[Fis] Fwd: Entropy, the Second Law, and Life

Arieh Ben-Naim ariehbennaim at gmail.com
Mon Jan 4 19:03:00 CET 2021

---------- Forwarded message ---------
From: Pedro C. Marijuan <pcmarijuan.iacs at aragon.es>
Date: Sat, Jan 2, 2021 at 8:57 PM
Subject: Entropy, the Second Law, and Life
To: Arieh Ben-Naim <ariehbennaim at gmail.com>

Dear FIS Discussants,

It is for me a great pleasure to impart this New Year Lecture. I will
address one of my favorite topics: the numerous and notable
misunderstandings that historically have accompanied, and continue to
accompany, the relationship between entropy and life. I have devoted many
years to the study of entropy and produced quite a few books and articles
about that (see the references below). It is amazing the persistence of so
many errors, misunderstandings and blunders around that fundamental
concept. As a guide to the present discussion, I have attached a chapter of
my new book on *"Entropy: The greatest Blunder in the History of Science"*.
In the excerpt that follows herein, I have dropped most of the formal
arguments, so let me emphasize reading the entire chapter --sent in a
separate mail (for list-server reasons).

Best wishes



*Entropy, the Second Law, and Life*

*Arieh Ben-Naim*

*Emeritus Professor, Department of Physical Chemistry**, The Hebrew
University of Jerusalem*


I would like to start this article with a quotation by Albert Einstein on

*“It is the only physical theory of universal content, which I am
convinced, that within the framework of applicability of its basic concepts
will never be overthrown.”*

Most people who use this quotation, emphasize the last part, namely, that
Thermodynamics will “*never be overthrown*.” Of course I agree with that
part. However, my emphasis, in this article is on the *“**framework of
applicability.” *My main point is that entropy and the Second Law were used
far beyond their* “**framework of applicability.” * One such application is
to living systems, which I will discuss in this article. The second is the
application of Entropy and the Second Law to the entire universe. This is
discussed in details in references [1,2].

The application of entropy and the Second Law to a living system is based
on two erroneous assumptions:

1.   Entropy is a measure of disorder (or disorganization)
Life is understood as a process towards organization and creation of order

>From these two assumptions it follows, almost naturally that life-processes
seem to be “a struggle against the Second Law of Thermodynamics.”

In this article we shall distinguish between two different questions: The
first one, the possibility of *defining *entropy; and the second, the
applicability of the Second Law to living systems. We shall start with the
general question on whether one can or cannot describe a living system by a
few thermodynamic variables such as temperature, pressure and composition.
This discussion will lead us to conclude that one cannot specify the
“thermodynamic state” of a living system. It follows that entropy is
undefinable for any living system. Next, we shall discuss the question of
the applicability of the Second Law to living systems. The answer to this
question is a definite, No!

*Can entropy be defined for any living system?*

This question is part of a more general question: Can physics, as we know
it today, be used to discuss and explain all aspects of life? In
particular, those aspects of life we call *mental processes* such as
thinking, feeling, consciousness, and the like. This question has been
discussed by numerous scientists, in particular by Schrödinger [3], Penrose
[4,5] and many others. Interestingly, some of these scientists raised
serious doubts about the general question stated above, yet they did not
shy away from applying entropy and the Second Law to living systems.

Everyone knows that life phenomena are the most complex, intricate,
interesting, wonderful, and whatever one wishes to ascribe to it. During
the 20th century science had achieved a great amount of knowledge and
understanding about the many aspects of life, from biochemical processes,
genetics, molecular biology, to brain functions, and many more. There are
however many more aspects of life that we do not understand. There are also
aspects of life that we might never understand.

Indeed, during the past century remarkable advances in understanding the
molecular basis of life have been achieved. A whole new branch of biology
was created: Molecular Biology. The mechanism of heredity was deciphered,
the so-called “genetic code” was discovered, the code which is responsible
for translating the message “written” in the DNA into synthesizing proteins
which are the so-called molecular robots in our cells.

There are many specific processes which have been studied by
thermodynamics. Examples: Chemical reactions, including metabolism where
energy stored in some chemical bonds are used to synthesize many molecules
which are vital to life. Photosynthesis, where energy from the sun rays is
used to convert carbon dioxide (CO2) and water (H2O) to high energy sugars.
In all of these cases the reactions could be studied *in vitro*, i.e. in a
laboratory setting, or in test tubes, isolated from the entire complicated
environment in the cell (*in vivo*). Clearly, thermodynamics was, and still
is, the main tool in understanding the energetics of these reactions.

There are other processes such as muscle contraction (i.e. converting
chemical energy into mechanical work) or “firing” of electrical signals
along the nerves’ axons which were studied thoroughly by thermodynamics and
statistical mechanics. In all of these specific processes one can isolate
the process and study it in well-defined environments and apply all the
tools of thermodynamics successfully. However, with all these remarkable
achievements which fill up countless textbooks on molecular biology,
biochemistry, energy transduction, neural networks and more, there is still
one phenomenon that was, and still is, inaccessible to study with the tools
of thermodynamics in particular, and in physics, in general. This is life

In fact, we still do not know how to define “life” or life related
phenomena such as consciousness, awareness, the mechanism underlying our
thinking, our feelings, and our ability to make decisions or create arts.
Notwithstanding the difficulty of defining “life,” it is clear that a
living system is far from equilibrium. As such the concept of entropy
cannot be applied. Simply because entropy is a *state function.* This means
that entropy is definable for a well-defined *thermodynamic system* at

We can easily describe the “state” of person sitting in a room. But this is
not a thermodynamic description which requires just a few thermodynamic
parameters. However, even if we could describe the *physical state* of the
body, there is still the question of how to describe the *state of the mind*
of the person? The last question brings us to the classical question about
the nature of the mind. It is possible that within some future extensions
of physical theories all mental activities could be discussed. However, at
this point in time it is appropriate to be cautious and refer to this
possibility as a “hypothesis.” In my view, statements such as Crick’s
“Astonishing Hypothesis” is very much a *hypothesis*, and it will remain a
hypothesis for a long time. If and when this hypothesis will be proven to
be correct, then it will be an enormously astonishing achievement,
particularly to all those who subscribe to the concept of dualism.

To conclude, we do not know whether or not living systems can be described
as purely material objects on which all the physical laws are applicable.
But even if such a description becomes feasible, one could not claim that
living systems are well-defined thermodynamic systems, i.e. macro-systems
describable by a few thermodynamic variables. Therefore, entropy may not be
applied to such systems. This conclusion very clearly follows from any
definition of entropy See Ben-Naim [1,2,8-10].

*The history of application of Entropy and the Second Law to living systems*

Perhaps the oldest association of Second Law with life is due to Boltzmann.
On May 29, 1886, Ludwig Boltzmann
<http://www.eoht.info/page/Ludwig+Boltzmann> presented a talk at the
Festive Session of the Imperial Academy of Sciences in Vienna where he
discussed “*The S**econd Law* <http://www.eoht.info/page/second+law>* of
Thermodynamics”* with special emphasis on its application in relation
to Charles
Darwin <http://www.eoht.info/page/Charles+Darwin>'s 1859 theory of evolution
<http://www.eoht.info/page/evolution> [16].

The most-quoted <http://www.eoht.info/page/Entropy+quotes> passage from
this lecture is that life <http://www.eoht.info/page/life> is a *struggle
for **entropy* <http://www.eoht.info/page/Entropy>:

“*The general **struggle for existence*
<http://www.eoht.info/page/Struggle+for+existence>* of **animate*
<http://www.eoht.info/page/Animate> *beings*
<http://www.eoht.info/page/Being>* is not **struggle*
<http://www.eoht.info/page/struggle>* for raw **materials*
<http://www.eoht.info/page/Matter>*, **these, for organisms, are air, water
and soil, all abundantly available, nor for **energy*
<http://www.eoht.info/page/Energy>*, which exists in plenty in anybody in
the form of **heat* <http://www.eoht.info/page/Heat>* Q, but of a **struggle
for entropy* <http://www.eoht.info/page/Struggle+for+entropy>*, which
becomes available through the transition of energy from the hot sun to the
cold earth.” *

As we have discussed above (see attached Chapter), Boltzmann believed that
a system proceeds from a low to a high probability, also he stated that
systems proceed from ordered to disordered states. Since living systems are
considered to proceed from disorganized to more organized he has used
essentially the argent in the abstract to conclude that life
<http://www.eoht.info/page/life> is a “*struggle for **entropy*

However, the most influential physicist who propagated the erroneous ideas
about entropy and life was Erwin Schrödinger. In his book “What is Life?”
published in (1944) [3], he discussed in greater detail the role of entropy
in living systems. We will provide some quotations from this book in the
next section.

*Schrödinger’s book: What is life?*

On the question: “What is life? one cannot avoid starting with the most
famous book written by Schrödinger [3].

This book is based on lectures delivered by Schrödinger in Dublin in 1943.
This book was most influential for a long time and probably laid the
cornerstone for the creation of the whole field of molecular biology. It
also has encouraged many physicists to apply the methods of physics to
biology. In this section we shall present only a few comments about some of
Schrödinger’s statement regarding entropy, more details may be found in
reference [2].

In Chapter 1 of his book, Schrödinger correctly pointed out that “the
physicist’s most dreaded weapon, mathematical deduction, would hardly be
utilized. The reason for this was not that the subject was simple enough to
be explained without mathematics, but rather it was too much involved to be
fully accessible to mathematics. As I noted above, it is not clear at all
which kind of mathematics or physics one would need to describe life. Then
Schrödinger outlines the plan of his lectures as follows:

*“The large and important and very much discussed question is: How can the
events in space and time which take place within the spatial boundary of a
living organism be accounted for by physics and chemistry?”*

His preliminary answer to this question:

 “*The preliminary answer which this little book will endeavor to expound
and establish can be summarized as follows: The obvious inability of
present-day physics and chemistry to account for such events is no reason
at all for doubting that they can be accounted for by those sciences*.”

Schrödinger attempts to explain the source of difficulty of applying the
methods of physics and chemistry to living systems. The fundamental
difference between a living system and any piece of matter that physicists
and chemists have ever handled is in the structure, or the arrangement of
atoms and molecules in the organism differs fundamentally from that of a
system dealt with physics and chemistry. It seems to me that Schrödinger,
at least in this stage of the book believed that once physicists enter into
biology and apply their powerful arsenal of physical methods and theories,
they shall be able to answer the question posed in the book.

On page 10 Schrödinger provides some hints about his intention to use the
Second Law:

 “*The reason for this is, that what we call thought (1) is itself an
orderly thing, and (2) can only be applied to material, i.e. to perception
or experiences, which have a certain degree of orderliness… Therefore, the
physical interactions between our system and others must, as a rule,
themselves possess a certain degree of physical orderliness, that is to
say, they too must obey strict physical laws to a certain degree of

My impression is that Schrödinger used the terms “*orderly thing,”
“orderliness,” “physical organization,” “well ordered organization,” *and
similar terms in anticipation of his usage of entropy and the Second Law of
thermodynamics in later chapters.

Chapter 6, of his book is titled: “Order, disorder and entropy.” He starts
with the common and erroneous statement of the Second Law in terms of the
“order” and “disorder.”

 “*It has been explained in Chapter 1 that the laws of physics, as we know
them, are statistical laws. They have a lot to do with the natural tendency
of things to go over into disorder.”*

There is of course, no such “natural tendency,” except in the minds of
those who have a distorted view of the Second Law. Then, he makes another
typical statement about life:

*Life seems to be orderly and lawful behavior of matter, not based
exclusively on its tendency to go over from order to disorder, but bases
partly on existing order that is kept up.*

The idea that life somehow withstands the “natural tendency to go from
order to disorder” is quite frequently found in the literature;” “life
withstands the ravages of entropy,” “life disobeyed the Second Law” and so
on. Unfortunately, all these statements are *meaningless*; there exists no
tendency of going from order to disorder in the first place. The tendency
of entropy to increase applies to some specific processes in isolated
systems, and not to a living system which is an open system, far from
equilibrium. It is only on page 74 that he explicitly relates the Second
Law with the behavior of living systems.

 “*The general principle involved is the famous Second Law of
Thermodynamics (entropy principle) and its equally famous statistical

His main claim is that “living matter evades the decay to equilibrium.”

 “*It is avoiding the rapid decay into the inert state of ‘equilibrium’
that an organism appears to be enigmatic; so much so, that from the
earliest times of human thought some special non-physical or supernatural
force (vis viva, entelechy) was claimed to be operative in the organism,
and in some quarters is still claimed.”*

Then he asks:

 “*How does the living organism avoid decay? The obvious answer is: By
eating, drinking, breathing and (in the case of plants) assimilating. The
technical term is **metabolism.”*

I believe that the book’s highlight is reflected on page 76:

 “*What then is that precious something contained in our food which keeps
us from death? That is easily answered. Every process, event, happening –
call it what you will; in a word, everything that is going on in Nature
means an increase of the entropy of the part of the world where it is going
on. Thus, a living organism continually increases its entropy – or, as you
may say, produces positive entropy – and thus tends to approach the
dangerous state of maximum entropy, which is death. It can only keep aloof
from it, i.e. alive, by continually drawing from its environment negative
entropy – which is something very positive as we shall immediately see.
What an organism feeds upon is negative entropy. Or, to put it less
paradoxically, the essential thing in metabolism is that organism succeeds
in freeing itself from all the entropy it cannot help producing while

First, I certainly do not agree that everything that goes on in Nature
means an “increase of the entropy,” second, that living things “produce
positive entropy,” and finally that the only way it can keep alive is by
drawing *negative entropy* from its environment. I, of course realize that
such assertions have been made by numerous scientists. Unfortunately, none
of these can be justified in terms of the entropy and the Second Law. Such
statements, in my opinion are meaningless. Entropy, by definition, is a
positive quantity. There is no negative entropy, as there is no negative
volume, negative mass or negative time.

Did Schrödinger have a bad slip of the tongue in this statement? It seems
to me that Schrödinger did believe in what he said. It is unfortunate
however, that many others, scientists as well as non-scientists fell into
the pitfall created by Schrödinger’s negative entropy. On page 78
Schrödinger concludes that “organization is maintained by extracting order
from the environment.”

*“Living organism… delays the decay into thermodynamic equilibrium (death),
by feeding upon negative entropy, attracting a stream of negative entropy
upon itself… and to maintain itself on a stationary and fairly low entropy

Since there is no way of measuring or calculating the “entropy level” of a
living system, all these impressive statements are outright meaningless.
They certainly do not answer the question posed in the title of
Schrödinger’s book.

In concluding, Schrödinger’s book was no doubt a very influential one
especially in encouraging many physicists to look into biology. Most people
praised the book, but some expressed their doubts about its content.

Perhaps, the most famous skeptic of Schrödinger’s contribution to
understanding of life, was Linus Pauling. In Hager’s (1995) biography of
Linus Pauling, he wrote about Pauling’s view about Schrödinger’s book [17].

*“Pauling thought the book was hogwash. No one had ever demonstrated the
existence of anything like “negative entropy… Schrödinger’s discussion of
thermodynamics is vague and superficial… Schrödinger made no contribution
to our understanding of life.”*

I fully agree!

Likewise, Perutz had a similar criticism of Schrodinger’s book, in 1987)

*“When I was invited to review the influence of What is Life? I accepted
with the intention of doing honor to Schrodinger's memory. To my
disappointment, a close study of his book and of the related literature has
shown me that what was true in his book was not original, and most of what
was original was known not to be true even when it was written.”*

In conclusion, in my view both comments by Pauling and Perutz were quite
mild. Regarding the involvement of entropy and the Second Law, I feel that
Schrödinger has miserably gone astray. In general, I was disappointed with
his book. My main reason is not because Schrödinger did not offer an answer
to the question posed in the title of the book, but because whatever
partial answers he offered are at best unconvincing and perhaps even

I should also add one personal comment about the very idea of invoking
entropy and the Second Law in connection with life phenomena. Personally, I
believe that if ever a “*complete theory of life*” will be available, it
will involve neither entropy nor the Second Law of thermodynamics. In light
of this belief, I think that Schrödinger’s book has unintentionally
encouraged people in making a lot of meaningless statements associating
entropy and the Second Law with life phenomena.

*More on Entropy, the Second Law and life*

Open any book discussing the question of “What is Life?” and you are likely
to read grandiose statements ranging from “life violates the Second Law of
Thermodynamics,” to “life emerges from the Second Law,” and that the Second
Law explains many aspects of life, perhaps life itself.

The involvement of the Second Law in life is based on the misconstrued (I
would even say, perverted) interpretation of entropy as a measure of
disorder, on one hand, and the view that life is a process towards more
order, more structure, more organization, etc. on the other hand.

Combining these two erroneous views inevitably leads us to the association
of life phenomena with a *decrease* in entropy. This in turn leads to the
erroneous (perhaps meaningless) conclusion that life is a “struggle”
against the Second Law. I should add that even if the two assumptions were
correct, the conclusion will still be wrong! The fact is that entropy *cannot
be defined forany living system*, and the Second Law, in its entropy
formulation does not apply to living systems.

Here is an example from Katchalsky[19] in (1963):

 “*Life is a constant struggle against the tendency to produce entropy by
irreversible process. The synthesis of large and
information-rich-macromolecules…all these are powerful anti-entropic
force…living organism choose the least evil. They produce entropy at a
minimal rate by maintaining a steady state.”*

This is a beautiful statement but devoid of any meaning. No one knows how
to define the entropy of a living system, and how much entropy is produced
by a living organism.

Volkenstein [20], comments on the “anti-entropic” by saying:

*“At least we understand that life is not “antientropic,” a word bereft of
meaning. On the contrary, life exists because there is entropy, the export
of which supports biological processes…”*

Indeed “anti-entropic” is as meaningless as “anti-volume,” (see also
reference [2]). Unfortunately, Volkenstein’s statement is far more
meaningless than the concept of “anti-entropic.”

Here is another outstanding example:

In Atkins’ (1984) introduction to his book [11] he writes:

 “*In Chapter 8 we also saw how the Second Law accounts for the emergence
of the intricately ordered forms characteristic of life.”*

Of course, this is an unfulfilled promise. No one has ever shown that the
Second Law accounts for the emergence of… life! At the end of Chapter 7,
Atkins writes:

*“We shall see how chaos can run apparently against Nature, and achieve
that most unnatural of ends, life itself.”*

Finally, after discussing some aspects of processes in a living organism,
Atkins concludes his book:

 “*We are the children of chaos, and the deep structure of change is decay.
At root, there is only corruption, and the unstemmable tide of chaos… This
is the bleakness we have to accept as we peer deeply and dispassionately
into the heart of the universe.*

*Yet, when we look around and see beauty, when we look within and
experience consciousness, and when we participate in the delights of life,
we know in our hearts that the heart of the universe is richer by far.”*

So beautiful and so empty combination of words!

*Do we feed on negative entropy?*

Brillouin [21], “*feeding on the negative entropy*” ideas pronounced by
Schrödinger, goes even further and claims that:

 “*If living organism needs food, it is only for the negentropy it can get
from it, and which is needed to make up for the losses due to mechanical
work done, or simple degradation processes in living systems. Energy
contained in food does not really matter: Since energy is conserved and
never gets lost, but negentropy is the important factor.”*

This is quite strange. If this is the case, why do all food products
reflect caloric value on their labels? The food manufacturers should
instead print the “important factor” of negentropy in units of calories per
degree or perhaps in bits, on their labels. Thus, next time you look at the
labels on food products you can ignore the “energy value” as they are not
important. What matters and the only important information to watch out for
is the meaningless *negentropy*!

While I am still baffled with the concept of *negative entropy*, or its
shorter version *negentropy*, I was greatly relieved to read Hoffmann’s
[22] explanation:

 “*Life uses a low-entropy source of energy (food or sunlight) and locally
decreases entropy (created order by growing) at the cost of creating a lot
of high-entropy “waste energy (heat and chemical waste).”*

In more modern books the meaningless notion of negative entropy (or
neg-entropy) is replaced by the more meaningful term of *low entropy*.

Is it meaningful to claim that we, living organisms feed on low entropy

If you are convinced that feeding on low entropy food is the thing that
keeps you alive you should take your soup (as well as your coffee and tea)
as cold as possible. This will assure you of feeding on the lowest possible
liquid food. As for solid food, you should try to eat frozen food (but be
careful not to put anything at very low temperatures into your mouth,
that’s going to be very dangerous). As we have noted before, the entropy of
a living system is not defined – not yet, or perhaps never. The main reason
is that we do not know how to define the *state* of a living system.

In a recent book by Rovelli [23], the nonsensical idea that “entropy is
more important than energy is elevated to highest peak. You will find there
a statement written in all capital letters:


This very sentence has been praised by some of Rovelli’s reviewers. Here, I
will briefly say that the entropy of the universe (or the world) is not
definable. Therefore, entropy does not, and cannot drive the universe. In
fact, (yes, it is a fact) entropy does not drive anything, not even
processes in systems for which the entropy is defined.

Besides this nonsensical statement, Rovelli goes on to discuss the idea of
living beings feeding on low entropy. In another copycat statement which is
attributed to Schrödinger, he suggests something which I think is
deceiving, irresponsible and dangerous. On page 164 he writes:

*“If all we needed was energy rather than entropy, we would head for the
heat of the Sahara rather than toward our meal.”*

First, I think it is unfair (to say the least) to say “if all we *needed*
was energy.” No one needs *only* energy. We need energy, for certain, but
we also need some minerals, vitamins, and more than anything, water is
essential for our general well-being. For the sake of argument, suppose
that we already have everything, and all the rest we need is energy. But
then, the author suggests that one should head for the heat of the Sahara.

This comment is dangerous because the energy that we need is energy stored
in some chemical compounds, not the “heat of the Sahara.” If one were to
believe that energy is important (and assuming that all other things
including water, are available) then going to the Sahara instead of having
the next meal, will kill you, so better not to heed the Rovelli’s advice.

Besides, the danger of the author’s suggestion is also an absurd one. As I
wrote above if you believe that entropy is more important than the energy
of food, then you should drink water as cold as possible (preferably iced)
which has a lower entropy than hot water. To paraphrase the author’s
suggestion (not to be taken seriously), I would say that if all we need is
entropy rather than energy, we should head for the cold arctic rather than
towards our next meal. I repeat that this is just to paraphrase the
author’s statement. I am not really suggesting that you do it.

If you swallow a cube of ice at 0 , or drink the equivalent amount of
liquid water at 0 , you will get the same benefit from the water molecules.
If you have a choice between the two options I recommend drinking water
(with a higher entropy) rather than the ice (with the lower entropy), not
because of the entropy difference between the two, but simply because the
latter might get stuck in your throat.

To conclude this section, it should be stressed that my objection to the
usage of entropy and the Second Law applies to the *entire* living system
and the whole life phenomena. There is no objection to studying specific
chemical, mechanical, or electrical processes occurring within a living
system. However, phenomena involving mental or conscious activities cannot
be included in such process.

*Some concluding remarks on Entropy, the Second Law and Life*

A great deal of knowledge (or information) has been accumulated on many
aspects of life. Yet, there is one aspect of life which is elusive and that
is, life itself. We do not know how to define life, how life was created
and whether or not life succumbs to the laws of physics. Specifically, we
do not know how to describe the state of being “alive,” for any living
organism. We can tell when something is alive or not alive, but we cannot
specify these states in any of the available physical terms. Therefore,
there is no point of applying the concept of *entropy*, or of the Second
Law to a living system.

We can still apply the concept of information both in its colloquial sense,
and in its informational theoretical sense. In spite of many claims in the
literature, the *information* we have about life is in general, not
measurable. On the other hand, we can use the Shannon Measure of
information (SMI) to many probability distributions associated with living
systems. We can define the probability distribution of compounds in a cell,
in an organ, or in the entire organism. We can assign distribution to the
letters in the DNA or the letters of proteins, and so on. To each of these
distributions we can define the corresponding SMI. All these SMI are
well-defined quantities but they are not entropy. Entropy, when viewed as a
particular case of a SMI is defined for a specific distribution at a
specific state of equilibrium. We know that a living system is not an
equilibrium state. We do not know whether a living system *tends* to an
equilibrium state, and whether it will ever reach an equilibrium state.
Therefore, as long as a living system is *alive*, it is meaningless to
apply to it the concept of entropy, nor the Second Law of thermodynamics.
It also follows that life does not violate the Second Law, nor does it
emerge from the Second Law. The Second Law does not apply to a living

At this stage of our knowledge of life we can be satisfied with applying
the SMI to well specified distribution functions associated with a living

Unfortunately, we do not know whether or not the SMI or information theory
can be applied to life itself. Certainly, it cannot be applied to explain
aspects of life that are far from being understood such as consciousness,
thoughts, feelings, creativity, etc. Yet again, statements claiming that
information theory can help us with the comprehension of these aspects of
life abound in the literature. These statements are no doubt very
impressive, but unfortunately they are far from being true.


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Institute for Advanced Studies at Trinity College, Dublin, in February 1943

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Pedro C. Marijuán
Grupo de Bioinformación / Bioinformation Group
pcmarijuan.iacs at aragon.eshttp://sites.google.com/site/pedrocmarijuan/

Prof. Arieh Ben-Naim
Department of Physical Chemistry
The Hebrew University of Jerusalem
Jerusalem, 91904
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