Beyond the Typical Set: Fluctuation Spectroscopy Cina Aghamohammadi - - PowerPoint PPT Presentation

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

An

ǫ ={(x1, x2, ..., xn) : 2−n(H(X)+ǫ)

≤ P(x1, x2, ..., xn) ≤ 2−n(H(X)−ǫ)} For large n, typical set is most probable, and the probability of each sequence in the typical set, An

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 hµ = H[X0|X:0] E = I[X:0; X0:] = I[S−; S+] rµ = H[X0|X:0, X1:] bµ = I[X0; X1:|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(hµL) and their probabilities decay as exp(−hµL). 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., Aβ)? If we could find a mapping that map our process T to new process Sβ in a way that it’s typical set be Aβ then we could calculate all the infor- mation measures for Sβ 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 Aβ and all the members of A∞ have the same decay rate for probability then all the members of the Aβ 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 ∈ Aℓ one associates an energy density: Uℓ

w := − log2 Pr(w)

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, Uℓ

w = Uℓ

• v. And so, in the set Uℓ =
• Uℓ

w : w ∈ Aℓ

, energy values may appear

• repeatedly. Let’s denote the frequency of equal Uℓ

ws by N(Uℓ w). Then, for the

thermodynamic macrostate at energy U, we define the thermodynamic entropy density: S(U) := lim

ℓ→∞

log2 N(Uℓ

w = U)

ℓ

Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits

The Map

The Map

• T (x)

β

• ij = eβ ln Pr(x|σi ) =
• Pr(x|σi)

β Tβ =

x∈A T (x) β

lβTβ = λβlβ, Tβrβ = λβrβ lβ · rβ = 1 We drove the correct mapping! Mβ : T → Sβ given by: (Sβ)ij = (Tβ)ij( rβ)j

• λβ(

rβ)i ,

• S(x)

β

• ij =
• T(x)

β

• ij(

rβ)j

• λβ(

rβ)i .

Introduction Resoloution Examples Large deviation Thermodynamic Classes in Process Space Beyond the limits

Biased Coin

0.7 0.8 0.9 1.0 1.1 1.2 1.3 U 0.0 0.2 0.4 0.6 0.8 1.0 S(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

0.60 0.65 0.70 0.75 0.80 0.85 0.90 U 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

S I

Large deviation rate (How each partition decay?): I(U) := limL→∞

• − log2 Pr(UL)

L

• It

could be shown that I(U) = U − S(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