On the Irreversibility of Digital Time
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
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[en français/in french]
Keywords: Floating Point Numbers, Nombres Flottants, Rounding-off Errors, Erreurs d'arrondi.
The laws of classical physics are independent of the direction of time flow. Let's consider a simple example using a moving object P with coordinate X
and velocity V on a straight line D, and examine its movement:
T=0: X0
T=dT: X1 = X0 + V.dT
At this moment, let's reverse the velocity V:
T=2.dT: X2 = X1 - V.dT = (X0 + V.dT) - V.dT = X0
Thus, point P has returned to its initial position...
The quantities Xi, V and dT are real numbers [01].
Obviously, the systems studied in physics, in particular, are much more complex
than the naive example presented above. In general, it is necessary to use a computer to analyze
them. However, unfortunately, real numbers do not exist in the finite world of digital computation,
which inevitably leads to rounding errors! So, what about the reversibility of "digital" time?
To answer this question, let's consider the example of a two-dimensional
billiard system [02]. We use a rectangular plane domain in which N balls of equal mass
are regularly arranged on a square grid and that are still except one whose velocity is random -middle right-
[03]. Over time, the particles collide, and the impacts are perfectly elastic, meaning that momentum is conserved.
At the midpoint of the simulation, all velocities are reversed [04]. In theory, this means that all the balls should return to their initial positions.
But does this still hold true in a computer?
:
With a small number of particles (N=192) , the final state is identical to the initial state. Rounding errors are minimal, and the process
is numerically reversible.
:
With more particles (N=336), the final state is no longer identical to the initial state. Rounding errors are numerous enough that the particles
do not return to their starting point: the process is numerically irreversible.
:
With many more particles (N=768 and N=1376) , the phenomenon of numerical irreversibility can only worsen!
We thus observe a kind of threshold effect: reversibility with a small number of particles and irreversibility when there are more.
In conclusion, and in a general sense, time is not reversible in our computers, whereas the underlying physics is...
- [01]
Recall that real numbers are the numbers of physics.
- [02]
Many examples of two-dimensional billiards can be found in this gallery.
- [03]
The color of each ball depends on its initial coordinates {X0,Y0}.
- [04]
This means that each velocity vector {Vx,Vy} is replaced by {-Vx,-Vy} at the same instant.
Copyright © Jean-François COLONNA, 2025-2025.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2025-2025.
