 + d^2x(t)/dt^2 + oo^2.x(t) = 0 Where h.o.t. = higher order terms, to be neglected. The equations of motion are solved numerically as follows. Start with: x(0) = 0 ; (x(dt) - x(0))/dt = oo ==> x(dt) = x(0) + oo.dt x(t + dt) - 2.x(t) + x(t - dt) ------------------------------ + oo^2.x(t) = 0 ==> dt^2 x(t + dt) = 2.x(t) - x(t - dt) - (dt.oo)^2.x(t) ==> x(0.dt) = 0 x(1.dt) = x(0) + oo.dt x(2.dt) = 2.x(1.dt) - x(0.dt) - (dt.oo)^2 x(1.dt) x(3.dt) = 2.x(2.dt) - x(1.dt) - (dt.oo)^2 x(2.dt) x(4.dt) = 2.x(3.dt) - x(2.dt) - (dt.oo)^2 x(3.dt) ......... Etcetera And similar equations if the time intervals (dt) have unequal sizes. In our numerical experiments, we have chosen oo = 1 and dt = 0.1 . So far so good. But there is an analytical approach as well. Solve: dt^2/12.d^4x(t)/dx^4 + d^2x(t)/dt^2 + oo^2.x(t) = 0 Characteristic equation: dt^2/12.L^4 + L^2 + oo^2 = 0 Substitute L^2 = M ==> dt^2/12.M^2 + M + oo^2 = 0 A quadratic equation. The solution can be written as: M = oo^2 / [ - 1/2 +/- 1/2.sqrt(1 - {dt.oo}^2/3) ] ==> L = +/- i.oo / sqrt(1/2 + 1/2*sqrt(1 - sqr(dt*oo)/3)) , +/- i.oo / sqrt(1/2 - 1/2*sqrt(1 - sqr(dt*oo)/3)) The latter frequencies are of order the discretization (dt) and hence will be neglected. The former frequencies imply a small correction on the (no longer) "exact" frequency (oo). Thus the "true" solution is a sine with frequency oo' = oo / sqrt(1/2 + 1/2*sqrt(1 - sqr(dt*oo)/3)) Meaning that, due to the _discrete_ nature of the differentials (dt) , it oscillates somewhat faster: as x(t) = sin(oo'.t) with oo' > oo . After a time lapse t = N.2.pi/oo the analytical solution x(t) = 0 . After t = (N.2.pi + pi/2)/oo' the true numerical solution x(t) = 1 . At the same time it holds that N.2.pi/oo = (N.2.pi + pi/2)/oo' ==> N.oo' = (N + 1/4).oo' ==> N = (1/4)/(oo'/oo - 1) So we can calculate the first moment at which the two curves differ a magnitude = 1 . But different time steps cause different divergences from the _ideal_ x(t) = sin(oo.t) curve. Therefore, in fact, we have a whole statistical _ensemble_ of true x(t) = sin(oo'.t) curves and their deviations from the ideal curve become larger as time proceeds. Needless to say that our numerical experiments are a confirmation of the above theory. We find that the ideal and true curves differ by an amount 1 in x after 37636 time steps dt = 0.1 . But, in reality, (dt) is supposed to be a stochastic variable with some mean and a spread, meaning that there is a bundle of solutions instead of an exact one. Now think about the "grand" consequences. Does the above imply that: TIME IS IRREVERSIBLE ? Han de Bruijn
From: Uncle Al on 22 Jun 2010 11:14 Han de Bruijn wrote: > > Hypothesis: > ---------- > | ALL LAWS OF NATURE ARE ONLY APPROXIMATELY TRUE Heisenberg, Goedel; Bugs Bunny, Invader Zim; Halliburton. > Just take it or leave it. I'm not going to defend this as a new kind > of dogma. Instead, we are going to explore some consequences of the > proposition, in a very specific mathematical sense. And that's it. One presumes your mathematics is more self-consistent than your polemics. > As an application of the above Hypothesis, consider the classical > _ideal_ harmonic oscillator, which is described analytically by the > following differential equation and boundary conditions: > > d^2x/dt^2 + oo^2.x = 0 ; x(0) = 0 ; dx/dt(0) = oo ; x = x(t) > > x = space coordinate, t = time, oo = (angular) frequency. > > The "exact", but I would rather say ANALYTICAL, solution of the above > equation is well known. And we call it the _ideal_ solution: > > x(t) = sin(oo.t) > > But according to our Hypothesis, the _true_ equations are not "exact"; > they are _approximations_ of the above ideal differential equation: > > x(t + dt) = x(t) + dt.x'(t) + dt^2/2.x''(t) > + dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (1) > > x(t - dt) = x(t) - dt.x'(t) + dt^2/2.x''(t) > - dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (2) > > - 2.x(t) = - 2.x(t) (3) > ------------------------------------------------------------------ + > x(t + dt) - 2.x(t) + x(t - dt) = dt^2.x''(t) + dt^4/12.x''''(t) .. > > So the TRUE equation of motion is, with dt = finite and uncertain: > > x(t + dt) - 2.x(t) + x(t - dt) > ------------------------------ + oo^2.x(t) = > dt^2 > > = h.o.t. + dt^2/12.d^4x(t)/dx^4 + d^2x(t)/dt^2 + oo^2.x(t) = 0 > > Where h.o.t. = higher order terms, to be neglected. The Golden Gate Bridge is Newtonian not relativistic. Do asstronauts in ISS FUBAR correct their wriswatches vs. ground clocks? Of course not. Does GPS? Of course it does. > The equations of motion are solved numerically as follows. [snip] > So we can calculate the first moment at which the two curves differ a > magnitude = 1 . But different time steps cause different divergences > from the _ideal_ x(t) = sin(oo.t) curve. Therefore, in fact, we have > a whole statistical _ensemble_ of true x(t) = sin(oo'.t) curves and > their deviations from the ideal curve become larger as time proceeds. Casinos do not always win events. So? At the end of the year, casinos always win. > Needless to say that our numerical experiments are a confirmation of > the above theory. We find that the ideal and true curves differ by an > amount 1 in x after 37636 time steps dt = 0.1 . But, in reality, (dt) > is supposed to be a stochastic variable with some mean and a spread, > meaning that there is a bundle of solutions instead of an exact one. Grandfather clocks work, weather forecasts and economics do not. So? > Now think about the "grand" consequences. Does the above imply that: > > TIME IS IRREVERSIBLE ? Dissipation is a Second Law, Large Numbers Hypothesis, weak arrow of time. Keep It Simple, Stupid: Feynman's sprinkler and conservation of angular momentum. Feynman's sprinkler absolutely does not play backwards. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz4.htm
From: Han de Bruijn on 23 Jun 2010 03:04 On Jun 22, 5:14 pm, Uncle Al <Uncle...(a)hate.spam.net> wrote: > Han de Bruijn wrote: > > > Hypothesis: > > ---------- > > | ALL LAWS OF NATURE ARE ONLY APPROXIMATELY TRUE > > Heisenberg, Goedel; Bugs Bunny, Invader Zim; Halliburton. > > > Just take it or leave it. I'm not going to defend this as a new kind > > of dogma. Instead, we are going to explore some consequences of the > > proposition, in a very specific mathematical sense. And that's it. > > One presumes your mathematics is more self-consistent than your > polemics. > > > > > > > As an application of the above Hypothesis, consider the classical > > _ideal_ harmonic oscillator, which is described analytically by the > > following differential equation and boundary conditions: > > > d^2x/dt^2 + oo^2.x = 0 ; x(0) = 0 ; dx/dt(0) = oo ; x = x(t) > > > x = space coordinate, t = time, oo = (angular) frequency. > > > The "exact", but I would rather say ANALYTICAL, solution of the above > > equation is well known. And we call it the _ideal_ solution: > > > x(t) = sin(oo.t) > > > But according to our Hypothesis, the _true_ equations are not "exact"; > > they are _approximations_ of the above ideal differential equation: > > > x(t + dt) = x(t) + dt.x'(t) + dt^2/2.x''(t) > > + dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (1) > > > x(t - dt) = x(t) - dt.x'(t) + dt^2/2.x''(t) > > - dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (2) > > > - 2.x(t) = - 2.x(t) (3) > > ------------------------------------------------------------------ + > > x(t + dt) - 2.x(t) + x(t - dt) = dt^2.x''(t) + dt^4/12.x''''(t) ... > > > So the TRUE equation of motion is, with dt = finite and uncertain: > > > x(t + dt) - 2.x(t) + x(t - dt) > > ------------------------------ + oo^2.x(t) = > > dt^2 > > > = h.o.t. + dt^2/12.d^4x(t)/dx^4 + d^2x(t)/dt^2 + oo^2.x(t) = 0 > > > Where h.o.t. = higher order terms, to be neglected. > > The Golden Gate Bridge is Newtonian not relativistic. Do asstronauts > in ISS FUBAR correct their wriswatches vs. ground clocks? Of course > not. Does GPS? Of course it does. > > > The equations of motion are solved numerically as follows. > > [snip] > > > So we can calculate the first moment at which the two curves differ a > > magnitude = 1 . But different time steps cause different divergences > > from the _ideal_ x(t) = sin(oo.t) curve. Therefore, in fact, we have > > a whole statistical _ensemble_ of true x(t) = sin(oo'.t) curves and > > their deviations from the ideal curve become larger as time proceeds. > > Casinos do not always win events. So? At the end of the year, > casinos always win. > > > Needless to say that our numerical experiments are a confirmation of > > the above theory. We find that the ideal and true curves differ by an > > amount 1 in x after 37636 time steps dt = 0.1 . But, in reality, (dt) > > is supposed to be a stochastic variable with some mean and a spread, > > meaning that there is a bundle of solutions instead of an exact one. > > Grandfather clocks work, weather forecasts and economics do not. So? > > > Now think about the "grand" consequences. Does the above imply that: > > > TIME IS IRREVERSIBLE ? > > Dissipation is a Second Law, Large Numbers Hypothesis, weak arrow of > time. Keep It Simple, Stupid: Feynman's sprinkler and conservation of > angular momentum. Feynman's sprinkler absolutely does not play > backwards. > > -- > Uncle Alhttp://www.mazepath.com/uncleal/ > (Toxic URL! Unsafe for children and most mammals)http://www.mazepath.com/uncleal/qz4.htm http://en.wikipedia.org/wiki/Feynman_sprinkler http://www.copernicusproject.ucr.edu/ssi/2007PhysicsRes/feynman-inverse-sprinkler1.pdf Thanks. Han de Bruijn
From: Han de Bruijn on 23 Jun 2010 06:38
On Jun 22, 11:22 am, Han de Bruijn <umum...(a)gmail.com> wrote: > Hypothesis: > ---------- > | ALL LAWS OF NATURE ARE ONLY APPROXIMATELY TRUE > > Just take it or leave it. I'm not going to defend this as a new kind > of dogma. Instead, we are going to explore some consequences of the > proposition, in a very specific mathematical sense. And that's it. [ .. snip first example .. ] Classical Free Particle Numerically =================================== Consider Newton's first law of motion for a free particle: d^2x/dt^2 = 0 ; x(0) = 0 ; dx/dt(0) = V ; x = x(t) x = space coordinate, t = time, V = initial velocity. The "exact", but I would rather say ANALYTICAL, solution of the above equation is well known. And we call it the _ideal_ solution: x(t) = V.t But according to our Hypothesis, the _true_ equations are not "exact"; they are _approximations_ of the above _ideal_ differential equation: x(t + dt) = x(t) + dt.x'(t) + dt^2/2.x''(t) + dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (1) x(t - dt) = x(t) - dt.x'(t) + dt^2/2.x''(t) - dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (2) - 2.x(t) = - 2.x(t) (3) ------------------------------------------------------------------ + x(t + dt) - 2.x(t) + x(t - dt) = dt^2.x''(t) + dt^4/12.x''''(t) .. So the TRUE equation of motion is, with dt = finite and uncertain: x(t + dt) - 2.x(t) + x(t - dt) ------------------------------ = dt^2 = h.o.t. + dt^2/12.d^4x(t)/dx^4 + d^2x(t)/dt^2 = 0 Where h.o.t. = higher order terms, to be neglected. The equations of motion are solved numerically as follows. Start with: x(0) = 0 ; (x(dt) - x(0))/dt = V ==> x(dt) = x(0) + V.dt x(t + dt) - 2.x(t) + x(t - dt) ------------------------------ = 0 ==> dt^2 x(t + dt) = 2.x(t) - x(t - dt) ==> x(0.dt) = 0 x(1.dt) = x(0) + V.(1.dt) x(2.dt) = 2.x(1.dt) - x(0.dt) = x(0) + V.(2.dt) x(3.dt) = 2.x(2.dt) - x(1.dt) = x(0) + V.(3.dt) x(4.dt) = 2.x(3.dt) - x(2.dt) = x(0) + V.(4.dt) ......... x(n.dt) = x(0) + V.(n.dt) , by mathematical induction eventually. This is exactly in concordance with the analytical _ideal_ solution: x(t) = V.t with t = k.dt , k = 0,1,2,3, ... So far so good. But there is an analytical approach to the numerics as well. Solve: dt^2/12.d^4x(t)/dx^4 + d^2x(t)/dt^2 = 0 Characteristic equation: dt^2/12.L^4 + L^2 = 0 Substitute L^2 = M ==> dt^2/12.M^2 + M = 0 A quadratic equation. The solution can be written as: M = 0 ; M = - 12/dt^2 ==> L = 0 ; L = +/- i.2.sqrt(3)/dt The latter solutions give rise to fast oscillations with a frequency more or less equal to the time discretization. Therefore they can be discarded. A solution of the form exp(0.x) = constant remains and the constant can be identified as the initial velocity V , resulting in: x(t) = x(0) + V.t So there is NOT a difference between the classical dynamics of a free particle analytically (ideally) or numerically (really). Mathematics, what else .. Han de Bruijn |