HOW TO SELECT A STABLE DISTRIBUTION OF PROBABILITY IN A A PHYSICAL - PowerPoint PPT Presentation

EXPONENTIAL OR POWER LAW - HOW TO SELECT A STABLE DISTRIBUTION OF PROBABILITY IN A A PHYSICAL SYSTEM Andrea Di Vita D.I.C.C.A., Università di Genova, Italy The 4th International Electronic Conference on Entropy and Its Applications (ECEA 2017), 21 November – 1st December 2017; Sciforum Electronic Conference Series, Vol. 4, 2017

Boltzmann’s exponential and Gibbs ’ thermodynamics

𝑇 = − 𝑞 𝑙 ln 𝑞 𝑙 Gibbs ’ entropy (normalized to 𝑙 𝐶 ) 𝑙 Once properly maximized, leads to 𝑞 𝑙 ∝ 𝑓 −𝛾𝜁 𝑙 Boltzmann’s exponential distribution of microstates in canonical systems (generalized to grand-canonical …) 𝑥  ∝ 𝑓 ∆𝑇  The probability of a fluctuation of an arbitrary parameter  around a 𝑇 = max state follows Einstein’s formula −1 𝜖 2 𝑇 and the variance of such fluctuation is 𝜖  2

𝑇 𝐵 ′ + 𝐵′′ = 𝑇 𝐵′ + 𝑇 𝐵′′ Gibbs ’ entropy is additive: if A’, A’’ are independent systems then Then, it can be written as the sum of the Gibbs ’ 𝑇 = 𝜍𝑡 𝑒𝑊 entropies of all small mass elements the system is made of  a local entropy density 𝑡 exists (  mass density) 𝑡 = 𝑛𝑏𝑦 If Local Thermodynamical Equilibrium (LTE): (  Gibbs-Duhem)

If LTE holds everywhere and at all times: General Evolution Criterion (GEC) If 𝑢 → ∞ and the system relaxes to a stable, steady ( relaxed ) state with Boltzmann exponential distribution of microstates in all small mass elements at all times, then GEC rules relaxation regardless of detailed model Glansdorff et al. 1964, Di Vita 2010

q-exponential and Tsallis ’ thermodynamics

𝑇 𝑟 = − 𝑞 𝑙 𝑟 ln 𝑟 𝑞 𝑙 Tsallis ’ entropy (normalized to 𝑙 𝐶 ) 𝑙 Tsallis 1988, Tsallis et al. 1998 𝑟→1 𝑇 𝑟 = 𝑇 lim 𝑟→1 𝑚𝑜 𝑟 𝑦 = 𝑚𝑜 (𝑦) lim Once properly maximized, leads to q-exponential distribution of microstates 𝑞 𝑙 ∝ 𝑓𝑦𝑞 𝑟−𝛾𝜁 𝑙 In canonical systems (generalized to grand- canonical …)  power law with exponent 1 1 − 𝑟 𝑟→1 𝑓𝑦𝑞 𝑟 𝑦 = 𝑓𝑦𝑞 (𝑦) lim What about 𝒓 ?

Tsallis ’ entropy is nonadditive: 𝑇 𝑟 𝐵 ′ + 𝐵′′ = 𝑇 𝑟 𝐵′ + 𝑇 𝑟 𝐵′′ + 1 − 𝑟 𝑇 𝑟 𝐵′ 𝑇 𝑟 𝐵′′ Then, it can not be written as the sum of the Gibbs ’ entropies of all small mass elements the system is made of  no local entropy density 𝑡 exists  NO LTE  NO GEC  only model-dependent info on relaxation (as 𝑢 → ∞ ) with power-law distribution of microstates What about 𝒓 ?

NLFP

E.g.: q-dependent, possibly nonlinear, 1D Fokker-Planck (NLFP) equation for continuous distribution function 𝑄 𝑦, 𝑢 , where a force 𝐵 = 𝐵 𝑦 is counteracted by a diffusion process, represented by a diffusion coefficient 𝐸 𝐵 = 𝐵 𝑦 Casas et al. 2012, Wedemann et al. 2016 What about 𝒓 ?

NFLP for 𝐾 = 0  Steady state solution is just the q-exponential:  is max. when the solution is Haubold et al. 2004, Ribeiro et al. 2012, Wedemann et al. 2016 What about 𝒓 ?

NFLP for 𝐾 ≠ 0  An H-theorem holds (provided that 𝐵 𝑦 is well-behaved at ∞ )  Relaxation does occur! Casas et al. 2012 What about 𝒓 ?

NFLP for 𝐾 ≠ 0  If 𝐾 ≠ 0 then: Casas et al. 2012 is the amount (> 0) of which is produced inside the bulk of the system in a time interval 𝑒𝑢 . is the amount of which is exchanged with the external world in 𝑒𝑢 .  𝐾 represents the interaction with the external world; in isolated system 𝐾 = 0 (its value is a boundary condition on NLFP) If a perturbation leaves the latter interaction unaffected then the increment 𝑒𝑇 𝑟′ of 𝑇 𝑟′ in a time interval 𝑒𝑢 is 𝑒𝑇 𝑟′ = 𝑒𝑢 ∙ What about 𝒓 ?

Mapping Tsallis onto Gibbs

A monotonically increasing, additive function of 𝑇 𝑟 exists even for 𝑟 ≠ 1 !! Tsallis 1988, Abe 2001, Vives et al. 2002 These facts have a lot of consequences … 

= 𝑛𝑏𝑦 if and only if 𝑇 𝑟 = 𝑛𝑏𝑦  𝑇 𝑟 …  Moreover: LTE, GEC formally unchanged provided that we replace 𝑇 𝑟 with 𝑇 𝑟 ( mapping of Tsallis ’ onto Gibbs ’ thermodynamics)  … and relaxation behaves formally the same way regardless of 𝑟 , in particular … −1 𝜖 2 𝑇 𝑟 = 𝑛𝑏𝑦 state is  … the variance of fluctuations of  around a 𝑇 𝑟 … 𝜖  2 −1 𝜖 2 𝑇 𝑟 𝑒𝑇 𝑟  … which implies (as 𝑒𝑇 𝑟 > 0 ) that the variance around a 𝑇 𝑟 = 𝑛𝑏𝑦 state is ∝ 𝜖  2 N.B. variance is always larger for Tsallis than for Gibbs! Vives et al. 2002

The quest for q: NLFP …with J  0

In NLFP? 𝑟 = const. However, nothing changes if 𝑟 = 𝑟 𝑢 provided that 𝑒 ln 𝑟 𝑒𝑢 ≫ 𝑒 ln 𝑄 𝑒𝑢 (slow evolution)  Slow evolution is a succession of relaxed states  If 𝐾 ≈ 0 (i.e., the interaction with the external world is weak) then the relaxed state at time 𝑢 corresponds just to 𝑇 𝑟 ′ =𝑟 ′ 𝑢 ≈ 𝑛𝑏𝑦 with 𝑄 ≈ 𝑄 𝐾 = 0,𝑟 … −1 𝜖 2 𝑇 𝑟′  … and the variance of fluctuations of  around a relaxed state is ∝ … 𝜖  2 −1  … which in a time interval 𝑒𝑢 is ∝ 𝑒𝑢 ∙ 𝜖 2 𝜖  2 The larger the variance, the larger the fluctuations of  which the relaxed state is stable against  the larger the variance, the more stable the relaxed state, the larger the fluctuations of  which the probability distribution of the relaxed state is stable against  is arbitrary  we may take 𝑒  = 𝑒𝑟′ , i.e. we deal with stability against (slow) fluctuations of the slope (depending on 𝑟′ ) of the probability distribution

𝜖 2 The most stable distribution function against fluctuations of 𝑟 ′ : 𝜖𝑟′ 2 = 0  This corresponds to an extremum of 𝜖 𝜖𝑟′  This is a minimum, as far as 𝐾 ≈ 0 at least. In the latter case, indeed: 𝑒  𝑒𝑇 𝑟 ′ = 𝑒𝑢  𝑒 = 𝑒𝑢  𝑒𝑟′  𝑒𝑟′ is the amount of produced in the time 𝑒𝑢 by the fluctuation 𝑒𝑟′ ; it is ≥ 0 for 𝑟′ = 1 (Gibbs ’ case!) as fluctuations involve irreversible physics and achieves its minimum value 0 at equilibrium (where 𝑒S = 0) of an isolated system (where 𝐾 = 0).  But GEC describes relaxation the same way regardless of 𝑟 ′ and the structure of 𝑒 the relaxed state is modified only slightly for 𝐾 ≈ 0 , hence 𝑒𝑟′ is still a minimum (even if non-zero), not a maximum! Allowable range for 𝑨 = 𝑟 ′ − 1 = 1 − 𝑟: 0 ≤ 𝑨 < 1 (𝑨 = 0 is Gibbs) Borland 1998

If 𝐾 ≈ 0 , Taylor-series development of 𝐾 in powers of 𝑨 lead to the following 𝑒 𝑒𝑨 once 𝐵 𝑦 and 𝐸 are known: useful formulas, which allow us to compute

A rule for finding 𝒓 in our NLFP! If NLFP leads to a relaxed state (well-behaved 𝐵 𝑦 ) and 𝐾 ≈ 0 then the probability distribution of microstates in the relaxed state which is more stable against slow fluctuations of its own slope is the q-exponential with 𝑟 = 1 − 𝑨 𝑒 (similar to a power law with exponent 𝑨 −1 ) and 𝑨 such that 𝑒𝑨 = min. and that 0 < 𝑨 < 1 . In this case, power-law is stable against larger fluctuations than Boltzmann epoxnential, because the variance of the latter is always lower  If such 𝑨 does not exist, then if a relaxed state exists then its probability distribution is a Boltzmann’s exponential . N.B. Variance of fluctuations around a power-law distribution are always larger. BUT… Why we have to depend on 𝑲 ≈ 𝟏 ? ? ?

The quest for q: noisy 1D maps

Application : 1D, discrete, autonomous map 𝑅 𝑗 = 𝑦 𝑢 ′ ∆𝑢′ The system evolves along a time interval ≫ ∆𝑢 ′ ( 𝑗 → ∞ ) 𝑅 𝑗+1 = 𝑦 𝑢 ′ + ∆𝑢′  ∆𝑢′

Noise ? Stochastic equation Borland 1998 N.B. 𝑨 unknown; noise may be either additive ( 𝑨 = 0 ) or multiplicative ; 𝐵 𝑦 and 𝐸 represent dynamics and noise level respectively; > 0 is arbitrary.

The stochastic equation is associated with NLFP (the probability distribution of the solution 𝑦 of the stochastic equation is the solution 𝑄 of NLFP): is arbitrary  we choose it in such a way that the approximation 𝐾 ≈ 0 applies  we need no more justification of 𝐾 ≈ 0 and our rules apply! Relaxed solutions of NLFP  the probability distributions for the noise affected 𝑅 𝑗 as 𝑗 → ∞  then …

A rule for noise-affected maps! Let a 1D, discrete, autonomus map 𝑅 𝑗+1 = 𝐻 𝑅 𝑗 be affected by noise (no matter if additive or multiplicative) and let the 𝑅 𝑗 ’s distribute as 𝑗 → ∞ along a probability distribution 𝑄 𝑅 𝑗 . Then: a) If 𝑨 exists such that 0 < 𝑨 < 1 and 𝑒 𝑒𝑨 = min then 𝑄 𝑅 𝑗 is a q-exponential with 𝑟 = 1 − 𝑨 (similar to a power law with exponent 𝑨 −1 ) b) Otherwise, 𝑄 𝑅 𝑗 is a Boltzmann’s exponential N.B. Variance of fluctuations around a power-law distribution are always larger. N.B. Only info on dynamics ( 𝐵 𝑦 = 𝐻 𝑦 − 𝑦 ) and level noise ( 𝐸 ) required!!!

Theory vs. (numerical) exp.