44

Does God Really Play Dice?

Der liebe Gott w¨

urfelt nicht. God does not play dice. (Einstein)

I still believe in the possibility of giving a model of reality which shall rep-
resent events themselves and not merely the probability of their occurrence.
(Einstein in his Spenser lectures 1933)

Neither Herr Boltzmann nor Herr Planck has given a definition of W .
(Einstein)

Usually W is put equal to the number of complexions. In order to calculate
W , one needs a complete (molecular-mechanical) theory of the system un-
der consideration. Therefore it is dubious whether the Boltzmann principle
has any meaning without a complete molecular-mechanical theory or some
other theory which describes the elementary processes (and such a theory
is missing). (Einstein)

44.1 Einstein and Modern Physics

In this concluding chapter we give some more perspectives on statistical me-
chanics, which today is considered as the foundation of thermodynamics.

Einstein could never accept the idea that physics ultimately is based

on microscopic games of dice as in Boltzmann’s gas dynamics and the
Copenhagen interpretation of quantum mechanics. Einstein’s criticism can
be given the following paraphrase in politics: Consider a politician who states
that he firmly believes that each individual voter plays dice when deciding
how to vote, and claims that his belief is based on the observation that he
cannot predict how individual voters will vote. Would you believe that this
politician has any contact with realities? Wouldn’t you say that most voters
probably try to make a rational choice based on the arguments in the election
debate, instead of simply playing dice?

Or on observing people walking around in a city, would you say that their

apparent erratic and unpredictable paths would be the results of playing dice

384

44 Does God Really Play Dice?

at each foot step? Would you say that it at least looks as if people are playing
dice, even if they don’t really do that, and that this would be a useful model?
Or would you say that you see no good reason to use such a model?

Well, if you believe that dice-playing voters belong to fiction (although

there may be some cynics using this approach), then you may also join Einstein
in his criticism of physics based on the idea that individual particles like
molecules, atoms or electrons play dice when deciding what to do next, based
on the observation that it is difficult to predict the position and velocity of
individual particles.

But from where does the idea come that physics is based on dice-playing

particles, which is a physics against the principles of Newtonian mechanics
based on cause-effect and which Einstein questioned long before his criticism
was dismissed as a sign of senility. Well, we know that the idea came out of
a need to develop a foundation of thermodynamics including the 2nd Law in
the late 19th century. In Newton’s mechanistic world the 2nd Law and arrow
of time is missing since Newton’s laws are time reversible, and so the scientists
had to come up with something: Large values were at stake, both scientific
and economical.

Fig. 44.1. Ludwig Boltzmann (1844–1906) and Jan Josef Loschmidt (1821–1895).

44.2 Boltzmann and Statistical Mechanics

The German mathematician and physicist Ludwig Boltzmann (Fig. 44.1) took
up the challenge to give thermodynamics a scientific mathematical basis in-
cluding an explanation of the 2nd Law of irreversibility [18]. Boltzmann
started with a molecular-deterministic model of a gas consisting of a very
large number of molecules colliding elastically following Newton’s laws. Boltz-
mann of course understood that this model is time reversible, so he had to

44.2 Boltzmann and Statistical Mechanics

385

modify it to make it irreversible. His modification was to let the molecules
play dice before collision to decide how to collide, but not after, which broke
the time reversibility of Newton’s laws and thus led to irreversibility. This was
the birth of statistical mechanics, which inspired Planck in the early evening
of Sunday October 7 1900, in an “act of despair”, to invent a solution to
the outstanding open problem of black-body radiation, based on statistics of
quanta. This opened the way to quantum mechanics with its statistical in-
terpretation of the Schr¨

odinger wave function as the probability of finding a

particle like an electron at a specific location in space-time. In short, this led
into forms of modern physics based on various forms of statistical mechan-
ics, which Einstein could never accept, even under strong pressure from the
scientific community.

Using particle statistics Boltzmann could demonstrate a version of the 2nd

Law stating that a certain quantity S named entropy could never decrease
and when increasing strictly would signify irreversibility and give the arrow of
time a direction forward: Reversal of a process with strictly increasing entropy
would violate the 2nd Law and thus be impossible. On Boltzmann’s tombstone
the famous formula

S = k log(W )

(44.1)

is engraved, where k is Boltzmann’s constant (k

≈ 10

−23

joules/kelvin) and W

denotes the probability of a certain state representing the number of “com-
plexions” corresponding to the state. Increasing entropy would then reflect
that Nature would tend to move from less probable to more probable states
or towards states with more complexions. Einstein could not accept these
ideas as healthy science, and the questions why and how Nature would seek
to increase entropy, was left without any answer. Modern physics has not
brought much of understanding, as is clear from the above citation of the
Nobel physics laureate Richard Feynman.

But is statistical mechanics scientific in the sense of Popper, so that it can

be falsified and is it based on sound scientific logic? Or is Einstein’s criticism
in fact relevant? After all, Einstein is considered to be the greatest scientist
of the 20th century, so how probable is it that he was on a completely wrong
track, all through the later half of his long life? We believe that he was not.

Is it possible to falsify statistical mechanics? Probably not, as it would

require a full molecular-mechanical model according to Einstein. Concern-
ing scientific logic, we recall that there are many particle systems based on
Newtonian deterministic mechanics with a very complex dynamics, like a tur-
bulent gas in an diesel engine or a galaxy of interacting stars, and also a game
of roulette or dice. In such complex systems the positions in space-time of
the particles are very sensitive to small perturbations, and thus are impos-
sible to predict or compute (although mean values in space-time may be as
we have seen in this book). We may refer to such systems as (macroscopic)
games of roulette, which thus are very complex. Now, statistical mechanics is
based on microscopic games of roulette, which leads to microscopics of micro-

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44 Does God Really Play Dice?

scopics in a never-ending chain, which is against scientific logic: By definition
microscopics is simple, because what is complex has its own microscopics and
thus is not microscopic. Further, to say that Nature seeks to move towards
more probable states, is a truism without scientific interest: How probable
could it be that a system moves towards a less probable state instead of a
more probable? We conclude that statistical mechanics is neither scientific
nor logical, and we present an alternative based on deterministic computation
with finite precision, without any form of statistics, which includes the 2nd
Law and irreversibility.

Note that we do not claim that it is impossible to set up models of elections

based on dice-playing voters, or models of thermodynamics based on dice-
playing particles, which sometimes may produce reasonable results. What we
say is that we see no reason to do so, since more precise results may be
obtained from a deterministic computational model.

We also recall that statistical mechanics grew out of a necessity to handle a

scientific dilemma without any computers available, and that with computers
Boltzmann probably would have chosen a solution based on computation less
open to criticism, to which he was very sensitive.

We claim that from a scientific point of view EG2 is better than

Boltzmann’s statistical mechanics, since the basic assumptions of Euler of
conservation of mass, momentum and energy can hardly be disputed and the
G2 computational methodology is transparent, while the basic assumptions
of statistical mechanics are both illogical and virtually impossible to either
prove or disprove.

44.3 Summary

We have outlined a new foundation of thermodynamics based on the 1st Law
in the form of the inviscid Euler equations expressing conservation of mass,
momentum and energy, combined with finite precision computation in the
form of G2, and we have shown that the resulting EG2 model satisfies a 2nd
Law implying irreversibility and an arrow of time. Further we have shown that
the irreversibility is a consequence of the impossibility to solve the Euler equa-
tions exactly, and the necessary occurrence of shocks/turbulence in inviscid
flow corresponding to G2 approximate weak solutions.

Thus we have resolved the main mystery of classical thermodynamics of

formulation and justification of the 2nd Law, by showing that the 2nd Law is
a consequence of the 1st Law and finite precision computation. We have thus
mathematically justified the 2nd Law, without using any form of statistics or
by referring to some property of physical systems to always increase entropy,
both of which have shown to be impossible to rationalize.

We have shown that if Nature follows the weak/strong midway of EG2 in

its own analog computation, then Nature will automatically (without know-

44.3 Summary

387

ing it) satisfy the 2nd Law, which opens to a scientific understanding of irre-
versibility and the arrow of time.

We may view the 2nd Law to reflect an interplay between stability and

finite precision stating that only processes which are stable under finite pre-
cision computation can be realized and thus exist. In particular, reversing a
transformation of kinetic energy to heat energy by friction or turbulent dissi-
pation is impossible, because it would require infinite precision.

In Vol 5 we develop further aspects of thermodynamics by applying EG2

to a variety of concrete problems. We also make a fresh attack on the fa-
mous problem of black-body radiation based on computation, with quantum
mechanics remaining as a veritable challenge.