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.
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