non-normalized boltzmann-gibbs statistics
TRANSCRIPT
Non-normalized Boltzmann-Gibbs statistics
Eli Barkai
Bar-Ilan University
Joint work with: David Kessler, Erez Aghion
Phys. Rev. Lett. 122, 010601(2019)
Tel-Aviv
Eli Barkai, Bar-Ilan Univ.
Outline
• Recap infinite ergodic theory in determinsitc setting.
• Recap on Boltzmann-Gibbs statistics and ergodic theory.
• Non-normalized Boltzmann-Gibbs states.
• Ensemble and time averages.
• Thermodynamics relations and entropy maximization.
Non-normalises states are found in an increasing number of physics papers:Kantz, Korabel, Akimoto (non-linear dynamics) Kessler, Lutz, Aghion (cold atoms)Rebenshtok, Denisov, Hänggi (Lévy walks).
Eli Barkai, Bar-Ilan Univ.
Pomeau Manneville Map
0 , 0 0 , 5 1 , 00 , 0
0 , 5
1 , 0
M(x)
xξ
x 0
xt+1 = M(xt) M(xt) ∼ xt + a(xt)z xt → 0
Marginally unstable fixed point on zero, yields anomalous dynamics, zeroLyapunov exponent, and infinite ergodic theory.
Eli Barkai, Bar-Ilan Univ.
Power-law sojourn times
In vicinity of unstable fixed point
dx
dt= axz
Easy to find time to reach b starting at x0.
Smooth injection of trajectories, x0 is uniformly distributed
ψ(t) ∝ t−(1+α), α = 1z−1.
When 0 < α < 1 (z > 2) mean sojourn time in vicinity of x ' 0diverges.
Eli Barkai, Bar-Ilan Univ.
Sub-exponential separation of nearby trajectories
0 2000 4000
0,0
0,5
1,0
1,5
1
ln(|M
'(xt)|)
t
S
0
∑t−1t=0 lnM ′(xt) ∝ number of renewals ∝ t∫ t
tt−1−αdt∝ tα.
Distribution of number of renewals in (0, t) yields distribution of λα.
λα =∑t−1t=0 lnM ′(xt)
tα
Eli Barkai, Bar-Ilan Univ.
Infinite Invariant Measure (Aaronson, Thaler, · · ·)
10-2 10-1 100
100
101
102
103
104
xc(t3)
xc(t4)
(x)
x
xc(t2)
~x-1/
ρ(x) = t1−αρ(x, t)
ρ(x) ∝ x−1/α Non Normalizable.
Eli Barkai, Bar-Ilan Univ.
Generalized Lyapunov Exp.
1 2 3 4 50
1
2
3
4
5
<>
z
〈λα〉 =
∫ 1
0
ln |M ′(x)|ρ(x)dx/α
Even though ρ(x) non normalizable, it yields the average.
Eli Barkai, Bar-Ilan Univ.
Recap
• Consider a particle in binding force field
Peq(x) =exp [−V (x)/kBT ]
Z.
• The normalizing partition function
Z =
∫ ∞−∞
exp [−V (x)/kBT ] dx.
• The ergodic hypothesis
O =
∫ t0O[x(t′)]dt′
t→ 〈O〉.
Eli Barkai, Bar-Ilan Univ.
Ergodicity
Ergodicity: time averages = ensemble averages.
Initial conditions are wiped out and forgotten
Eli Barkai, Bar-Ilan Univ.
The Hydrogen Paradox (Herzfeld, Fowler, Wigner,...)
The barometric formula ρ(x) = ρ0 exp(−mgh/kBT ) means that we teachstudents that earth is flat. Same problems for black holes.
Eli Barkai, Bar-Ilan Univ.
Non-normalizable
• Bouchuad trap model for glass dynamics
Z ∝∫ ∞
0
exp(−E/Tg + E/T )dE =∞.
• The log potential.
• In physics:A trick is to introduce cutoffs, consider a box philosophy.
• In math:Infinite ergodic theory: Aaronson, Thaler, Zweimuller has an elegant appeal.
Eli Barkai, Bar-Ilan Univ.
Single Molecule Experiments
Particle immersed in a bath with temperature T .Force field vanishes for large x.
Eli Barkai, Bar-Ilan Univ.
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 + ∂∂x
V ′(x)kBT
]Pt(x).
• In illustration we will consider the Lennard Jones potential
VLJ(x) = ....
• And later a potential which for x >> 1 is logarithmic.
Eli Barkai, Bar-Ilan Univ.
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?
For a large class of force fields this is not advisable.
• But in what sense is the normalized solution Pt(x) attracted to anon-normalised state?
Eli Barkai, Bar-Ilan Univ.
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.
Eli Barkai, Bar-Ilan Univ.
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.
Eli Barkai, Bar-Ilan Univ.
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.
• Similar to ordinary stat. mech. averages are obtained with respect to theBoltzmann factor.
• Provided that the integral is finite. And then O is called integrable.
• One example is the potential energy of the particles, Ep = V (x).
• Integrable observables are the super common, for example the occupationfraction (also entropy, virial theorem, need non-normalizable stat. mech.).
• We can treat also the non-integrable observables, e.g. O = x2.
Eli Barkai, Bar-Ilan Univ.
Time averages
The mean of the time average is obtained with the non-normalizedstate
〈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.
Eli Barkai, Bar-Ilan Univ.
Entropy extremum principle
S[P (x, t)] = −kB∫ ∞
0
P (x, t) lnP (x, t)dx−
βkB
(∫ ∞0
U(x)P (x, t)dx− 〈U〉)− λkB
(∫ ∞0
P (x, t)dx− 1
)−ζkb
(∫ ∞0
x2P (x, t)dx− 2Dt
).
• First term: Equal probability.
• Second: Average energy (canonical like ensemble).
• Normalization.
• The Gaussian central limit theorem.
Pt(x) ' 1√πDt
exp[−V (x)/kBT − x2/(4Dt)
]Eli Barkai, Bar-Ilan Univ.
Thermodynamic relation: S(t) = kBln(Zt) + Ep/T + kBζ〈x2〉
Entropy-energy relation:
•(∂S∂Ep
)t
= 1T so fixed volume is replaced with t
Eli Barkai, Bar-Ilan Univ.
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.
Eli Barkai, Bar-Ilan Univ.
Aaronson-Darling-Kac in a thermal setting
Then ξ = O/〈O〉 = n/〈n〉.
The number of events 1 yield the statistics of the time averages.
The time in the interval (xa, xb) is n〈τin〉.
How do we get the PDF of n? First fix n and let t =∑ni τout(i). Here
PDF of t is a one sided Lévy distribution.
Lévy statistics describes the distribution of the timeaverage
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!
Eli Barkai, Bar-Ilan Univ.
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).
Eli Barkai, Bar-Ilan Univ.
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.
Eli Barkai, Bar-Ilan Univ.
Transition from BG to NNBG
First return calculation α = 12 + 1
T .
• When α < 1: i) mean return time ii) Z and iii) SBG, blow up.
Eli Barkai, Bar-Ilan Univ.
Ratio between time and ensemble averages is 1/α
• Infinite-ergodic theory ->
Eli Barkai, Bar-Ilan Univ.
The force field determines the nature of stat. mech.
M. Siler et al. PRL (2018).Aghion, Kessler, Barkai PRL (2019).
Eli Barkai, Bar-Ilan Univ.
Virial Theorem (flat potentials)
• The machinery of stat. mech. can be extended to infinite ergodic theory.
• The virial theorem:
〈xF (x)〉 ∼ 2B2ZtkBT
• B2/Zt ratio of two length scales, with B2 = 12
∫∞0{1− exp[−V (x)/kBT ]} dx.
• Use 〈xF (x)〉 = kBT∫∞
0x∂x[exp(−V (x)/kBT )− 1]dx/Zt.
• B2 is the second virial coefficient.
Eli Barkai, Bar-Ilan Univ.
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.
Aghion, Kessler, Barkai Phys. Rev. Lett. 122, 010601 (2018)
Eli Barkai, Bar-Ilan Univ.