Dear Michel and Sung,
Your discussion is way above my head in the jargon and background
knowledge. Please bear with me while a non-mathematician tries to express
some observations that regard symmetry.
Two almost symmetrical spaces appear as Gestalts, expressed by numbers, if
one orders and reorders the expression *a+b=c. *One uses natural numbers –
in the range of 1..16 – to create a demo collection, which one then sorts
and re-sorts ad libitum / ad nauseam. The setup of the whole exercise does
not take longer than 1, max 2 hours. Then one can observe patterns.
The patterns here specifically referred to are two – almost – symmetrical
rectangular, orthogonal spaces. As these patterns are derived from simple
sorting operations on natural numbers, one can well argue that they
represent fundamental pictures.
The generating algorithm is 5 lines of code. Here it is.
*#d=16*
*begin outer loop, i:1,d*
*begin inner loop, j:i,d*
*append new record*
*write*
* a=i, b=j, c=a+b, k=b-2a, u=b-a, t=2b-3a,*
*q=a-2b, s=(d+1)-(a+b), w=2a-3b*
*end inner loop*
*end outer loop*
The next step is to *sequence* (sort, order) the rows. We use 2 sorting
criteria: as first, any one of {a,b,c,k,u,t,q,s,w}, and as 2nd sorting
criterium any of the remaining 8. This makes each of the 9 aspects of
*a+b=c* to be once a first, and once a second sorting key. We register the
linear sequential number of each element in a column for each of the 72
catalogued sorting orders..
Do you think the idea of symmetry is somehow connected to some very basic
truths of logic? Then maybe the small effort to create a database with 136
rows and 9+72 columns is possible.
The trick begins with the next step:
We go through the 72 sorting orders and re-sort from each of them into all
and each of the remaining 71. We register the sequential place of the
element in the order αβ while being resorted into order γδ. This gives each
element a value (a linear place, 1..136) “from” and a value “to”. The
element is given the attributes: Element: *a,b, *“Old Order”: αβ, from
place nr *i*, “New Order” γδ, to place nr. j. While doing this, one will
realise, that reorganisations happen by means of *cycles, *and will add
attributes : Cycle nr: *k, *Within cycle step nr:. *l.* This is simple
counting and using logical flags.
The cycles, that we have now arrived at, give a very useful skeleton for
any and all theories about order. You will find the two Euclid-type spaces
by filtering out those reorganisations that consist of 46 cycles, of which
45 have 3 elements in their corpus, where each of the 45 cycles has
Σa=18, Σb=33.
The two rectangular spaces – created by paths of elements during resorting
– are not quite symmetrical. As an outsider, I’d believe that there is
something to awake the natural curiosity of mathematicians.
Hoping to have caught your interest.
Karl
2018-05-07 15:06 GMT+02:00 Michel Petitjean <petitjean.chi...@gmail.com>:
> Dear Sung,
>
> The formula of the Planckian information in Table 1 is intriguing.
> The argument of the log_2 function was proposed in 1895 by Karl Pearson as
> a measure of asymmetry of a distribution (see [1], p. 370).
> In general the mean can be smaller than the mode (so the log cannot
> exist), but I assume that in your context that cannot happen.
> Also, I assume that this context excludes distributions such as a mixture
> of two well separated unit variance Gaussian laws, for which the mean is
> located at an antimode, and not at a mode.
>
> The skewness, which is also used as an asymmetry coefficient, is the
> reduced third order centered moment (may be positive or negative).
> The square of this latter quantity was also introduced by Karl Pearson as
> a measure of asymmetry of a distribution (see [1], p. 351).
>
> So, all these quantities are used as asymmetry measures.
>
> Two questions arise:
> 1. Has the Planckian information some relations with symmetry or asymmetry?
> If yes, which ones?
> That would not be shocking: Shu-Kun Lin (refs [2,3]) discussed about
> relations between information and symmetry.
> 2. The asymmetry measures above have a major drawback: a null value can be
> observed for some families of asymmetric distributions, and not only for
> symmetric distributions.
>
> In the case you indeed need to consider the log of a non negative quantity
> measuring the asymmetry of a distribution, which vanishes if and only if
> the distribution is symmetric, you may consider the chiral index \chi
> (section 2.9, ref [4]).
> \chi index takes values in [0..1] (in fact, in [0..1/2]) for univariate
> probability distributions, and it is null if and only if the distribution
> is symmetric.
> It has other properties, but that falls out of the scope of this
> discussion.
> Then, simply replace [ (\mu-mode) / \sigma ] by \chi as the argument of
> log_2.
>
> [1] Pearson, K.
> Contributions to the Mathematical Theory of Evolution,-II. Skew Variation
> in Homogeneous Material.
> Phil. Trans. Roy. Soc. London (A.), 1895, 186, 343-414.
>
> [2] Lin, S.K.
> Correlation of Entropy with Similarity and Symmetry.
> J. Chem. Inf. Comput. Sci. 1996, 36, 367--376
>
> [3] Lin, S.K.
> The Nature of the Chemical Process. 1. Symmetry Evolution –Revised
> Information Theory, Similarity Principle and Ugly Symmetry.
> Int. J. Mol. Sci. 2001, 2, 10--39
> (available in open access)
>
> [4] Petitjean, M.
> Chirality and Symmetry Measures: A Transdisciplinary Review.
> Entropy, 2003, 5[3], 271--312.
> (available in open access)
>
> Best regards,
>
> Michel.
>
> Michel Petitjean
> MTi, INSERM UMR-S 973, University Paris 7,
> CNRS SNC 9079
> 35 rue Helene Brion, 75205 Paris Cedex 13, France.
> Phone: +331 5727 8434; Fax: +331 5727 8372
> E-mail: petitjean.chi...@gmail.com (preferred),
> michel.petitj...@univ-paris-diderot.fr
> http://petitjeanmichel.free.fr/itoweb.petitjean.symmetry.html
>
>
> 2018-05-07 4:08 GMT+02:00 Sungchul Ji <s...@pharmacy.rutgers.edu>:
>
>> Hi FISers,
>>
>> I think information and energy are inseparable in reality. Hence to
>> understand what information is, it may be helpful to understand what energy
>> (and the associated concept of motion) is. In this spirit, I am forwarding
>> the following email that I wrote motivated by the lecture given by Dr.
>> Grossberg this afternoon at the 119th Statistical Mechanics Conference. In
>> *Table
>> 1* in the email, I divided particle motions studied in physics and
>> biology into three classes -- (i) *random*, (ii) *passive*, and (iii)
>> *active*, and identified the field of specialization wherein these
>> motions are studied as (i) *statistical mechanics*, (ii) *stochastic
>> mechanics*, and (iii) *info-statistical mechanics*. The last term was
>> coined by me in 2012 in [1]. I will be presenting a short talk (5
>> minutes) on* Info-statistical mechanics* on Wednesday, May 9, at the
>> above meeting. The abstract of the short talk is given below:
>>
>> Short talk to be presented at the *119th Statistical Mechanics
>> Conference*, Rutgers University, Piscataway, N.J., May 6-9, 2018).
>>
>>
>>
>> *Planckian Information** may be to Info-Statistical Mechanics what
>> Boltzmann Entropy is to Statistical Mechanics. *
>>
>> Sungchul Ji, Department of Pharmacology and Toxicology, Ernest Mario
>> School of Pharmacy, Rutgers University, Piscataway, N.J. 08854
>>
>> Traditionally, the dynamics of any N-particle systems in statistical
>> mechanics is completely described in terms of the 6-dimensional *phase
>> space* consisting of the 3N positional coordinates and 3N momenta, where
>> N is the number of particles in the system [1]. Unlike the particles dealt
>> with in statistical mechanics which are featureless and shapeless, the
>> particles in biology have characteristic shapes and internal structures
>> that determine their biological properties. The particles in physics
>> are completely described in terms of energy and matter in the phase space
>> but the description of the particles in living systems require not only the
>> energy and matter of the particle but also their genetic information,
>> consistent with the information-energy complementarity (or gnergy)
>> postulate discussed in [2, Section 2.3.2]. Thus, it seems necessary to
>> expand the dimensionality of the traditional phase space to accommodate the
>> *information
>> *dimension, which includes the three coordinates encoding the *amount *(in
>> bits), *meaning* (e.g., recognizability), and *value* (e.g., practical
>> effects) of information [2, Section 4.3]. Similar views were expressed by
>> Bellomo et al. [3] and Mamontov et al. [4]. The expanded “phase space”
>> would comprise the 6N phase space of traditional statistical mechanics plus
>> the 3N information space entailed by molecular biology. The new space
>> (to be called the “gnergy space”) composed of these two subspaces would
>> have 9N dimensions as indicated in Eq. (1). This equation also makes
>> contact with the concepts of *synchronic* and *diachronic* informations
>> discussed in [2, Section 4.5]. It was suggested therein that the
>> traditional 6N-dimensional phase space deals with the *synchronic
>> information* and hence was referred to as the *Synchronic Space* while
>> the 3N-dimensional information space is concerned with the consequences of
>> history and evolution encoded in each particle and thus was referred to as
>> the *Diachronic Space*. The resulting space was called the *gnergy
>> space* (since it encodes not only *energy* but also *information*).
>>
>>
>>
>> *Gnergy Space* = *6N-D Phase Space* + *3N-D Information
>> Space* (1)
>>
>> (*Synchronic Space*)
>> (*Diachronic
>> Space*)
>>
>>
>>
>> The study of both *energy* and *information* was defined as
>> “info-statistical mechanics” in 2012 [2, pp. 102-106, 297-301]. The
>> Planckian information of the second kind, IPS, [5] was defined as the
>> negative of the binary logarithm of the skewness of the long-tailed
>> histogram that fits the Planckian Distribution Equation (PDE) [6]. In
>> *Table
>> 1*, the Planckian information is compared to the Boltzmann entropy in
>> the context of the complexity theory of Weaver [8]. The inseparable
>> relation between *energy *and *information* that underlies
>> “info-statistical mechanics” may be expressed by the following aphorism:
>>
>>
>>
>> *“Information without energy is useless; Energy without information is
>> valueless.”*
>>
>>
>>
>> *Table 1.* A comparison between Planckian Information (of the second
>> kind) and Boltzmann entropy. Adopted from [6, Table 8.3].
>>
>> *Order*
>>
>> *Disorder*
>>
>> IPS = - log2 [(µ - mode)/σ]
>>
>>
>>
>> (2008-2018)
>>
>> S = k log W
>>
>>
>>
>> (1872-75)
>>
>> *Planckian Information *
>>
>> *Boltzmann entropy *[7]
>>
>> *Organized Complexity *[8]
>>
>> *Disorganized Complexity *[8]
>>
>> *Info-Statistical Mechanics* [2, pp. 102-106]
>>
>> *Statistical Mechanics *[1]
>>
>>
>>
>>
>>
>> *References:*
>>
>> [1] Tolman, R. C. (1979). *The Principles of Statistical Mechanics, *
>> Dover
>> Publications, Inc.,
>>
>> New York, pp. 42-46.
>>
>> [2] Ji, S. (2012) *Molecular Theory of the Living Cell: Concepts,
>> Molecular Mechanisms, and *
>>
>> *Biomedical Applications*. Springer, New York.
>>
>> [3] Bellomo, N., Bellouquid, A. and Harrero, M. A. (2007). From
>> microscopic to macroscopic
>>
>> descriptions of multicellular systems and biological growing tissues. *Comp.
>> Math. Applications*
>>
>> *53*: 647-663.
>>
>> [4] Mamontov, E., Psiuk-Maksymowitcz, K. and Koptioug, A. (2006).
>> Stochastic
>> mechanics
>>
>> in the context of the properties of living systems. *Math. Comp.
>> Modeling* *44*(7-8): 595-607.
>>
>> [5] Ji, S. (2018). Mathematical (*Quantitative*) and Cell Linguistic
>> (*Qualitative*) Evidence for
>>
>> *Hypermetabolic Pathways* as [SJ1] Potential Drug Targets. *J. Mol.
>> Genet. Medicine *(in press).
>>
>> [6] Ji, S. (2018). *The Cell Language Theory: Connecting Mind and
>> Matter.* World Scientific
>>
>> Publishing, New Jersey. Chapter 8.
>>
>> [7] Boltzmann distribution law. https://en.wikipedia.org/wiki/
>> Boltzmann_distribution.
>>
>> [8] Weaver, W. (1948) Science and Complexity. *American Scientist*
>> *36*:536-544.
>>
>>
>>
>>
>>
>> - - - - - - - - - - - - - -the Email to Dr. Grossberg
>> --------------------- - - - --- - -- - - - - - - - - - - - - -
>>
>> ------------------------------
>>
>> Hi Dr. Grossberg,
>>
>> Thank you for your thought-provoking lecture (entitled "From Sisyphus to
>> Boltzmann: an example of repulsive depletion interaction") this
>> afternoon at the 119th Statistical Mechanics Conference at Rutgers.
>>
>> Your lecture prompted me to construct the following table based on my
>> experience as a theoretical cell biologist over the last 4 decades as
>> summarized in [1, 2].
>>
>> According to *Table 1*, we can recognize three distinct kinds of *particle
>> motions* in physics and biology that may be studied in 3 distinct fields
>> of specialization, tentatively identified with (i) statistical mechanics,
>> (ii) stochastic mechanics, and (iii) what I recently referred to as
>> "info-statistical mechanics" [*1*,pp. 102-107; *2*, pp. 371-374].
>>
>>
>> *Table 1.* The trichotomy of particle motions in physics and biology
>>
>>
>> *Particle Motions **(**Discipline**)*
>>
>> *Random motion (1)*
>>
>> (*Statistical mechanics*)
>>
>> *Passive motion (2)*
>> (*Stochastic mechanics* ?)
>>
>> *Active motion (3)*
>>
>> (*Info-statistical mechanics* ?)
>>
>> [*1*, *2*]
>>
>> *Energy Source*
>>
>> homogeneous thermal environment
>>
>> external
>>
>> (e.g., magnetic field)
>>
>> internal
>> (e.g., chemical reactions)
>>
>> *Physics*
>>
>>
>>
>> dust particles in water
>>
>> at equilibrium
>>
>> dust particles in water in flow
>>
>> a piece of sodium metal in water at equilibrium
>>
>> Brownian motion
>>
>> e.g., Orenstein-Uhlenbeck process
>>
>> artificial molecular machines
>>
>> *Biology*
>>
>>
>>
>> dead bacteria in water at equilibrium
>>
>> live bacteria moving down a gradient
>>
>> live bacteria swimming against a gradient
>>
>> Brownian motion
>>
>> diffusion driven by gradient
>>
>> Chemotaxis driven by ATP hydroly
>>
>> - In your lecture today,. you referred to "active motions" of
>> inanimate particles, which seems to correspond to "passive motions" in the
>> above table, since the driving force for your particle motion is external
>> to the particle. If you wish to preserve the term "active motion", it
>> seems
>> to me necessary to differentiate between the "externally driven active
>> motion" and the "internally driven active motion". On the other hand, if
>> we adopt the terms,"active" vs. "passive" particle motions, we have a
>> precedence in biology where "active transport" (e.g., the Na/K pump) and
>> "passive transport" (e.g., the Na/Ca exchange channel) are well known. In
>> fact, the triadic classification of particle motions defined in *Table
>> 1* can be applied to ion movements across biomembranes with an equal
>> force.
>>
>>
>> If you have any questions or comments on the suggestions made in Table 1,
>> I would appreciate hearing from you.
>>
>> All the best.
>>
>> Sung
>>
>> *References:*
>>
>> [*1*] Ji, S. (2012). *Molecular Theory of the Living Cell: Concepts,
>> Molecular Mechanisms, and Biomedical Applications*. Springer, New York.
>>
>> [*2*] Ji, S. (2018). *The Cell Language Theory: Connecting Mind and
>> Matter*. World Scientific Publishing, New Jersey.
>>
>>
>> With all the best.
>>
>>
>> Sung
>>
>>
>> _______________________________________________
>> Fis mailing list
>> Fis@listas.unizar.es
>> http://listas.unizar.es/cgi-bin/mailman/listinfo/fis
>>
>>
>
> _______________________________________________
> Fis mailing list
> Fis@listas.unizar.es
> http://listas.unizar.es/cgi-bin/mailman/listinfo/fis
>
>
_______________________________________________
Fis mailing list
Fis@listas.unizar.es
http://listas.unizar.es/cgi-bin/mailman/listinfo/fis
