Infinite Ergodic Theory meets Boltzmann-Gibbs statisticsbarkaie/BadWildBad.pdf · Outline...

Infinite Ergodic Theory meetsBoltzmann-Gibbs statistics

Eli Barkai

Bar-Ilan University

Joint work with: David Kessler, Erez Aghion

Phys. Rev. Lett. 122, 010601(2019)

Bad Wildbad

Eli Barkai, Bar-Ilan Univ.

Outline

• Non-normalized Boltzmann-Gibbs states.

• Ensemble and time averages.

• Thermodynamics relations virial theorem.

Infinite ergodic theory formulated by: Aaronson, Thaler,Zweimuller....

Non-normalises states are found in an increasing number of physical works: Fermi...(Hydrogen atom paradox) Bouchaud (trap model) Klages, Kantz, Korabel, Akimoto(non-linear dynamics) Kessler, Lutz, Aghion (cold atoms) Rebenshtok, Denisov,Hänggi, Fouxon, Radons (Lévy walks, Lorentz gas) Leibovich, Deng (multiplicativenoise), Farago (periodic potentials), Ryabov (unstable fields), Burioni, Vezzani, Wang(Big Jump).


Normalizable and non-Normalizable fields


BG for non-binding potential Z =∞


Entropy extremum principle

S[P (x, t)] = −kB∫ ∞

0

P (x, t) lnP (x, t)dx−

βkB

(∫ ∞0

V (x)P (x, t)dx− 〈V 〉)− λkB

(∫ ∞0

P (x, t)dx− 1

)−ζkb

(∫ ∞0

x2P (x, t)dx− 2Dt

).

• Equal probability.

• Averaged energy fixed, Canonical like ensemble.

• Normal diffusion.

P (x, t) =exp

[−V (x)/kBT − x2/(4Dt)

]Zt

.

limt→∞ZtP (x, t) = exp(−V (x)kBT

).


Model

• The Langevin equation with fluctuation dissipation D = kBT/γ

x(t) = −V ′(x)/γ +√

2DΓ(t).

• The Fokker-Planck equation

∂Pt(x)∂t = D

[∂2

∂x2+ ∂

∂xV ′(x)kBT

]Pt(x).

• In illustration we will consider the Lennard Jones potential VLJ(x).


Towards a non normalized state

• Let us consider fixed point solutions ∂Pt(x)∂t = 0.

• A mathematical solution of the Fokker-Planck equation

Pfp(x) = Const exp[−V (x)/kBT ]

• If the potential is normalising then this is a Boltzmann state.If not trash the non-normalised solution?


Back to the black-board

We consider the Fokker-Planck equation for asymptotically flatpotential V (∞) = 0. e.g. the LJ potential.

The force field is non binding, the density close to the minimum ofthe potential decays in time.

At long times and finite x <<√2Dt

Pt(x) ∝ t−α exp[−V (x)/kBT ]

Since V (x) is small for x >> 1

Pt(x) ' t−1/2 exp(−x2/4Dt)/√πD.

Ahhh... α = 1/2, normal diffusion.

The uniform solution, found by matching or by eigenfunctionexpansion

Pt(x) ' 1√πDt

exp[−V (x)/kBT − x2/(4Dt)

]Eli Barkai, Bar-Ilan Univ.

From normalized density to non-normalized BG state

• In the long time limit limt→∞ exp(−x2/4Dt) = 1.

• For asymptotically flat potentials

limt→∞ZtPt(x) = exp[−V (x)/kBT ]

where Zt =√πDt.

• This solution is independent of the initial state.

• All force fields in nature decay at large distances, so asymptoticallyflat potentials are common.


Ensemble averages

• The ensemble average, by definition

〈O(x)〉t =∫∞

0O(x)Pt(x)dx.

• In the long time limit,

〈O(x)〉t ∼ 1Zt

∫∞0O(x)e−V (x)/kBTdx.

• Averages are obtained with respect to the Boltzmann factor.

• Provided that the integral is finite. And then O is called integrable.

• Integrable observables are common, for example: the occupation fraction,energy, and the virial theorem.


Simulation in LJ potential

Non-normalizable Boltzmann-Gibbs state


Time averages

Mean of the time average is obtained with the non-normalized state

〈O[x(t)]〉 ∼ 1t

∫ t0〈O(x)〉t′dt′ = 2〈O(x)〉

A doubling effect is found from a time integration of 1/Zt ∼ t−1/2.

More generally 〈O〉/〈O〉 = 1/α, where 0 < α < 1.

More on α soon.


Ratio between time and ensemble average is 2

Infinite-ergodic theory


Distribution of time average

• Time averages remain random even in the long time limit.

• Let ξ = O/〈O〉.

• For example consider the indicator θ(xa < x(t) < xb) = 1 if condition holds,otherwise zero.

• The sequence 1, 0, 1, 0, , ... with τin, τout, τin, ....

PDF(τout) ∝ (τout)−(1+α)

• α is the first return exponent.

• Mean return time diverges.

• The process is recurrent.

• α = 1/2 for asymptotically flat potentials in dimension one.


Aaronson-Darling-Kac in a thermal setting

Then ξ = O/〈O〉 = n/〈n〉.

Number of returns/renewals yield the fluctuations of O.

Lévy statistics describes the distribution of time averages.

The magic: this holds true for any integrable observable.

PDF(ξ) = Γ1/α(1+α)

αξ1+1/α Lα

[Γ1/α(1+α)

αξ1/α

]Eli Barkai, Bar-Ilan Univ.

Aaronson-Darlin-Kac: Mittag-Leffler distribution of time-averages

The energy resembles a renewal process!


Virial Theorem

• The machinery of stat. mech. can be extended to infinite ergodic theory.

• The virial theorem:

limt→∞Zt〈xF (x)〉 = l0kBT .

• l0 is the second virial coefficient

l0 =

∫ ∞0

{1− exp[−V (x)/kBT ]} dx.

• To see this:

〈xF (x)〉 =kBT

∫∞0 x∂x[exp(−V (x)/kBT )−1]dx

Zt.


• The importance of the virial

P (x, t) = e−x2/4Dt√πDt

[1− l0x

2Dt + · · ·].

for large x.


Virial Theorem


Eigen function expansion, U(x) = V (x)/kBT

• eigen function expansion

Pt(x) = e−U(x)/2+U(x0)/2

∑N

2kΨk(x0)Ψk(x)e

−Dk2t.

• The Schrodinger like equation

HΨ(x) = Dk2Ψ(x)

• The energy spectrum, is k2, as for a free particle. k = 0 ground state.

• Explicitly

−∂2

∂x2Ψk(x) +

[U ′(x)2

4−U ′′(x)

2

]Ψk(x) = k

2Ψk(x).

• Standard: Ground state is exp[−U(x)/2]. It is not normalized.


• In that sense the infinite density is the ground states of the system.

• Here all the states k are non-normalizable (basic QM).

• Small k corrections

Ψk(x) ' e−U(x)/2[cos(kx)(1− k2

g(x))− kl0sin(kx)].


• For x ' 1

P (x, t) =exp [−U(x)]√πDt

[1−

h(x, x0)

t...

]

• Small k large t so∫

0dk exp(−k2t) ∼ 1/t1/2.

• The correction are non-universal: they depend on x0.

• For x� 1

P (x, t) =e−x

2/4Dt

√πDt

[1−

l0x

2Dt+ · · ·

].

• Corrections are universal.

• Leading term∫∞

0e−k

2t cos(kx) ∝ exp(−x2/4t).




Summary

M. Siler et al. PRL (2018).Aghion, Kessler, Barkai PRL (2019).


Summary

Stochastic thermodynamics of single particle trajectories in thepresence of asymptotically flat (or log) potential fields, uses non-normalised Boltzmann-Gibbs statistics. Both time and ensembleaverages of integrable observables are calculated using the non-normalised Boltzmann state. If the process is recurrent, with aninfinite mean return time, standard ergodic theories obviously fail,however the resampling of the phase space implies that we maystill construct a stat. mech. framework which is independent ofthe initial condition. Lévy statistics describes the fluctuations of thetime averages (ADK theorem). An extremum principle yields a newensemble, where the Gaussian central limit describes the dynamicsfor x >> 1. This leads to non-equilibrium thermodynamic relations,e.g. between entropy and energy, the virial theorem etc.

Aghion, Kessler, Barkai Phys. Rev. Lett. 122, 010601 (2018)


Thermodynamic relation: S(t) = kBln(Zt) + Ep/T + kBζ〈x2〉

Entropy-energy relation:

•(∂S∂Ep

)t

= 1T so fixed volume is replaced with t


Single Molecule Experiments

Particle immersed in a bath with temperature T .Force field vanishes for large x.


Particle in a log potential

X0

• V (x) = 0.5 ln(1 + x2), PBG(x) = (1 + x2)−1/2T/Z for T < 1.

• The normal BG phase Kessler Barkai PRL 2010.

• Transition point T = 1.


Transition from BG to NNBG

First return calculation α = 12 +

1T .

• When α < 1: i) mean return time ii) Z and iii) SBG, blow up.


Ratio between time and ensemble averages is 1/α

• Infinite-ergodic theory ->


Higher dimensions?


Normalized BG in log potential

x

P(x)

BG

ICD


Trajectory


Distribution of generalized Lyapunov Exp.

ζ = λα/〈λα〉.Renewal Theory: distribution of λα is Mittag-Leffler.Aaronson-Darling-Kac Theorem.Korabel Barkai Phys. Rev. Lett. 102, 050601 (2009).


The importance of being flat

Consider an inverted harmonic potential field V (x) = −x2/2.

Also here fixed point solution is non-normalized.

However

P (x, t) =√

12π(1−e−2t) exp

[−(xe−t−x0)2

2(1−e−2t)

]e−t

So

limt→∞√2π exp(t)P (x, t) = exp

[−(x0)

2/2].

Here initial conditions remain for ever.

No Boltzmann Gibbs state.


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