<div dir="ltr">Dear Joseph,<div>You wrote:</div><div><span style="color:rgb(0,0,128);font-family:"Lucida Sans Unicode";font-size:16px">It is thus in opposition to the operation of the Second Law, which drives everything down-hill toward a featureless </span><i style="color:rgb(0,0,128);font-family:"Lucida Sans Unicode";font-size:16px">identity </i><span style="color:rgb(0,0,128);font-family:"Lucida Sans Unicode";font-size:16px">(heat death, perhaps with black holes on the way, doomed to evaporate).</span> </div><div> I do not agree with this common view of the Second Law. This is explained in my book on Entropy: the Greatest Blunder...</div><div>The Second Law does not drive everything downhill...</div><div>Best</div><div>Arieh<br><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">---------- Forwarded message ---------<br>From: <strong class="gmail_sendername" dir="auto">Joseph Brenner</strong> <span dir="auto"><<a href="mailto:joe.brenner@bluewin.ch">joe.brenner@bluewin.ch</a>></span><br>Date: Tue, Jan 5, 2021 at 5:47 PM<br>Subject: RE: [Fis] Fwd: Fwd: Entropy, the Second Law, and Life (OFF -LINE ). The Pauli Exclusion Principle<br>To: Arieh Ben-Naim <<a href="mailto:ariehbennaim@gmail.com">ariehbennaim@gmail.com</a>><br></div><br><br>
<u></u>
<u></u>
<u></u>
<u></u>
<u></u>
<u></u>
<div lang="FR" link="blue" vlink="blue">
<div class="m_9208343107615574449Section1">
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">Dear Arieh,<u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy"><u></u> <u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">I have been working with Pedro Marijuan and the FIS for over 12
years. I was also a chemist, an organic one, and I recycled myself after
retirement from Du Pont in logic, information, and philosophy. I have been
trying to mine the logical philosophy of the Franco-Romanian thinker Stéphane
Lupasco (Bucharest, 1900 – Paris, 1988), working also with <u></u>Basarab Nicolescu<u></u>, Professor (em.) of Theoretical
Physics at the <u></u><u></u>University<u></u>
of <u></u>Paris VI<u></u><u></u>. My second
book, <i><span style="font-style:italic">Philosophy in Reality; a New Book of
Changes</span></i>, was published in the Springer SAPERE Series in December.<u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy"><u></u> <u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">Based on what I have just read, I am looking forward to reading
your Chapter and your book. However, since I believe in being as focused as
possible on this List, based on common understanding of some basics, I limit
myself in this first note to one specific question: in your approach, do you
assign hermeneutic character to the operation of the Pauli Exclusion Principle?
Lupasco saw the existence of the Principle as the origin or source of <i><span style="font-style:italic">diversity </span></i>and emergence of new complex
entities in the universe. It is thus in opposition to the operation of the
Second Law, which drives everything down-hill toward a featureless <i><span style="font-style:italic">identity </span></i>(heat death, perhaps with black
holes on the way, doomed to evaporate). <u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy"><u></u> <u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">The dialectic, as I am sure you see, is ubiquitous in nature,
instantiated in the other dualities of physics. I look forward to your answer
which I would of course ‘process’ and circulate to the group.<u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy"><u></u> <u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">Best wishes,<u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">Joseph<u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy"><u></u> <u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">Joseph E. Brenner, Ph. D. University of Wisconsin 1958 (ugh). <u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">Associate Director, <u></u>International<u></u>
<u></u>Center<u></u> for the Philosophy of
Information, <u></u>Xi’An<u></u> Jiaotong (Social Sciences)
University, <u></u>Xi’An<u></u>, <u></u><u></u>China<u></u><u></u><u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy">Vice-President Transdisciplinary Projects, International Society
for Study of Information (IS4IS), <u></u>Technical<u></u>
<u></u>University<u></u> of <u></u><u></u>Vienna<u></u><u></u> <u></u><u></u></span></font></p>
<p class="MsoNormal"><font size="3" color="navy" face="Lucida Sans Unicode"><span lang="EN-GB" style="font-size:12.0pt;font-family:"Lucida Sans Unicode";color:navy"><u></u> <u></u></span></font></p>
<p class="MsoNormal"><font size="2" color="navy" face="Arial"><span lang="EN-GB" style="font-size:10.0pt;font-family:Arial;color:navy"><u></u> <u></u></span></font></p>
<p class="MsoNormal"><font size="2" color="navy" face="Arial"><span lang="EN-GB" style="font-size:10.0pt;font-family:Arial;color:navy"><u></u> <u></u></span></font></p>
<div>
<div class="MsoNormal" align="center" style="text-align:center"><font size="3" face="Times New Roman"><span lang="EN-US" style="font-size:12.0pt">
<hr size="2" width="100%" align="center">
</span></font></div>
<p class="MsoNormal"><b><font size="2" face="Tahoma"><span lang="EN-US" style="font-size:10.0pt;font-family:Tahoma;font-weight:bold">From:</span></font></b><font size="2" face="Tahoma"><span lang="EN-US" style="font-size:10.0pt;font-family:Tahoma">
Fis [mailto:<a href="mailto:fis-bounces@listas.unizar.es" target="_blank">fis-bounces@listas.unizar.es</a>] <b><span style="font-weight:bold">On
Behalf Of </span></b>Arieh Ben-Naim<br>
<b><span style="font-weight:bold">Sent:</span></b> mardi, 5 janvier 2021 14:50<br>
<b><span style="font-weight:bold">To:</span></b> <a href="mailto:fis@listas.unizar.es" target="_blank">fis@listas.unizar.es</a><br>
<b><span style="font-weight:bold">Subject:</span></b> [Fis] Fwd: Fwd: Entropy,
the Second Law, and Life</span></font><span lang="EN-US"><u></u><u></u></span></p>
</div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
</div>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">---------- Forwarded message ---------<br>
From: Arieh Ben-Naim <<a href="mailto:ariehbennaim@gmail.com" target="_blank">ariehbennaim@gmail.com</a>><br>
Date: Tue, Jan 5, 2021 at 15:49<br>
Subject: Re: [Fis] Fwd: Entropy, the Second Law, and Life<br>
To: <<a href="mailto:karl.javorszky@gmail.com" target="_blank">karl.javorszky@gmail.com</a>><u></u><u></u></span></font></p>
</div>
<p class="MsoNormal" style="margin-bottom:12.0pt"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Dear Karl,<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">May I correct your “tacheles” into “ tachles” or better “
takhles”<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">the origin of which is in Hebrew “tachlit” ( תכלית) which
was used in Yiddish as “tachles” or better “takhles”<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">You are right in its meaning “to get to the point or to the bottom
line”<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Best<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Arieh<u></u><u></u></span></font></p>
</div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">On Tue, Jan 5, 2021 at 15:10 Karl Javorszky <<a href="mailto:karl.javorszky@gmail.com" target="_blank">karl.javorszky@gmail.com</a>> wrote:<u></u><u></u></span></font></p>
</div>
<blockquote style="border:none;border-left:solid #cccccc 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-right:0cm">
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Dear Arieh,<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">my auto-correct had mis-spelled your name, which I didn't notice.
Please excuse the inadvertent mistake.<u></u><u></u></span></font></p>
</div>
</div>
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Karl<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
</div>
</div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Am Di., 5. Jan. 2021 um 13:14 Uhr schrieb Arieh Ben-Naim <<a href="mailto:ariehbennaim@gmail.com" target="_blank">ariehbennaim@gmail.com</a>>:<u></u><u></u></span></font></p>
</div>
<blockquote style="border:none;border-left:solid #cccccc 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-right:0cm">
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Dear Michel, and whoever interested,<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">I fully agree that there is no definition of order for a thermodynamic
system.<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">However, even if there was a definition it will not be relevant to
entropy (except for on “definition” by Callen, who “defines” disorder by
Shannon measure of information.<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">I have criticized this “definition” in several of my books,
specifically in Entropy: The Greatest Blunder in the History of Science).<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">In addition, and independently of the availability of a definition to
order, or disorder, I do not think one can apply the concept of Entropy, or the
Second Law to living organisms <u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Best wishes and a happy new year.<u></u><u></u></span></font></p>
</div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Arieh<u></u><u></u></span></font></p>
</div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt"><u></u> <u></u></span></font></p>
<div>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">On Mon, Jan 4, 2021 at 21:47 Michel Petitjean <<a href="mailto:petitjean.chiral@gmail.com" target="_blank">petitjean.chiral@gmail.com</a>>
wrote:<u></u><u></u></span></font></p>
</div>
<blockquote style="border:none;border-left:solid #cccccc 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-right:0cm">
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Dear Arieh,<br>
<br>
Many thanks for your great contribution.<br>
Eisnstein was right, the framework of applicability is the main point.<br>
And it is often neglecetd, about the universe, and about life.<br>
We can find in many books about thermodynamics a definition of entropy.<br>
I don't discuss it, even if there is much to say about it.<br>
An important point is, what is disorder?<br>
Does anybody can show me a definition of disorder?<br>
May be the negation of order.<br>
But what is order ?<br>
Not the mathematical definition in set theory (partial order, total<br>
order), it seems to be something else.<br>
Does anybody can show me a definition of order?<br>
If somebody can tell me what is order or what is disorder<br>
(microstates? randomness? else?), my next question follows.<br>
Is it possible to define what is the maximal order (or minimal<br>
disorder), and conversely, what is the minimal order (or maximal<br>
disorder)?<br>
Once we get clear concepts in simple situations, we may discuss their<br>
application to life.<br>
If the concepts are unclear in simple situations, disussing their<br>
application to life will never end.<br>
<br>
Best regards,<br>
<br>
Michel.<br>
<br>
Michel Petitjean<br>
Université de Paris, BFA, CNRS UMR 8251, INSERM ERL U1133, F-75013 Paris,
France<br>
Phone: +331 5727 8434; Fax: +331 5727 8372<br>
E-mail: <a href="mailto:petitjean.chiral@gmail.com" target="_blank">petitjean.chiral@gmail.com</a>
(preferred),<br>
<a href="mailto:michel.petitjean@univ-paris-diderot.fr" target="_blank">michel.petitjean@univ-paris-diderot.fr</a><br>
<a href="http://petitjeanmichel.free.fr/itoweb.petitjean.html" target="_blank">http://petitjeanmichel.free.fr/itoweb.petitjean.html</a><br>
<br>
<br>
Le lun. 4 janv. 2021 à 19:03, Arieh Ben-Naim <<a href="mailto:ariehbennaim@gmail.com" target="_blank">ariehbennaim@gmail.com</a>>
a écrit :<br>
><br>
><br>
><br>
> ---------- Forwarded message ---------<br>
> From: Pedro C. Marijuan <<a href="mailto:pcmarijuan.iacs@aragon.es" target="_blank">pcmarijuan.iacs@aragon.es</a>><br>
> Date: Sat, Jan 2, 2021 at 8:57 PM<br>
> Subject: Entropy, the Second Law, and Life<br>
> To: Arieh Ben-Naim <<a href="mailto:ariehbennaim@gmail.com" target="_blank">ariehbennaim@gmail.com</a>><br>
><br>
><br>
> Dear FIS Discussants,<br>
><br>
> 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).<br>
><br>
> Best wishes<br>
><br>
> Arieh<br>
><br>
>
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------<br>
><br>
> Entropy, the Second Law, and Life<br>
><br>
> Arieh Ben-Naim<br>
><br>
> Emeritus Professor, Department of Physical Chemistry, The Hebrew
University of Jerusalem<br>
><br>
><br>
><br>
> Introduction<br>
><br>
> I would like to start this article with a quotation by Albert Einstein on
thermodynamics:<br>
><br>
> “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.”<br>
><br>
> 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].<br>
><br>
> The application of entropy and the Second Law to a living system is based
on two erroneous assumptions:<br>
><br>
> 1. Entropy is a measure of disorder (or disorganization)<br>
><br>
> Life is understood as a process towards organization and creation of order<br>
><br>
> From these two assumptions it follows, almost naturally that
life-processes seem to be “a struggle against the Second Law of
Thermodynamics.”<br>
><br>
> 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!<br>
><br>
> Can entropy be defined for any living system?<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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 itself.<br>
><br>
> 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 equilibrium.<br>
><br>
> 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.<br>
><br>
> 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].<br>
><br>
> The history of application of Entropy and the Second Law to living systems<br>
><br>
> Perhaps the oldest association of Second Law with life is due to
Boltzmann. On May 29, 1886, Ludwig Boltzmann presented a talk at the Festive
Session of the Imperial Academy of Sciences in Vienna where he discussed “The
Second Law of Thermodynamics” with special emphasis on its application in
relation to Charles Darwin's 1859 theory of evolution [16].<br>
><br>
> The most-quoted passage from this lecture is that life is a struggle for
entropy:<br>
><br>
> “The general struggle for existence of animate beings is not struggle for
raw materials, these, for organisms, are air, water and soil, all abundantly
available, nor for energy, which exists in plenty in anybody in the form of
heat Q, but of a struggle for entropy, which becomes available through the
transition of energy from the hot sun to the cold earth.”<br>
><br>
> 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 is a “struggle for entropy”<br>
><br>
> 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.<br>
><br>
> Schrödinger’s book: What is life?<br>
><br>
> On the question: “What is life? one cannot avoid starting with the most
famous book written by Schrödinger [3].<br>
><br>
> 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].<br>
><br>
> 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:<br>
><br>
> “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?”<br>
><br>
> His preliminary answer to this question:<br>
><br>
> “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.”<br>
><br>
> 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.<br>
><br>
> On page 10 Schrödinger provides some hints about his intention to use the
Second Law:<br>
><br>
> “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 accuracy.”<br>
><br>
> 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.<br>
><br>
> 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.”<br>
><br>
> “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.”<br>
><br>
> 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:<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> “The general principle involved is the famous Second Law of
Thermodynamics (entropy principle) and its equally famous statistical
foundation.”<br>
><br>
> His main claim is that “living matter evades the decay to equilibrium.”<br>
><br>
> “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.”<br>
><br>
> Then he asks:<br>
><br>
> “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.”<br>
><br>
> I believe that the book’s highlight is reflected on page 76:<br>
><br>
> “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 alive.”<br>
><br>
> 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.<br>
><br>
> 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.”<br>
><br>
> “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 level.”<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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].<br>
><br>
> “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.”<br>
><br>
> I fully agree!<br>
><br>
> Likewise, Perutz had a similar criticism of Schrodinger’s book, in 1987)
[18]:<br>
><br>
> “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.”<br>
><br>
> 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 meaningless.<br>
><br>
> 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.<br>
><br>
> More on Entropy, the Second Law and life<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> Here is an example from Katchalsky[19] in (1963):<br>
><br>
> “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.”<br>
><br>
> 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.<br>
><br>
> Volkenstein [20], comments on the “anti-entropic” by saying:<br>
><br>
> “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…”<br>
><br>
> 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.”<br>
><br>
> Here is another outstanding example:<br>
><br>
> In Atkins’ (1984) introduction to his book [11] he writes:<br>
><br>
> “In Chapter 8 we also saw how the Second Law accounts for the
emergence of the intricately ordered forms characteristic of life.”<br>
><br>
> 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:<br>
><br>
> “We shall see how chaos can run apparently against Nature, and achieve
that most unnatural of ends, life itself.”<br>
><br>
> Finally, after discussing some aspects of processes in a living organism,
Atkins concludes his book:<br>
><br>
> “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.<br>
><br>
> 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.”<br>
><br>
> So beautiful and so empty combination of words!<br>
><br>
> Do we feed on negative entropy?<br>
><br>
> Brillouin [21], “feeding on the negative entropy” ideas pronounced by
Schrödinger, goes even further and claims that:<br>
><br>
> “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.”<br>
><br>
> 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!<br>
><br>
> 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:<br>
><br>
> “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).”<br>
><br>
> In more modern books the meaningless notion of negative entropy (or
neg-entropy) is replaced by the more meaningful term of low entropy.<br>
><br>
> Is it meaningful to claim that we, living organisms feed on low entropy
food?<br>
><br>
> 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.<br>
><br>
> 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:<br>
><br>
> “IT IS ENTROPY, NOT ENERGY THAT DRIVES THE WORLD”<br>
><br>
> 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.<br>
><br>
> 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:<br>
><br>
> “If all we needed was energy rather than entropy, we would head for the
heat of the Sahara rather than toward our meal.”<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> Some concluding remarks on Entropy, the Second Law and Life<br>
><br>
> 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.<br>
><br>
> 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 system.<br>
><br>
> 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
system.<br>
><br>
> 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.<br>
><br>
><br>
><br>
> References<br>
><br>
> 1. Ben-Naim, A. (2016), Entropy the Truth the Whole Truth and Nothing but
the Truth, World Scientific Publishing, Singapore<br>
><br>
> 2. Ben-Naim, A. (2020), The Greatest Blunder in the History of
Science, involving Entropy, Time, Life and the Universe. Independently
Publisher, Amazon.<br>
><br>
> 3. Schrödinger (1944), What Is Life? : The Physical Aspect of the Living
Cell, Based on lectures delivered under the auspices of the Dublin Institute
for Advanced Studies at Trinity College, Dublin, in February 1943<br>
><br>
> 4. Penrose, R. (1989), The Emperor’s Mind. Concerning Computers, Minds and
the Law of Physics, Penguin Books, New York<br>
><br>
> 5. Penrose, R. (1994)Penrose, R. (1994), Shadows of the Mind: An Approach
to the Missing Science of Consciousness, Oxford University Press, Oxford<br>
><br>
> 6. Crick, F. (1994), “The Astonishing Hypothesis,” The Scientific Search
For the Soul,” Touchstone, Simon and Shuster, New York<br>
><br>
> 7. Dennett, D. (2017), “From Bacteria to Bach and Back,” The Evolution of
Minds,” W. W. Norton, Inc. USA, Henry Holt and Co., New York (2018)<br>
><br>
> 8. Ben-Naim, A. (2017), The Four Laws that do not drive the Universe,
World Scientific Publishing, Singapore<br>
><br>
> 9. Ben-Naim A. and Casadei D. (2017), Modern Thermodynamics, World Scientific
Publishing, Singapore<br>
><br>
> 10. Ben-Naim, A. (2018), Time’s Arrow?The Timeless Nature of Entropy and
the Second Law of Thermodynamics. Lulu Publishing Services<br>
><br>
> 11. Atkins, P. (1984),The Second Law, Scientific American Books, W. H.
Freeman and Co., New York<br>
><br>
> 12. Atkins, P. (2007), Four Laws That Drive The Universe, Oxford
University Press<br>
><br>
> 13. Brush, S. G. (1976), The Kind of Motion We Call Heat. A History of the
Kinetic Theory of Gases in The 19th Century, Book 2: Statistical Physics and
Irreversible Processes. North-Holland Publishing Company<br>
><br>
> 14. Brush, S. G. (1983), Statistical Physics and the Atomic Theory of
Matter, from Boyle and Newton to Landau and Onsager. Princeton University
Press, Princeton.<br>
><br>
> 15. Ben-Naim, A. (2020),Time for Everyone and Time for Everything,
Independent Publisher, Amazon<br>
><br>
> 16. Boltzmann, L. (1877), Vienna Academy. 42, “Gesammelte Werke” p. 193.<br>
><br>
> 17. Hager, T. (1995), Forces of Nature, the Life of Linus Pauling, Simon
and Schuster, New York<br>
><br>
> 18. Perutz, M.F. (1987),Physics and the riddle of life, Nature, 326,
555–558<br>
><br>
> 19. Katchalsky, A. (1963), Nonequilibrium Thermodynamics, Int. Sci.
Technol, 43<br>
><br>
> 20. Volkenstein, M. V. (2009), Entropy and Information, translated by A.
Shenitzer and A. G. Burns, Birkhauser, Berlin<br>
><br>
> 21. Brillouin, L. (1962), Science and Information Theory, Academic Press,
New York<br>
><br>
> 22. Hoffman, P.M. (2012), Life’s Ratchet, How Molecular Machines Extract
Order from Chaos, Basic Books, New York<br>
><br>
> 23. Rovelli, C. (2018) The Order of Time, Riverhead books, New York<br>
><br>
> 24. Styer, D.F. (2008), Entropy and Evolution, Am. Journal of Physics, 76,
1031<br>
><br>
> 25. Sanford, J. C. (2005), Genetic Entropy and the Mystery of the Genome,
Ivan Press, a division of Elim Publishin<br>
><br>
><br>
><br>
> --<br>
> -------------------------------------------------<br>
> Pedro C. Marijuán<br>
> Grupo de Bioinformación / Bioinformation Group<br>
><br>
> <a href="mailto:pcmarijuan.iacs@aragon.es" target="_blank">pcmarijuan.iacs@aragon.es</a><br>
> <a href="http://sites.google.com/site/pedrocmarijuan/" target="_blank">http://sites.google.com/site/pedrocmarijuan/</a><br>
> -------------------------------------------------<br>
><br>
><br>
><br>
> --<br>
> Prof. Arieh Ben-Naim<br>
> Department of Physical Chemistry<br>
> The Hebrew University of Jerusalem<br>
> Jerusalem, 91904<br>
> Israel<br>
> _______________________________________________<br>
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</blockquote>
</div>
</div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">-- <u></u><u></u></span></font></p>
<div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Prof. Arieh Ben-Naim<br>
Department of Physical Chemistry<br>
The Hebrew University of Jerusalem<br>
Jerusalem, 91904<br>
Israel<u></u><u></u></span></font></p>
</div>
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">_______________________________________________<br>
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<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">-- <u></u><u></u></span></font></p>
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<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Prof. Arieh Ben-Naim<br>
Department of Physical Chemistry<br>
The Hebrew University of Jerusalem<br>
Jerusalem, 91904<br>
Israel<u></u><u></u></span></font></p>
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<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">-- <u></u><u></u></span></font></p>
<div data-smartmail="gmail_signature">
<p class="MsoNormal"><font size="3" face="Times New Roman"><span style="font-size:12.0pt">Prof. Arieh Ben-Naim<br>
Department of Physical Chemistry<br>
The Hebrew University of Jerusalem<br>
Jerusalem, 91904<br>
Israel<u></u><u></u></span></font></p>
</div>
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</div><br clear="all"><div><br></div>-- <br><div dir="ltr" class="gmail_signature" data-smartmail="gmail_signature">Prof. Arieh Ben-Naim<br>Department of Physical Chemistry<br>The Hebrew University of Jerusalem<br>Jerusalem, 91904<br>Israel</div></div></div>