#  Simulating Physics with Computers

Søren Toxvaerd  Department of Science and Environment, Roskilde University,
Postbox 260, DK-4000 Roskilde, Denmark<EMAIL_ADDRESS>Submitted to Symmetry

###### Abstract

Feynman gave in 1982 a keynote speech Simulating Physics with Computers (Int.
J. Theor. Phys. 21, 467 (1982)) in which he talked “…about the
possibility…that the computer will do exactly the same as nature”. The
motivation was that: “…the physical world is quantum mechanical, and therefore
the proper problem is the simulation of quantum physics”. Here I try after
more than forty years to answer Feynman’s question of whether it is possible
to perform exact computer simulations. Many computer simulations are not
exact, they contain mean field approximations that disobey the symmetry in the
quantum dynamics with Newton’s third law, e.g. almost all astrophysical
simulations of galaxy systems. After a review of computer simulations and the
problems of simulating real systems, I argue that Newton’s discrete dynamics,
which is used in almost all computer simulations and which is exact in the
same sense as Newton’s analytic dynamics, is the classical limit path of
Feynman’s quantum paths. However, the physical world is not known exactly and
it is much more complex than any simulated systems, and so far no real systems
have been simulated exactly. Hence, more than forty years later, and after
hundreds of thousands of computer simulations of the physical system’s
dynamics the answer to Feynman’s question is still negative. But although it
is not possible to simulate the dynamics exactly for any real systems, the
simulations have been and will be of great use in Natural Science.

##  I Introduction

Richard P. Feynman gave in 1982 a keynote speech Simulating Physics with
Computers Feynman1981  in which he wanted “…to talk about the possibility that
there is to be an $exact$ simulation, that the computer will do $exactly$ the
same as nature.” The motivation was that: “the physical world is quantum
mechanical, and therefore the proper problem is the simulation of quantum
physics”. This article tries after more than forty years, to answer Feynman’s
question of whether it is possible to perform an exact computer simulation of
the dynamics of nature.

Almost all computer simulations of the dynamics of real systems are with
algorithms for Newton’s classical dynamics. Newton’s classical $analytic$
dynamics can, however, only be obtained exactly for a few systems, e.g. the
harmonic oscillator, but the exact analytic dynamics of a conservative system
of objects with classical dynamics has some characteristics: the dynamics is
time reversible, symplectic, and with three dynamical invariances (momentum,
angular momentum, and energy). In Feynman’s talk he points out that $exact$
dynamics is quantum dynamics (QD), and Feynman also noted that there is a
difference between quantizing and discretizing, and that QD is in the coherent
space citef . However, so are almost all computer simulations for classical
dynamics. Furthermore, the simulations are not with analytic but discrete
dynamics as QD, and where the actions of force quants result in discrete
changes of positions in a coherent space. The formulation of exact classical
discrete dynamics was solved a long time ago by I. Newton in Proposition I in
the first part of Newton’s book $Principia$ Newton1687 . Proposition I is
Newton’s algorithm for discrete classical dynamics Toxvaerd2023 . In computer
simulations, the algorithm appears with a variety of names, e.g. the Verlet
algorithm Verlet1967 , or the leap-frog algorithm, and it is used in almost
all Molecular Dynamics (MD) simulations of classical, as well as semiclassical
MD simulations. Proposition I is given in the next section, and the algebraic
formulation and the proof that Newton’s discrete dynamics is exact in the same
manner as Newton’s analytic Classical Mechanics is given in Toxvaerd2023 .

Here I will argue that Newton’s discrete dynamics is the classical discrete
limit path in the coherent space of Feynman’s quantum paths. That it is a
discrete dynamics, and not an analytic dynamics, which is the limit path of QD
was in fact suggested shortly after Feynman’s talk by Nobel Laurate T. D. Lee
and coworkers Lee1983 ; Friedberg1983 .

More than forty years after Feynman’s talk, the article tries to answer his
question based on today’s knowledge of computer simulations of real systems.

Figure 1:  Newton’s figure at $Proposition$ $I$ in Principia, with the
formulations of the discrete dynamics. The discrete positions are A:
$\bf{r}$$(t-\delta t)$; B: $\textbf{r}(t)$; C: $\textbf{r}(t+\delta t)$, etc..
The deviation cC from the straight line ABc (Newton’s first law) is caused by
a force from the position S at time $t$.

##  II Proposition I

Newton begins Principia by Proposition I. The English translation is

Of the Invention of Centripetal Forces.
PROPOSITION I. Theorem I.
The areas, which revolving bodies describe by radii drawn to an immovable
centre of force do lie in the same immovable planes, and are proportional to
the times in which they are described.
For suppose the time to be divided into equal parts, and in the first part of
time let the body by its innate force describe the right line AB. In the
second part of that time, the same would (by Law I.), if not hindered, proceed
directly to c, along the line Bc equal to AB; so that the radii AS, BS, cS,
drawn to the centre, the equal areas ASB, BSc, would be described. But when
the body is arrived at B, suppose that a centripetal force acts at once with a
great impulse, and, turning aside the body from the right line Bc, compels it
afterwards to continue its motion along the right line BC. Draw cC parallel to
BS meeting BC in C; and at the end of the second part of the time, the body
(by Cor. I of Laws) will be found in C, in the same plane with the triangle
ASB. Join SC, and, because SB and Cs are parallel, the triangle SBC will be
equal to the triangle SBc, and therefore also to the triangle SAB. By the like
argument, if the centripetal force acts successively in C, D, E, & c., and
makes the body, in each single particle of time, to describe the right lines
CD, DE, EF, & c., they will all lie in the same plane; and the triangle SCD
will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE. And
therefore, in equal times, equal areas are described in on immovable plane:
and, by composition, any sums SADS, SAFS, of those areas, are one to the other
as the times in which they are described. Now let the number of those
triangles be augmented; and their breadth diminished in infinitum; and (by
Cor. 4, Lem III) their ultimate perimeter ADF will be a curve line: and
therefore the centripetal force, by which the body is perpetually drawn back
from the tangent of this curve, will act continually; and any described areas
SADS, SAFS, which are always proportional to the times of description, will,
in this case also, be proportional to those times. Q. E. D.

Newton illustrated Proposition I by a figure (Fig. 1). The assumption
..suppose that a centripetal force acts at once with a great impulse.. is
highlighted here. It is a central part in the proposition. The force at times
between a discrete time step is zero, time and forces are quantized, and the
change in positions in a coherent space is discretized proportional to the
actions of the forces.

According to Proposition I, a new position in Newton’s discrete dynamics,
$\textbf{r}_{i}(t+\delta t)$, at time $t+\delta t$ of an object $i$ with the
mass $m_{i}$ is determined by the force quant $\textbf{f}_{i}(t)$ acting on
the object at the discrete position $\textbf{r}_{i}(t)$ at time $t$. The quant
causes a discrete change of the direction, and it changes the momenta and the
positions $\delta\textbf{r}_{i}$ proportional to $\delta t$ with $\delta
t\textbf{f}_{i}(t)=\overrightarrow{cC}$ (see Figure 1). The positions
inbetween the force quants are changed from
$\overrightarrow{AB}=\textbf{r}_{i}(t)-\textbf{r}_{i}(t-\delta
t)=\overrightarrow{Bc}$ to $\overrightarrow{BC}=\textbf{r}_{i}(t+\delta
t)-\textbf{r}_{i}(t)$, i.e

$\overrightarrow{BC}=\overrightarrow{AB}+\overrightarrow{cC},$ (1)

or

$m_{i}\frac{\textbf{r}_{i}(t+\delta t)-\textbf{r}_{i}(t)}{\delta
t}=m_{i}\frac{\textbf{r}_{i}(t)-\textbf{r}_{i}(t-\delta t)}{\delta t}+\delta
t\textbf{f}_{i}(t).$ (2)

The algorithm, Eq. (2), is usual presented as the “the Verlet algorithm”

$\textbf{r}_{i}(t+\delta t)=2\textbf{r}_{i}(t)-\textbf{r}_{i}(t-\delta
t)+\delta t^{2}/m_{i}\textbf{f}_{i}(t),$ (3)

or the “leap frog” algorithm for the velocities

$\displaystyle\textbf{v}_{i}(t+\delta t/2)=\textbf{v}_{i}(t-\delta t/2)+\delta
t/m_{i}\textbf{f}_{i}(t)$ $\displaystyle\textbf{r}_{i}(t+\delta
t)=\textbf{r}_{i}(t)+\delta t\textbf{v}_{i}(t+\delta t/2).$ (4)

Newton’s discrete dynamics, Eq. (3) or Eq. (4,) is used in almost all
simulations of classical, as well as semiclassical simulations. The trajectory
A,B,C,D,.. in Figure 1 in Proposition I is a discrete classical QD path but
the difference between analytic- and discrete dynamics is proportional to the
square of $\delta t=$ Planck time ($\approx 10^{-44}$ sec) and negligible
Toxvaerd2014 .

##  III Simulating with Computers

The title of Feynman’s talk is Simulating Physics with Computers, but today
computer simulations are used not only in physics but in almost all
subdisciplines in Natural Science. The first section after Feynman’s
Introduction is  2\. Simulating Time, and thus he limited this talk to cover
the part of computer simulations that concerns the simulations of the time
evolutions of physical objects. These simulations are often named Molecular
Dynamics, with a misleading name since they cover all kinds of simulations
with objects from atoms to galaxies. Feynman excludes the many statistical
mechanics simulations, which often are named Monte Carlo simulations (MC) with
a name that refers to the statistical probability of an event in MC.

It appears from his speech that Feynman has a supercomputer in mind and with
an enormous hardware capability, and these computers exist today and might be
extended to quantum computers in the future (which probably will bring
humanity to a new era in the “artificial intelligence” world). But computer
simulations are not hardware, it is software and for the dynamic evolution of
a physical system, they are algorithms. Newton was the first to formulate an
algorithm in Proposition I and he (and his students) used it to calculate the
future of the Solar system. Newton lived in the rational era and the evolution
of the Solar system was believed to be “exactly” predicted by Newton’s
“simulations”, a fact which revolutionized the philosophy. The first MD
simulations appeared in the sixties Rahman1964 ; Verlet1967 , almost three
hundred years after $Proposition$’s publication and today there have appeared
at least many hundred thousand articles with MD, or might be more than a
million.

An MD simulation is usually for a system with $N\approx 1000-10000$ objects,
with periodical boundaries to simulate a continuum, with approximated force
fields, with truncated force fields at long-range interactions Toxvaerd2011 ,
and with discrete time increments $\delta t\approx 10^{-14}$ sec much larger
than Planck time. In astrophysics, the simulations are for a much larger
system, but with mean-field approximations for long-range interactions. These
approximations disobey fundamental physical laws and destroy the exactness of
the discrete classical dynamics (see later). Many simulations are often
performed using packages with “black box” software to the simulations. In
general, the simulations are not exact since they appear with a series of
approximations of the models of the real systems.

The many computer simulations have, however, changed our insight into the
dynamic evolution of real systems, and the success of MD was acknowledged by a
Nobel Prize in 2013 Nobel2013 .

##  IV  Newton’s third law, Feynman’s quantum paths and QD entanglements

Feynman’s talk was about simulating QD in the physical world and several
properties must be clarified before one can answer Feynman’s question of
whether it is possible to perform an exact computer simulation of a real
system, e.g.:

A. Is Newton’s third law valid in QD?

B. Is gravity quantized, and with gravitational waves?

C. Does the interaction between baryonic object pairs with gravitation follow
QD, with Feynman’s quantum paths and QD entanglement?

All three questions have been heavily debated in the past.

###  IV.1 Newton’s third law

Newton began $Principia$ by postulating his three laws, and he justified the
third law from daily life experience in the subsequent $Scholium$ in
$Principia$. Newton’s law of the inverse square gravitational attraction
(ISL), which entangles pairs of baryonic objects according to his third law
is, however, formulated much later in $Principia$ on page 401 in the Third
Book Mundi Systemate, and where Newton first formulated four philosophic
principles for Natural Science. The first and in an English translation reads:
We are to admit no more causes of natural things than such as are both true
and sufficient to explain their appearances.
This principle might not be fulfilled by assuming the existence of dark matter
in the Universe (see later).

The validity of the third law has been debated right up to our days, for a
historical review see Domski2022 . The symmetry of the interactions ensures
the conservations of momentum and angular momentum for a conservative system
(Noerthe’s theorem Noerthe1918 ). It has been questioned whether Newton’s
third law is valid for conservative electromagnetic systems. It was doubted by
P. Cornille in his review of the applications of Newton’s third law in physics
Cornille1999 . According to Cornille, the law is not valid for electromagnetic
systems, and Cornille claims that it can be disproved for a charged conductor
at rest which is predicted to set itself in motion and accelerate its center
of mass or rotate without external help. It is a breach of the second law of
thermodynamics, but it is a quality which would be of great technical
importance. However, no such conductors have been reported and a recent
article confirms the validity of Newton’s third law for these electromagnetic
systems Epp2023 .

Newton’s third law ensures the conservation of momentum and angular momentum
for a conservative system Noerthe1918 . However, many computer simulations
e.g. simulations of galaxies, use mean field approximations which disobey
Newton’s third law. Simulations of galaxies with the ensemble of “suns” with
gravitational forces have been performed for many decades. The dynamics of the
galaxies are obtained by the Particle-Particle/Particle-Mess (PPPM) method
Hockney1974 ; Klypin1983 ; Centrella1983 , which is a mean-field approximation
of the long-range forces in the system where a mass unit is moving in the
collective field of these forces, and the Poison equation for the PPPM grid is
solved numerically. The galaxies are believed to be unstable unless they are
stabilized by dark matter and the indirect proof of its existence is in fact
based on extensive computer simulations Springel2005 ; Schaye2010 ; Dubois2014
; Vogelsberger2014 ; Dubois2016 ; Price2018 ; Weinberger2020 ; Ludlow2021 .
Cosmological simulations of galaxies are reviewed in Vogelsberger2020 . The
justification for assuming the existence of dark matter is reviewed in
Bahcall1995 ; Bahcall2015 . The justification is not only based on the
simulations of galaxy systems but on several experimental observations of
galaxies. The mass distribution and the rotational velocities are non-
Newtoinian/Kepler-like with a substantial baryonic matter in the halos of the
galaxies and with a higher rotational velocity than in a system with classical
dynamics. These facts have supported the hypothesis of the existence of dark
matter.

Also the alternative MOND formulation Milgrom1983  for classical celestial
dynamics disobeys Newton’s third law Felten1984 ; Toxvaerd2024 , and computer
simulations of galaxies reveal that the lack of conserved angular momentum
destabilizes the galaxy systems Toxvaerd2024 . But the recent exact
simulations of models of galaxy systems Toxvaerd2024  demonstrate that the
stability and a correct rotation profile of the galaxies can be obtained
purely from classical dynamics that obey Newton’s laws by modifing Newton’s
law of universal inverse square attraction.

In conclusion: So far no experimental evidence exists that Newton’s third law
is not valid for any systems, including the dynamics of galaxy systems and
quantum electromagnetic dynamics (QED). Concerning computer simulations, the
simulations with mean field attractions or MOND disobey Newton’s third law and
they are not exact.

###  IV.2  Gravity quantization and gavitational waves

Feynman assumed that the dynamics appear with positions in a coherent space,
and so do the computer-generated dynamics, but with discrete changes of the
positions. Lee and Friedberg suggested quantization of changes in positions
Lee1987 ; Friedberg1994 , and quantization of the gravitational attraction has
been suggested for a long time ago to overcome several problems with a
continuum formulation of general relativity Regge1961 . Since then a series of
articles with theories of quantum gravity based on the path integral approach
of Feynman have appeared. For reviews see Ambjorn2012 ; Loll2020 ; Ambjorn2021
.

Stephen Hawking wrote in 1999: So what the singularity theorems are really
telling us, is that the universe had a quantum origin, and that we need a
theory of quantum cosmology, if we are to predict the present state of the
universe Wiki . Quantum cosmology is now a discipline in astrophysics with
countless articles QuantumCosmos2022 .

Einstein predicted the existence of gravitational waves caused by acceleration
of mass units in the Einstein-de Sitter Universe. For a review of
gravitational waves see Cota2016 . One of the greatest success of the Hubble
telescope was the experimental detection of gravity waves Abbott2016 ;
Abbott2016a , and the accelerated expansion of the Universe Riess1998 ;
Perlmutter1999 . The equal behavior of electromagnetic and gravitational waves
was the experimental verification of gravity quantization. The Universe
expands with the relative Hubble expansion, which is an accelerating expansion
with an intensive expansion at every point in our three-dimensional space
(3D). The accelerated expansion of the space with baryonic units will create
and propagate gravitational waves with an inverse square intensity from the
baryonic units in a 3D space. This fact explains the common behavior of
gravitation and the Coulomb ISL attraction. The gravitation maintains a long-
ranged ISL attraction due to the symmetry break at Hawking’s QD origin of the
Universe with the lack of antimatter.

Newton’s discrete algorithm, which is a central difference algorithm has been
extended to include the central difference effect on positions due to the
Hubble expansion Toxvaerd2022a , and the extended algorithm is used to
simulate models of galaxies Toxvaerd2022a ; Toxvaerd2024 .

In conclusion: There is no conflict in assuming quantum gravity and space-time
quantization in connection with Feynman’s path integral formulation of QD.

###  IV.3  Barionic object pairs with gravitation QD, with Feynman’s quantum
paths and QD entanglements

Newton’s third law ensures the QD entanglement between two baryonic units $i$
and $j$, and if Newton’s law of gravitational attraction is caused by
gravitational waves which, with the speed of light moves from $i$ and $j$,
respectively, then the attraction on $i$ and $j$ appears synchronously and
ensures the entanglement between $i$ and $j$. However, although gravitation
might be quantized and with QD entanglement, Newton’s universal gravitational
attraction is not included in the standard model. This fact has given rise to
a series of proposed modifications of Newton’s inverse square law (ISL) for
gravitational attraction to an inverse attraction (IA) for longer distances
with the intention of unifying gravity with particle physics, so far without
success Fischbach2001 ; Adelberger2003 ; Lee2020 ; Henrichs2021 .

Einstein also proposed a modification of the ISL law Einstein1928 , and series
of modified gravity in cosmology including IA attractions have been proposed,
for a review see Capozziello2011 ; Clifton2012 . The ISL might also be
modified by gravitational lensing Renn1997 ; Toxvaerd2024 . The cosmological
effect of gravitation must be very long-range in favor of e.g. an IA
attraction since galaxies are known to appear in clusters with bound
rotations. The Milky Way and Andromeda galaxies belong to the Local Group
Redd2018 , and these galaxy clusters are again collected in the Virgo super
collection of galaxies clusters Mei2009 .

In conclusion: Although there is nothing exclusive in assuming quantum
gravity, the gravitational attraction is not included in the standard model
and there is a need to explain the stability, mass distribution, and
rotational velocity of galaxies either by including dark matter in the
universe or by modifying Newton’s ISL, e.g. to an IA long-range attraction.

##  V Reply to Feynman’s question

When Feynman in his talk “wants to talk about the possibility that there is to
be an $exact$ simulation, that the computer will do $exactly$ the same as
nature”, there is a fundamental problem because we do not exactly know what
nature does. Feynman lists a series of problems concerning performing exact
computer simulations, and no doubt his answer to the question was negative, it
is not possible. Nevertheless, hundreds of thousands of computer simulations
have been published since his talk, and they have led to a much better
understanding of the dynamics of real systems. However, almost all computer
simulations contain a series of approximations, e.g., mean field
approximations and approximations of the range of the forces. They are not
strictly exact, but in the best case, they are qualitatively correct.

Feynman pointed out that: “…the physical world is quantum mechanical, and
therefore the proper problem is the simulation of quantum physics”. Newton
formulated in $Principia$ his three laws, the ISL law for gravitation, and
found the analytic solution for Kepler’s equation for pairs of objects with
the ISL. The correct dynamics is, however, QD as Feynman pointed out, and if
so then Newton’s analytic dynamics is not the classical limit path in QD. It
is Newton’s discrete dynamics, formulated in Proposition I at the beginning of
$Principia$ which is the classical limit path of Feynman’s quantum paths, and
with the QD entanglement given by Newton’s third law. The difference between
the discrete and the analytic classical dynamics is, however, marginal, and
furthermore, there exists a “shadow Hamiltonian” Toxvaerd1994  where the
discrete positions are located on the analytic trajectories for the analytic
dynamics for the shadow Hamiltonian. So there is no qualitative difference
between Newton’s analytic and discrete classical dynamics Toxvaerd2023 .
Newton lived 1643-1727 in the Age of Enlightenment, and he believed that his
analytic dynamics were God’s exact dynamics.

Nature consists of elementary particles, atoms, molecules,…, planets, solar
systems, galaxies, and perhaps black matter, and with different forces where
not all are given by the standard model. All the computer simulations for real
systems are with drastical simplifications of the systems. The simulations of
models of dwarf galaxies in Toxvaerd2024 are claimed to be exact simulations.
But they are only exact in the sense that they are without any approximation
of the classical Feynman path of QD and they are still a drastic
simplification of even the smallest dwarf galaxy Simon2019 . Hence, more than
forty years after Feynman formulated his question the answer is still
negative, the computer simulations are not exact. But in the real world,
simulations with Newton’s discrete algorithm have been and will be of great
use in natural science.

## Acknowledgements

Niccolõ Guicciardini is gratefully acknowledged.

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