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**Boltzmann, Shannon, and (Missing) Information**

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**Entropy of a gas. Entropy of a message. Information?**

Second Law of Thermodynamics. Entropy of a gas. Entropy of a message. Information?

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**B.B. (before Boltzmann): Carnot, Kelvin, Clausius, (19th c.)**

Second Law of Thermodynamics: The entropy of an isolated system never decreases. Entropy defined in terms of heat exchange: Change in entropy = (Heat absorbed)/(Absolute temp). (+ if absorbed, - if emitted). (Molecules unnecessary).

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**Q Hot (Th) Cold (Tc) Isolated system. Has some structure (ordered).**

Heat, Q, extracted from hot, same amount absorbed by cold – energy conserved, 1st Law. Entropy of hot decreases by Q/Th; entropy of cold increases by Q/Tc > Q/Th, 2d Law. In the fullness of time … Lukewarm No structure (no order).

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**Stuff releases heat q, gets more organized entropy decreases**

Paul’s entropy picture Sun releases heat Q at high temp entropy decreases Surroundings absorb q, gets more disorganized entropy increases … Living stuff absorb heat Q at lower temp larger entropy increases Overall, entropy increases

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**2d Law of Thermodynamics does not forbid emergence of local complexity (e.g., life, brain, …).**

2d Law of Thermodynamics does not require emergence of local complexity (e.g., life , brain, …).

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**Boltzmann (1872)) Entropy of a dilute gas.**

N molecules obeying Newtonian physics (time reversible). State of each molecule given by its position and momentum. Molecules may collide – i.e., transfer energy and momentum among each other. colliding

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**pk = fraction of particles whose positions and momenta are in bin k.**

Represent system in a space whose coordinates are positions and momenta = mv (phase space). momentum position Subdivide space into B bins. pk = fraction of particles whose positions and momenta are in bin k.

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**Build a histogram of the pk’s.**

pk’s change because of Motion Collisions External forces

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**Given the pk’s, how much information do you need to locate a molecule in phase space?**

All in 1 bin – highly structured, highly ordered no missing information, no uncertainty. Uniformly distributed – unstructured, disordered, random. maximum uncertainty, maximum missing information. In-between case intermediate amount of missing information (uncertainty) Any flattening of histogram (phase space landscape) increases uncertainty.

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Boltzmann: Amount of uncertainty, or missing information, or randomness, of the distribution of the pk’s, can be measured by HB = pk log(pk)

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**pk histogram revisited.**

All in 1 bin – highly structured, highly ordered HB = 0. Maximum HB. Uniformly distributed – unstructured, disorder, random. HB = - log B. Minimum HB. In-between case intermediate amount of missing information (uncertainty). In – between value of HB.

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**Boltzmann’s Famous H Theorem**

Define: HB = pklog(pk) Assume: Molecules obey Newton’s Laws of motion. Show: HB never increases. AHA! HB never decreases: behaves like entropy!! If it looks like a duck … Identify entropy with – HB : S = - kBHB Boltzmann’s constant

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**New version of Second Law:**

The phase space landscape either does not change or it becomes flatter. life? It may peak locally provided it flattens overall.

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Two “paradoxes” 1. Reversal (Loschmidt, Zermelo). Irreversible phenomena (2d Law, arrow of time) emerge from reversible molecular dynamics. (How can this be? – cf Tony Rothman).

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**2. Recurrence (Poincaré)**

2. Recurrence (Poincaré). Sooner or later, you are back where you started. (So, what does approach to equilibrium mean?) Graphic from: J. P. Crutchfield et al., “Chaos,” Sci. Am., Dec., 1986.

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Well … Interpret H theorem probabilistically. Boltzmann’s treatment of collisions is really probabilistic,…, molecular chaos, coarse-graining, indeterminacy – anticipating quantum mechanics? Entropy is probability of a macrostate – is it something that emerges in the transition from the micro to the macro? Poincaré recurrence time is really very, very long for real systems – longer than the age of the universe, even. Anyhow, entropy does not decrease! … on to Shannon

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**AB (After Boltzmann): Shannon (1949) Entropy of a message **

Message encoded in an alphabet of B symbols, e.g., English sentence (26 letters + space + punctuations) Morse code (dot, dash, space) DNA (A, T, G, C) pk = fraction of the time that symbol k occurs (~ probability that symbol k occurs).

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**pick a symbol – any symbol …**

Shannon’s problem: Want a quantity that measures missing information: how much information is needed to establish what the symbol is, or uncertainty about what the symbol is, or how many yes-no questions need to be asked to establish what the symbol is. Shannon’s answer: HS = - k pk log(pk) A positive number

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**Morse code example: All dots: p1 = 1, p2 = p3 = 0.**

Take any symbol – it’s a dot; no uncertainty, no question needed, no missing information, HS = 0. 50-50 chance that it’s a dot or a dash: p1 = p2 = ½, pk = 0. Given the p’s, need to ask one question (what question?), one piece of missing information, HS = log(2) = 0.69 Random: all symbols equally likely, p1 = p2 = p3 = 1/3. Given the p’s, need to ask as many as 2 questions -- 2 pieces of missing information, HS = log(3) = 1.1

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**Two comments: 1. It looks like a duck … but does it quack?**

There’s no H theorem for Shannon’s HS. 2. H is insensitive to meaning. Shannon: “ [The] semantic aspects of communication are irrelevant to the engineering problem.”

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**On H theorems: Q: What did Boltzmann have that Shannon didn’t?**

A: Newton (or equivalent dynamical rules for the evolution of the pk’s). Does Shannon have rules for how the pk’s evolve? In a communications system, the pk’s may change because of transmission errors. In genetics, is it mutation? Is the result always a flattening of the pk landscape, or an increase in missing information? Is Shannon’s HS just a metaphor? What about Maxwell’s demon?

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**On dynamical rules. Is a neuron like a refrigerator?**

Entropy of fridge decreases. Entropy of signal decreases.

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**General Electric designs refrigerators.**

The entropy of a refrigerator may increase, but it needs electricity. The entropy of the message passing through a neuron may increase, but it needs nutrients. General Electric designs refrigerators. Who designs neurons?

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**Same pk’s, same entropies – same “missing information.”**

Insensitive to meaning: Morse revisited X={… } H E L L O W O R L D Y={.- -… … } A B C D E F G H I J M Same pk’s, same entropies – same “missing information.”

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**If X and Y are separately scrambled – still same pk’s, same “missing **

information” – same entropy. The message is in the sequence? What do geneticists say? Information – as entropy – is not a very useful way to characterize the genetic code?

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**Do Boltzmann and Shannon mix?**

Boltzmann’s entropy of a gas, SB = - kB Spklog pk kB relates temperature to energy: E = kBT relates temperature of a gas to PV. Shannon’s entropy of a message, SS = - kSpklog pk k is some positive constant – no reason to be kB. Does SB + SS mean anything? Does the sum never decrease? Can an increase in one make up for a decrease in the other?

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**Maxwell’s demon yet once more.**

Demon measures velocity of molecule by bouncing light on it and absorbing reflected light; process transfers energy to demon; increases demon’s entropy – makes up for entropy decrease of gas.

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