Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Beyond the Typical Set: Fluctuation Spectroscopy Cina Aghamohammadi Complexity Sciences Center Department of Physics University of California, Davis Adviser: James P . Crutchfield June 2, 2015
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Typical Set A n ǫ = { ( x 1 , x 2 , ..., x n ) : 2 − n ( H ( X )+ ǫ ) ≤ P ( x 1 , x 2 , ..., x n ) ≤ 2 − n ( H ( X ) − ǫ ) } For large n , typical set is most probable, and the probability of each sequence in the typical set, A n ǫ , have almost the same value 2 − nH ( X ) .
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Information Measures All the information measures are defined on the typical set! Info Measures h µ = H [ X 0 | X : 0 ] E = I [ X : 0 ; X 0 : ] = I [ S − ; S + ] r µ = H [ X 0 | X : 0 , X 1 : ] b µ = I [ X 0 ; X 1 : | X : 0 ] But what about the non typical part? There are really rare. Events and their probabilities lying outside typical set are fluctuations or, sometimes, deviations.
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Non Typical Sets ∼ Rare events Goal: we want to have information measures for all of the parts of the whole sets. What is the meaning of that? For example we know number of words in typical set grows as exp ( h µ L ) and their probabilities decay as exp ( − h µ L ) . we can ask a same question for other parts of the whole set.
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Idea: β mapping How to calculate information measures for a subset of A ∞ (e.g., A β )? If we could find a mapping that map our process T to new process S β in a way that it’s typical set be A β then we could calculate all the infor- mation measures for S β and that gives us the answer.
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Partitioning the whole set. How? because we map A ∞ to A β and all the members of A ∞ have the same decay rate for probability then all the members of the A β should have the same decay rate too. so? We put all the words with same decay rate of probability in a same partition and label that partition with β
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Fluctuation spectroscopy To each word w ∈ A ℓ one associates an energy density: w := − log 2 Pr ( w ) U ℓ , ℓ mirroring the Boltzmann weight common in statistical physics: Pr ( w ) ∝ e − U ( w ) . Naturally, different words w and v may lead to same energy density, � w : w ∈ A ℓ � v . And so, in the set U ℓ = U ℓ w = U ℓ U ℓ , energy values may appear repeatedly. Let’s denote the frequency of equal U ℓ w s by N ( U ℓ w ) . Then, for the thermodynamic macrostate at energy U , we define the thermodynamic entropy density : log 2 N ( U ℓ w = U ) S ( U ) := lim ℓ ℓ →∞
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits The Map The Map � � � � β ij = e β ln Pr ( x | σ i ) = T ( x ) Pr ( x | σ i ) β T β = � x ∈A T ( x ) β l β T β = λ β l β , T β r β = λ β r β l β · r β = 1 We drove the correct mapping! M β : T → S β given by: ( S β ) ij = ( T β ) ij ( � r β ) j , � λ β ( � r β ) i � � T ( x ) ij ( � r β ) j � � β S ( x ) ij = . β � λ β ( � r β ) i
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Biased Coin 1.0 0.8 0.6 S ( U ) 0.4 0.2 0.0 0.7 0.8 0.9 1.0 1.1 1.2 1.3 U
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Nemo ∼ persistent symmetry
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits RRIP ∼ hidden symmetry
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Large deviation 0.8 S 0.7 I Large deviation rate (How 0.6 each partition decay?): 0.5 � � − log 2 Pr ( U L ) 0.4 I ( U ) := lim L →∞ L 0.3 It could be shown that 0.2 I ( U ) = U − S ( U ) 0.1 0.0 0.60 0.65 0.70 0.75 0.80 0.85 0.90 U
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Thermodynamic Classes in Process Space
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Infinite-State Processes
Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits Non Ergodicity
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