# The thermodynamic cost of choosing

Carlo Rovelli  Aix-Marseille University, Université de Toulon, CPT-CNRS,
F-13288 Marseille, France.  Department of Philosophy and Rotman Institute,
1151 Richmond St.N London, ON N6A5B7  Perimeter Institute, 31 Caroline Street
N, Waterloo ON, N2L2Y5, Canada

###### Abstract

Choice can be defined in thermodynamical terms and be shown to have a
thermodynamic cost: choosing between a binary alternative at temperature $T$
dissipates an energy $E\geq kT\ln 2$.

In a celebrated paper, Landauer noticed that erasing information requires
dissipation Landauer . The precise status of this observation is controversial
(see a discussion and extended references in Norton ), but the idea has had a
large impact on our understanding of the thermodynamics involved in
computation and information. Here I observe that there is a related connection
between choosing (which generates  infrmation) and dissipation. This statement
can be given a precise meaning under a definition of choice, discussed below.
With this definition, a choice has a thermodynamic cost: choosing between a
binary alternative at temperature $T$ dissipates an energy $E\geq kT\ln 2$,
where $k$ is the Boltzmann constant.

The verb “to choose” is used in a variety of contexts. One might say that a
thermostat has chosen to turn the heating on right now. This is not the kind
of choice that here I am interested in, because a thermostat functions on the
basis of a predictable mechanism. Here I rather reserve the term “choice” for
the situations when the outcome of the choice is not predictable.

I consider the classical context first, then the quantum one. In the context
of classical mechanics, lack of predictability is necessarily associated with
incomplete information. Therefore when when we talk about choice we are always
talking about our ignorance of an underlying dynamical process determining the
outcome Spinoza .

We can formalize this situation by considering a system $S$ with a phase space
$\Gamma$ and an incomplete set of $N$ variables $A_{n}:\Gamma\to I\\!\\!R$,
with $n=1,...,N$, that capture our incomplete knowledge. Here ‘incomplete’
means that the map into $I\\!\\!R^{n}$ defined by the set of the $A_{n}$’s is
not injective.

This structure defines a thermodynamic-like context, because we can
distinguish ‘micro-states’ $s\in\Gamma$, from ‘macro-states’ $a_{n}$ that are
the possible values that the observables $A_{n}$ can take. Each macro-state
$a_{n}$ defines a region $R(a_{n})$ in $\Gamma$ as its inverse image
$R(a_{n})=\\{s\in\Gamma,A_{n}(s)=a_{n}\\}$ and if a phase space has a natural
associated volume $V$, each macro-states has an associated entropy $S(a_{n})$
defined by $S=k\ln V(R(a_{n}))$. The entropy of a macro-state measures how
many micro-states it contains, and therefore the amount of ignorance about the
details, if we only the macro-state is known.

Figure 1: Landauer memory erasure. Black lines represent micro-histories;
thick grey lines macro-histories. The system is initially in one of two macro-
states; time merges the two..

A microscopic history of a system is described by a function $s(t)$, where $t$
is time. A macroscopic history of a system is described by the functions
$a_{n}(t)$. A given macro-history $a_{n}(t)$ is the ensemble of all the micro-
histories $s(t)$ such that $A_{n}(s(t))=a_{n}(t)$. Because of the determinism
of classical mechanics, knowledge of $s(t)$ for $t<0$ is sufficient to
determine $s(t)$ uniquely also for $t>0$. But not so for macroscopic
histories. That is, it is possible to have two macroscopic histories
$a_{n}(t)$ and $a^{\prime}_{n}(t)$, both compatible with the dynamics of the
system such that

$\displaystyle a_{n}(t)$ $\displaystyle\neq$ $\displaystyle
a^{\prime}_{n}(t),\ {\rm for}\ t<-\epsilon<0,$ $\displaystyle a_{n}(t)$
$\displaystyle=$ $\displaystyle a^{\prime}_{n}(t),\ {\rm for}\ t>\epsilon>0$
(1)

or

$\displaystyle a_{n}(t)$ $\displaystyle=$ $\displaystyle a^{\prime}_{n}(t),\
{\rm for}\ t<-\epsilon<0,$ $\displaystyle a_{n}(t)$ $\displaystyle\neq$
$\displaystyle a^{\prime}_{n}(t),\ {\rm for}\ t>\epsilon>0$ (2)

Figure 2: A choice. A macro-state splits into two macro-states, each carrying
a fraction of the micro-states.

These two processes are represented in Figures 1 and 2, respectively. The
first of these processes (Fig 1) represents a Landauer memory erasure: erasure
is a process that merges two macro-histories. Information is not lost at the
micro-level, where evolution is deterministic. Where information is lost is at
the macro-level.

In the second process (Fig 2), the macroscopic system chooses between two
different alternatives, both compatible with the dynamics, during the interval
$[-\epsilon,\epsilon]$. This is what happens when we make a choice by flipping
a coin: the macroscopic future is determined by a small fluctuation in the
air’s density, that determines whether the coin ends up head or cross. The
micro-history, of course, is deterministic.

The thermodynamics of a Landauer erasure can be understood in this language.
Consider Figure 1. Say that there are $N_{1}>0$ micro-histories that start in
the left macro-history and $N_{2}>0$ micro-histories that start in the right
macro-history. Say that the world is in one of the micro-histories, for
instance one that starts in the left macro-state. Before the erasure, the
entropy of the actual macro-state of the world is $S_{i}=k\ln{N_{1}}$; after
the erasure, the entropy of the actual macro-state of the world is
$S_{f}=k\ln{(N_{1}+N_{2})}>S_{i}$. Hence erasure increases entropy. If
$N_{1}=N_{2}=1$, the increase of entropy is $\Delta S=k\ln 2$. This is the
Landauer principle, that states the minimal cost for erasing a memory. This
statistical interpretation of the Landauer principle is ultimately equivalent
to the thermodynamical ones.

The simple argument above might lead one to expect that a choice decreases
entropy. This is not the case, and this is the main point I make in this
article.

The reason the entropy does not actually decrease in a choice is that
something needs to stabilize the information transferred from the microphysics
to the macro-physics by the choice. This differentiates the erasure from the
choice. In the erasure, the entropy grows, therefore standard irreversibility
implied by equilibration is sufficient to stabilize the erasure. In a choice,
described by the second figure, the entropy decreases, and therefore
equilibration (“the second principle”) tends to take it back immediately. That
is, the two macroscopic states in which the process in the right panel end up
are immediately merged again by equilibration unless something stabilizes
their outcome. In the absence of something that stabilizes the choice, the
right panel only describes the first phase of an arbitrary (undetected)
thermal fluctuation. The flipped coin needs to hit the ground and dissipate
its kinetic energy in order for providing us with a choice. Let us see this
phenomenon in detail in an example.

Consider the device depicted in Figure 3. It is formed by a single particle
moving fast in a cavity with two small windows. At each window there is a
pendulum. If the ball hits a pendulum, it may set it in motion. This can be
considered as a devise that makes a choice, if we take the position and
velocity of the particle to be micro-variables and the energy in the pendulums
to be macro-variables. In the macro-world evolution is unpredictable: it is
determined by the unknown micro-history: if the pendulums are initially at
rest, which one of the two is set in motion first is determined by the actual
position and velocity of the particle, which play here the role of the
flipping coin.

Let us analyze the full thermodynamics of the process.

Figure 3: A mechanical choice device. The ball, whose exact position is not
known, hits first one or the other of the two penduli.

We can associate a temperature $T_{h}$ to the particle (which has three
degrees of freedom), defined by $E=3\times\frac{1}{2}kT_{h}$, where $E$ is the
energy of the particle. If the pendulums are initially exactly at rest, they
have zero energy hence zero temperature. A collision transfers some energy
from the higher temperature system to the lower temperature system, hence
raises entropy. To be more realistic, imagine that the pendulums themselves
have an ambient temperature $T$. As long as $T_{h}>T$ it is more probable that
in the collision energy is transferred from the particle to the pendulum,
because of energy equipartition. Therefore the pendulum hit first can receive
energy and register the choice in the macro-world. For this to happen, the
transferred $\Delta E$ energy must be larger than the thermal energy of the
pendulum $E=\frac{1}{2}kT$, otherwise the oscillation of the pendulum caused
by the collision cannot be distinguished from the thermal fluctuations. The
increase in entropy due to the transfer of energy from a hot to a cold system
is

$\Delta S=\frac{\Delta E}{T}-\frac{\Delta E}{T+\Delta T}$ (3)

where $T_{h}=T+\Delta T$. The increase of information due to the outcome of
the binary choice is a single bit. For this to be consistent with the second
principle, the overall entropy increase must at least compensate it, namely
$\Delta S>k\ln 2$, giving

$\Delta E\left(\frac{1}{T}-\frac{1}{T+\Delta T}\right)>k\ln 2$ (4)

By taking $\Delta T$ arbitrary large we can minimize the parenthesis, leaving

$\Delta E>kT\ln 2,$ (5)

which is the relation we were seeking. That is, the minimal energy that must
be dissipated in the binary choice at temperature $T$ is $\Delta E=kT\ln 2$.
It is easy to see that the mechanism is general.

Choosing appears to be a phenomenon that is in a sense inverse of erasing
information (because branching is the inverse of merging). Therefore one may
be puzzled by the above result and rather be tempted to expect that if erasing
macroscopic information raises thermodynamical entropy, then choosing would
decrease it. But this is wrong. The reason is that there is no reversibility
in dissipative phenomena. A cyclic sequence that produces a choice and then
erases it repeatedly is not reversible: it continuously increase entropy at
every step, like most real macroscopic phenomena. Entropy is raised both in
choosing and in erasing the choice. The information (negative-entropy )gained
by the split of the macroscopic state must be compensate by a larger entropy
increase (diffusion, energy equipartition,…), in order for the process to
happen and stabilize.

Consider next quantum theory. At first sight, it seems that quantum theory can
give us a free ride, circumventing the result above. For instance, a Stern-
Gerlach apparatus can give rise to a binary choice. But a moment of reflection
shows that this is wrong: it disregards the physics of the apparatus and the
environment. This cannot be disregarded if we are concerned with the
thermodynamical balance.

In fact, as nicely pointed out in Max , the conventional textbook description
of a quantum measurement appears to violate all three laws of thermodynamics!
Thermodynamical legality is restored by considering the thermal properties of
the apparatus and the environment, where entropy raises in order to store the
result of the measurement into the ‘classical’ macro-physics. See Max  for a
detailed discussion of this point.

Equivalently, for an actual quantum measurement to be completed, it is
necessary for the branches corresponding to the different outcomes to
decohere, and this implies that information is lost into the environment,
hence the entropy relative to the observer increases. See andrea  for a
similar discussion in the context of the Relational Interpretation and Bohmian
for the role of dissipation in Bohmian Mechanics’ measurement. I shall not
pursue a quantitative analysis f he quantum case here, leaving it to future
work, but the above notes are sufficient to show that quantum measurement
cannot produce a choice without raising entropy either.

A general observation underpins the fact that a choice has a thermodynamical
cost. Thermal fluctuations are not predictable because they refer to the
dynamics of the microscopic degrees of freedom and these are by definition
characterized by their general inaccessibility. If we measure a specific
thermal or quantum fluctuation, we get a way to extract novel information from
a thermal or quantum system; but any measurement implies dissipation. The same
basic physical principle (also rooted in the second principle of
thermodynamics) underpins the thermodynamical cost of learning Milburn .

The first to notice the basic and –seems to me– under-appreciated fact that
any measurement is dissipative is, as far as I know, Reichenbach in
Reichenbach . In turn, the fact that measurement implies dissipation follows
from an even simpler observation: a measurement records information in the
future of the measurement interaction. This breaks time reversal invariance,
and the only physical source for the breaking of time reversal invariance is
thermodynamical irreversibility, namely dissipation.

In summary, choosing can be modelled as bringing information from the micro
(or quantum) physics to the macro (and classical) world. This process
necessitates to be stabilized, and this can only happen via dissipation. The
overall entropic balance is necessarily positive: the information gained in
the choice is over-compensated by what is lost in the associated dissipation.
The second law is not circumvented. As the Germans say ‘Wahl ist Leiden’, to
choose is to suffer: perhaps not sufferance, but certainly dissipation.

***

This work was made possible through the support of the ID#62312 grant from the
John Templeton Foundation, as part of the “The Quantum Information Structure
of Spacetime (QISS)” Project (qiss.fr). I thank all the participant to the
August 2023 FQXi retreat, where I have found key ideas for completing this
note.

## References

  * (1) Rolf Landauer, “Irreversibility and heat generation in the computing process”, IBM Journal of Research and Development, 5 (1961)183–191.
  * (2) John D., Norton, “Waiting for Landauer”, Studies in History and Philosophy of Modern Physics 42 (2011) 184–198
  * (3) Baruk Spinoza, Ethics, Third Part, Proposition II and Scolio, 1677, Wikisource, Ethics, Part 3.
  * (4) Emanuel Schwarzhans, Felix C. Binder, Marcus Huber, and Maximilian P. E. Lock, “Quantum measurements and equilibration: the emergence of objective reality via entropy maximisation”, arXiv:2302.11253.
  * (5) Hans Reichenbach, The Direction of Time (Dover, New York, 1956).
  * (6) G. J. Milburn, S. Shrapnel and P. W. Evans, “The Physics of Learning”, arXiv:2209.11954.
  * (7) Andrea Di Biagio, Carlo Rovelli, “Stable Facts, Relative Facts”, Foundations of Physics, 51, 30 (2021), arXiv:2006.15543.
  * (8) C Rovelli, “Preparation in Bohmian Mechanics”, arXiv:2104.11339. Foundations of Physics, to appear.