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Hierarchical Thermodynamics James P Crutchfield Complexity Sciences Center Physics Department University of California, Davis http://csc.ucdavis.edu/~chaos/ Workshop on Information Engines at the Frontier of Nanoscale Thermodynamics Telluride Science Research Center 3- August 27 Joint work with Cina Aghamohammadi, Alec Boyd, Dibyendu Mandal, Sarah Marzen, Paul Riechers, and Greg Wimsatt

Structural Hierarchy in Biology In what ways does nature organize? (Phenomenology) How does it organize? (Mechanism) Are these levels real or merely convenient? (Objectivity) Why does nature organize? (Optimization versus chance versus.) Does thermodynamics play a role? Physics Today, March 26

What s new in biology, but not in physics?

What s new in biology, but not in physics? Meaning, Purpose, & Functionality

What s new in biology, but not in physics? Meaning, Purpose, & Functionality Calculi of Emergence (992) but with energetics

What s new in biology, but not in physics? Meaning, Purpose, & Functionality Calculi of Emergence (992) but with energetics Punch Line Meaning, purpose, and functionality arise from Organization Thermodynamics

Problem Statement = Ecological population dynamics of structurally complex adapting agents + Reproduction (evolutionary population dynamics) JP Crutchfield, The Calculi of Emergence: Computation, Dynamics, and Induction, Physica D 75 (994) -54. In Proceedings of the Oji International Seminar: Complex Systems from Complex Dynamics to Artificial Reality 5-9 April 993, Numazu, Japan.

Agenda Level Thermodynamics Level Organization Level Thermo-Semantics Hierarchical Thermodynamics Hierarchical Organization

Level Thermodynamics Level Organization Level Thermo-Semantics Hierarchical Thermodynamics Hierarchical Organization

Thermodynamics of Organization: Information Processing Second Law of Thermodynamics (IPSL)

Information Ratchets Beyond Maxwell+Szilard: Net Work Extraction! Z N X N Thermal Reservoir T Q W Mass output string Ratchet input string. Y :N Output h µ Information Reservoir Y N: h µ Input A. Boyd, D. Mandal, and JPC, Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets. New Journal of Physics 8 (26) 2349. D. Mandal and C. Jarzynski. Work and information processing in a solvable model of Maxwell s demon. Proc. Natl. Acad. Sci. USA, 9(29):64 645, 22.

Information Ratchets Beyond Maxwell+Szilard: Net Work Extraction! Stochastic Turing Machine Z N X N Thermal Reservoir T Q W Mass output string Ratchet input string. Y :N Output h µ Information Reservoir Y N: h µ Input A. Boyd, D. Mandal, and JPC, Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets. New Journal of Physics 8 (26) 2349. D. Mandal and C. Jarzynski. Work and information processing in a solvable model of Maxwell s demon. Proc. Natl. Acad. Sci. USA, 9(29):64 645, 22.

Information Processing Second Law of thermodynamics Asymptotic IPSL: hw i apple k B T ln 2 h µ h µ Information is fuel: (Ordered inputs are a thermodynamic resource.) Generalizes Landauer Principle; cf.: Q erase k B T ln 2 (h µ = ; h µ = ) output input A. B. Boyd, D. Mandal, and JPC, Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets, New Journal of Physics 8 (26) 2349.

Information Processing Second Law of thermodynamics IPSL constrains information processing done by a thermodynamic system. Upper bound on the maximum average work hw i extracted per cycle. Lower bounds the amount hw i of input work required for a physical system to support a given rate of intrinsic computation.

Information Processing Second Law of thermodynamics IPSL determines Thermodynamic Functionality

Information Ratchets Second Law for Intrinsic Computation hw i apple k B T ln 2 h µ h µ Thermodynamic Functions Eraser Eraser p Dud Engine Engine q Eraser Eraser Ratchet parameters: (p, q)

Information Ratchets Second Law for Intrinsic Computation. H q.5. hw i kb T ln 2 hµ.4.4 versus hw i kb T ln 2.2 hµ H() H..2 -.2..5 p. Entropy Rates H() = H () H() hµ = hµ hµ Lesson: Kolmogorov-Sinai entropy rates (hµ and hµ ) account for all correlations.

Information Ratchets Second Law for Intrinsic Computation. H q.5. hw i kb T ln 2 hµ.4.4 versus hw i kb T ln 2.2 hµ H() H..2 -.2..5 Pure-correlation powered erasure p. Entropy Rates H() = H () H() hµ = hµ hµ Lesson: Kolmogorov-Sinai entropy rates (hµ and hµ ) account for all correlations.

Requisite Complexity Lessons: Information processing thermodynamic systems should match the complexity of their inputs/environment: Memoryless ratchets optimal for uncorrelated environments. Memoryful ratchets optimal for correlated environments. A. B. Boyd, D. Mandal, and JPC, Leveraging Environmental Correlations: The Thermodynamics of Requisite Variety, Journal of Statistical Physics (27) in press. arxiv.org:69.5353. W. Ross Asbhy, An Introduction to Cybernetics. John Wiley and Sons, New York, second edition, 96.

Functional Fluctuations J. P. Crutchfield and C. Aghamohammadi, Not All Fluctuations are Created Equal: Spontaneous Variations in Thermodynamic Function. arxiv.org:69.259.

Functional Fluctuations

Functional Fluctuations When is an Engine an Eraser?

Functional Fluctuations When is an Engine an Eraser? Information Processing Second Law + Large Deviation Theory

Fluctuations? Biased Coin (6% Heads/4% Tails) 5 l = l =2 l = 3 log 2 Pr( w :l ) log 2 Pr( w :l ) -3 5 l = 4 l = 5 l = 6-3 5 l = 7 l = 8 l = 9 log 2 Pr( w :l ) -3 w :l w :l w :l Distributions of length l words

Fluctuations? Biased Coin: Fluctuations in sequence probabilities.8.6 * l = 2.8.6.4.4.2. U.2. -log Pr( ) w :l 2.8.8.6.4 *.6.4..8.6.4 s l (U).2. w :l

Fluctuations in Intrinsic Computation Spectrum of Fluctuations.8.6.4 * l = 2.8.6.4 Large Deviations.2. U.2. -log Pr( ) w :l 2 Typical Set.8.6.4 *.8.6.4 Large Deviations..8.6.4 s l (U).2. w :l

Information Ratchets (A. Boyd, D. Mandal, and JPC, Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets, New Journal of Physics 8 (26) 2349.) Z N X N Thermal Reservoir Q W Mass output string Ratchet input string. Y :N Output h µ Information Reservoir Y N: h µ Input

Fluctuations in Thermodynamic Function h µ h µ h µ

Fluctuations in Thermodynamic Function h µ h µ h µ Informational Second Law Thermodynamic Function

Fluctuations in Thermodynamic Function h µ h µ h µ Information Ratchet + Fluctuation Spectroscopy + Informational Second Law (IPSL) Informational Second Law Thermodynamic Function

Fluctuations in Thermodynamic Function Observable? Length input sequences: Engine: 8% Dud: 7.8% Eraser: 2.2%

Fluctuations in Intrinsic Computation What s new: Determine S(U) and other measures for any structured process from

Lessons Function emerges from the Information Processing Second Law of Thermodynamics Computing fluctuates Function fluctuates

Level Thermodynamics Level Organization Level Thermo-Semantics Hierarchical Thermodynamics Hierarchical Organization

Information-Theoretic Analysis of Complex Systems... Process Pr( X, X ) is a communication channel X X from the past to the future : Past Present Future Channel

Information-Theoretic Analysis of Complex Systems... Process Pr( X, X ) is a communication channel X X from the past to the future : Past Present Future Information Rate h µ Channel Capacity C

Information-Theoretic Analysis of Complex Systems... Process Pr( X, X ) is a communication channel X X from the past to the future : Past Present Future Information Rate h µ Channel Capacity C Channel Utilization: Excess Entropy E = I[ X ; X ]

Foundations: Computational Mechanics A B Environment C µ = C... Sensory Data Pr(s ) 3 Agent Model ε-machine: Unique, minimal, & optimal predictor X 2S Causal Equivalence: x x, Pr(! X x )=Pr(! X x ) Structure versus Randomness B A 2/3 /3 Stored versus Generated Information Pr( ) log 2 Pr( ) versus h µ = X 2S /2 /2 Pr( ) X 2S Pr( ) log 2 Pr( ) C

Foundations: Computational Mechanics A B Environment C µ = C... Sensory Data Pr(s ) 3 Agent Model ε-machine: Unique, minimal, & optimal predictor X 2S Causal Equivalence: x x, Pr(! X x )=Pr(! X x ) B A 2/3 /3 Stored versus Generated Information Pr( ) log 2 Pr( ) versus h µ = X 2S /2 /2 Pr( ) X 2S Pr( ) log 2 Pr( ) C Intrinsic Computation:. How much historical information does a process store? 2. In what architecture is it stored? 3. How is it used to produce future behavior? J.P. Crutchfield & K. Young, Inferring Statistical Complexity, Physical Review Letters 63 (989) 5-8. J.P. Crutchfield, Between Order and Chaos, Nature Physics 8 (January 22) 7-24.

Computational Mechanics ε-machine: M = S, {T (x) : x A} Dynamic: Start State A 2 3 B Transient States T (x), = Pr(,x) 2 3 C 4 3 4, S 2 Recurrent States D

Varieties of ε-machine Denumerable Causal States Fractal J. P. Crutchfield, Calculi of Emergence: Computation, Dynamics, and Induction, Physica D 75 (994) -54. Continuous

Intrinsic Computation: Consequences A system is unpredictable if it has positive entropy rate: h µ > A system is complex if it has positive structural complexity measures: C µ > A system is emergent if its structural complexity increases over time: C µ (t ) >C µ (t), if t >t A system is hidden if its crypticity is positive: = C µ E >

What is a Level?

What is a Level? Pattern discovery: Learn the world s hidden states Pr(R X) Causal shielding: Pr(X X) = Pr(X R)Pr( X R) Search in the space of models: R M Objective function min I[X; R] + Pr(R X) I[X; X R] Model: Map from histories to states Info states contain about histories Reduce info history has about future Long history: N.H. Packard, J.P. Crutchfield, J.D. Farmer, R. S. Shaw, Geometry from a Time Series, Phys. Rev. Lett. 45 (98) 72-76. J. P. Crutchfield and B.S. McNamara, Equations of Motion from a Data Series, Complex Systems (987) 47-452. S. Still, C.J. Ellison, J.P. Crutchfield: arxiv.org: 78.654 [physics.gen-ph] & 78.58[cs.IT]

What is a Level? Optimal states where Pr opt (R X) = E(R, X) = D Pr( X R) = Pr(R) = Pr(R X) Pr(R) X Pr(R) Z( X, e ) are Gibbs distributions: E(R,X) Pr( X X) Pr( X R) X Pr( X X)Pr(R X)Pr(X) Pr(R X)Pr(X) Now, solve these self-consistently

What is a Level? Optimal balance structure & error At each level of approximation In theory Causal Rate Distortion Curve H[Past] εm limit R(D) I[Past;Rivals] Slope = inverse T = Beta Distortion IID limit I[Past;Future Rivals] = E - I[Future;Rivals] E = I[Past;Future]

What is a Level? Optimal balance structure & error At each level of approximation In theory Causal Rate Distortion Curve In practice: Learn an -state world H[Past] εm limit 3.5 R(D) I[Past;Rivals] Slope = inverse T = Beta I[ X 5 ;R] 3. 2.5 2..5..5. 6 states 5 states 4 states 3 states 2 states Infeasible Feasible P() P() 5 2 P() P( X x ) P() state Distortion IID limit I[Past;Future Rivals] = E - I[Future;Rivals] E = I[Past;Future]..2.4.6.8..2.4 I[ X 2 ; X 5 R]

Level Thermodynamics Level Organization Level Thermo-Semantics Hierarchical Thermodynamics Hierarchical Organization

Beyond Structure to Meaning, Purpose, & Functionality Semiotic Hierarchy of Information: Syntax Semantics Pragmatics Function For physical systems

Measurement Semantics

Measurement Semantics What does a particular measurement mean? A B C... α γ β δ System Instrument Process Modeller Measurement Channel

Measurement Semantics An ϵmcaptures Pattern : Measurement semantics: Prediction level What is the meaning of a particular measurement? Shannon says the amount of information is log 2 Pr(observing s) Given ϵm (assuming you re sync d): log 2 Pr(observing s)= log 2 Pr(S! s S )

Measurement Semantics An ϵmcaptures Pattern : Measurement semantics: Prediction level... Example: t = 2 3 4 5 6 7 8 9 s = At t = measure s = How much information does this give? H(s s =,s 9 =,...) h µ (.585 bits) Degree of observer s surprise (predictability) Does not say what the event s = means to the observer!

Measurement Semantics Meaning: Tension between representations of same event at different levels; e.g., Level is data stream and the event is a measurement Level 2 is the agent and the event updates it s model Degree of meaning of observing s 2 A Θ(s)= log 2 Pr(! s S) where S is the causal state to which s brings observer. Meaning content: State selected from anticipated palette.

Measurement Semantics Meaningless: Start state (all futures possible) Θ(s)= log 2 Pr(S )= log 2 = s = Action on disallowed transition: Reset to state of total ignorance (start state) Disallowed transition is meaningless. Meaningless measurements are informative, though: log 2 Pr(S! s S )= log 2 =

Measurement Semantics Theorem: h (s)i = C µ Average amount of meaning is the Statistical Complexity.

Thermodynamic Cost of Extracting Meaning

Thermodynamic Cost of Extracting Meaning Two cases: o In NESS o Out of NESS

Thermodynamic Cost of Extracting Meaning in NESS Learning about environment Agent predicts environment to leverage possible resources Z N X N Thermal Reservoir Q W Mass Output output string Ratchet input string. h µ Y :N Information Reservoir Y N: h µ Input (A. Boyd, D. Mandal, and JPC, Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets, New Journal of Physics 8 (26) 2349.)

Thermodynamic Cost of Extracting Meaning in NESS Thermo cost of implementation that predicts: Q implement min = k BT ln 2 I[S; Y ] = k B T ln 2 C µ Recall Theorem on Total Semantic Content h (s)i = C µ Agent memory about environment. Thermo-semantic cost: Q implement min = k BT ln 2 (s)

Thermodynamics of Meaning and Function beyond NESS

Thermodynamics of Meaning and Function beyond NESS Environment: Periodic with random phase slips Z N Thermal Reservoir output string Ratchet input string. Y :N Q X N W Mass Y N: Error Correction Engine +e Ratchet Functioning Dud/Eraser start D :.5 :.5 : c :c F : c E :c c

Thermodynamics of Meaning and Function beyond NESS Semiotics of Information Engines: Syntactic information = Measurements Semantic information = Envt l Phase, Sync/No Sync Functional information = Error Correction

Summary Level Thermodynamics Level Organization Level Semantics Hierarchical Thermodynamics Hierarchical Organization Thanks!

Thermodynamics of Organization Bibliography Functional computation & optimal prediction: A.B. Boyd, D. Mandal, and J.P. Crutchfield, Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets, New Journal of Physics 8 (26) 2349. Information creation: C. Aghamohammadi and J.P. Crutchfield, Thermodynamics of Random Number Generation, Physical Review E 95:6 (27) 6239. Synchronization & error correction: A.B. Boyd, D. Mandal, and J.P. Crutchfield, Correlation-powered Information Engines and the Thermodynamics of Self-Correction, Physical Review E 95: (27) 252. Homeostasis: A.B. Boyd, D. Mandal, and J.P. Crutchfield, Leveraging Environmental Correlations: Thermodynamics of Requisite Variety, Journal Statistical Physics 67:6 (27) 555-585. Functional fluctuations: J. P. Crutchfield and C. Aghamohammadi, Not All Fluctuations are Created Equal: Spontaneous Variations in Thermodynamic Function. arxiv.org:69.259. Controller structure: A.B. Boyd, D. Mandal, and J.P. Crutchfield, Transient Dissipation and Structural Costs of Physical Information Transduction, Physical Review Letters 8 (27) 2262. Intelligent control: A.B. Boyd and J.P. Crutchfield, Maxwell Demon Dynamics: Deterministic Chaos, the Szilard Map, and the Intelligence of Thermodynamic Systems, Physical Review Letters 6 (26) 96. NESS transitions: P.M. Riechers and J.P. Crutchfield, Fluctuations When Driving Between Nonequilibrium Steady States, Journal of Statistical Physics (27) in press. arxiv.org:6.9444. Sensors: S. E. Marzen and J. P. Crutchfield, Prediction and Power in Molecular Sensors: Uncertainty and Dissipation When Conditionally Markovian Channels Are Driven by Semi-Markov Environments, arxiv.org: 77.3962. Modularity: A.B. Boyd, D. Mandal, and J.P. Crutchfield, Above and Beyond the Landauer Bound: Thermodynamics of Modularity, arxiv.org, submitted.