Statistical description of non-equilibrium systems
Eric Bertin
Laboratoire de Physique, ENS Lyon
Habilitation defence
ENS Lyon, 15 November 2011
Eric Bertin
HDR defence
CV
2000-2003: PhD supervised by J.-P. Bouchaud, SPEC, CEA
Saclay
2003-2006: post-doc, group of M. Droz, Dep. Theo. Phys.,
Univ. Geneva
Since 2006: Chargé de Recherche CNRS, Laboratoire de
Physique, ENS Lyon
Eric Bertin
HDR defence
Research interests
Different types of non-equilibrium systems
Non-stationarity states: relaxation to equilibrium
(Most interesting cases: glasses, long relaxation)
Driven steady states (two heat baths, external forces, sheared
fluids,...)
Systems composed of macroscopic (non-conservative)
“entities” (grains, agents,...)
Eric Bertin
HDR defence
Research interests
Main questions
What are the relevant macroscopic parameters to describe
non-equilibrium systems?
Beyond average values, how to characterize the (often
non-Gaussian) fluctuations of global observables like energy,
magnetization,...?
How far can statistical physics approaches be applied to
systems composed of many macroscopic particles or agents?
Eric Bertin
HDR defence
OUTLINE
Main Results
Relevant macroscopic parameters
Fluctuations of global observables
Statistical physics of many-agent systems
Outlook
Research projects
A word on teaching
Eric Bertin
HDR defence
OUTLINE
Main Results
Relevant macroscopic parameters
Fluctuations of global observables
Statistical physics of many-agent systems
Outlook
Research projects
A word on teaching
Eric Bertin
HDR defence
Relevant macroscopic parameters
Can one generalize the equilibrium notions of temperature,
pressure or chemical potential?
Still possible to define thermodynamic parameters conjugated
to conserved quantities (e.g., number of particles), but do not
necessarily equalize between two different systems
Bertin, Dauchot, Droz, PRL 2006
Bertin, Martens, Dauchot, Droz, PRE 2007
Martens, Bertin, JSTAT 2011
Energy generally not conserved: need for another approach to
define a non-equilibrium temperature
⇒ Fluctuation-dissipation relations
Eric Bertin
HDR defence
Fluctuation-dissipation relation
Basic ingredients: response and correlation
Response
External field h conjugated to a
physical observable M
Arbitrary second observable B
∂
hB(t)i to the
Response χ(t) = ∂h
field h switched off at t = 0
χ(t)
h(t)
Correlation
C (t) = hB(t) M(0)i − hB(t)ihM(0)i
Eric Bertin
HDR defence
Fluctuation-dissipation relation
Equilibrium fluctuation-dissipation relation
Response and correlation are proportional
χ(t) =
1
C (t)
T
(kB = 1)
Proportionality factor = temperature,
whatever the choice of the observable B
Eric Bertin
HDR defence
Fluctuation-dissipation relation
Non-equilibrium generalization
If linear relation between χ(t) and C (t), proportionality factor
interpreted as an effective temperature
χ(t) =
1
C (t)
Teff
Relevant definition only if Teff is the same for all observables
Independence of the observable found for instance in
mean-field spin-glasses in the aging regime
Cugliandolo, Kurchan, Peliti, PRE 1997
Is the FD temperature independent of the observable for
steady-state non-equilibrium systems?
Eric Bertin
HDR defence
Fluctuation-dissipation relation
Test of the observable dependence
Family of observables Bp , p ≥ 0 integer
Response χp (t) =
∂
∂h hBp (t)i
to the field h
Correlation Cp (t) = hBp (t) M(0)i
(hMi = 0)
Generalized fluctuation-dissipation relation
For systems close to equilibrium:
1
Cp (t)
χp (t) =
Tp
under some simplifying assumptions
Eric Bertin
HDR defence
Fluctuation-dissipation relation
Main result
Effective temperature Tp depends on the observable
√
|Tp − T0 |
≈ κp ∆S
T0
with ∆S = Seq − Sneq ≥ 0 the entropy difference between
the non-equilibrium state and the equilibrium state with the
same average energy
[Entropy S = −
P
c
P(C ) ln P(C )]
∆S may be interpreted as a distance to equilibrium
Generically, no unique effective temperature for steady-state
non-equilibrium systems
Martens, Bertin, Droz, PRL 2009 & PRE 2010
Eric Bertin
HDR defence
OUTLINE
Main Results
Relevant macroscopic parameters
Fluctuations of global observables
Statistical physics of many-agent systems
Outlook
Research projects
A word on teaching
Eric Bertin
HDR defence
Fluctuations of global observables
Motivation: Going beyond average values
How to characterize the fluctuations of global observables
(energy, magnetization) in non-equilibrium systems?
More specifically:
How to understand the surprising observations of asymmetric
distributions reported by differents groups
Bramwell, Holdsworth, Pinton, Nature 1998
Bramwell et.al., PRL 2000, PRE 2001, EPL 2002
Antal, Györgyi, Droz, Rácz, PRL 2001
Joubaud, Petrosyan, Ciliberto, Garnier, PRL 2008
...
Eric Bertin
HDR defence
Fluctuations of global observables
100
0
−1.0
10 1
-1
10 2
log(Π(θ ))
P (h)
log(σp(x))
−3.0
0.5
0.45
−5.0
0.4
P (h)
0.35
10 3
-2
Re=70791
Re=212370
Re=318560
Re=111240
Re=333750
Re=500590
χ2 model
KO62 model
0.3
0.25
-3
0.2
0.15
−7.0
0.1
0.05
0
−9.0
−10.0
0.0
−5.0
5.0
10 4
(x−<x>)/σ
Bramwell et.al.,
PRL 2000
-4
-2
-6
-4
-2
0
0
h h
h h
4
6
2
4
-4
6
-6
-4
-2
θ
0
2
Bramwell et.al.,
Portelli, Holdsworth,
EPL 2002
Pinton, PRL 2003
Well described by generalized Gumbel
distribution Ga (x) of continuous
parameter a
Ga (x) = C e −a(x+e
2
10
10
−x )
Origin of this Gumbel distribution?
Eric Bertin
10
G1(x)
G3(x)
G5(x)
-1
-2
10
HDR defence
0
-3
-2
0
x
2
4
6
Fluctuations of global observables
Gumbel distribution with a = 1 originates from extreme
statistics = statistics of the max(x1 , ..., xN ) of N identical
random variables xi .
Gumbel distribution with Gk (x), k integer
= statistics of the k th largest value
To be contrasted with fluctuations of global observables:
often statistics of a sum of non-identical and/or correlated
random variables
Link between extremes and sums?
Eric Bertin
HDR defence
Fluctuations of global observables
Mapping between extreme values and sums
P(z)
yn
z’N
z’2
z’1
z
Maximum value = sum of intervals between successive values
Eric Bertin
HDR defence
Fluctuations of global observables
Using this mapping, one can show that the GumbelP
distribution Ga (x) describes the statistics of sums n ρn with
p(ρn ) = (λn + β) e −(λn+β)ρn ,
a=1+
β
λ
Bertin, PRL 2005
Basin of attraction of Ga (x) = a specific class of correlated
sums
Bertin, Clusel, J. Phys. A 2006
Bertin, Clusel, Holdsworth, JSTAT 2008
Clusel, Bertin, Int. J. Mod. Phys. B 2008 (Review)
Eric Bertin
HDR defence
Fluctuations of global observables
Schematic representation of the mapping
Equivalence
kth largest value
statistics
G (x), k integer
Σ un with pn,k(u n)
G (x), a > 0 real
Σ un with pn,a(u n)
k
a
Eric Bertin
HDR defence
Fluctuations of global observables
An exactly solvable cascade model with the generalized
Gumbel distribution Ga (x)
J(µ1)
0
10
φ(µi)
1
ρi
-1
10
µj
0.4
-2
PE(x)
ρ
µi
PE(x)
µ1
10
0.3
0.2
-3
10
∆(µj )
p(ρn ) = (λn+β) e −(λn+β)ρn ,
-2
a = 1+
β
λ
λ characterizes dissipation, β injection
[Bertin, PRL 2005]
Eric Bertin
HDR defence
x
-1
0
0
1
x
2
4
If ρn = |cq |2 (Fourier)
Correlation length
ξ=
L
2π(a − 1)
OUTLINE
Main Results
Relevant macroscopic parameters
Fluctuations of global observables
Statistical physics of many-agent systems
Outlook
Research projects
A word on teaching
Eric Bertin
HDR defence
Statistical physics of many agents systems
Main questions
How to describe such systems of interacting agents?
Can concepts and methods from statistical physics be helpful?
Possible analytical approaches in simplified situations
Define an effective energy if possible
If interactions very localized in space and time:
kinetic theory (Boltzmann equation)
...
Eric Bertin
HDR defence
Effective energy approach
A model for the dynamics of residential moves
Agent moves if gain G > 0
effective gain =
(individual gain)
+ α (gains of other agents)
α = cooperativity
parameter
U ∗ = 1: homogeneous
U ∗ < 1: phase separation
(segregation)
Grauwin, Bertin, Lemoy, Jensen, PNAS (2009)
Eric Bertin
HDR defence
A simple model of self-propelled particles
θ
(a)
θ’
η
θ
θ1
(b)
θ2
θ’1
θ2’
η1
η2
(a) Self-diffusion
New angle θ′ = θ + η [2π]
η a Gaussian noise with variance σ02 , distribution p0 (η)
(b) Binary collisions
Define the average angle θ = Arg(e iθ1 + e iθ2 )
New angles θ1′ = θ + η1 and θ2′ = θ + η2
η a Gaussian noise with variance σ 2 that may differ from σ02 ,
and distribution p(η)
Eric Bertin
HDR defence
Boltzmann approach
Principle of the description
Evolution equation for the one-particle phase-space
distribution f (r, θ, t) = probability to find a particle at time t
in r, with a velocity angle θ
Approximation scheme: factorize the two-particle distribution
as a product of one-particle distributions
Boltzmann equation
∂f
(r, θ, t) + v0 e(θ) · ∇f (r, θ, t) = Idif [f ] + Icol [f ]
∂t
Eric Bertin
HDR defence
Integral terms in the Boltzmann equation
Self-diffusion term
Idif [f ] = −λf (r, θ, t)
Z ∞
Z π
∞
X
′
δ(θ′ + η − θ + 2mπ) f (r, θ′ , t)
dη p0 (η)
dθ
+λ
−∞
−π
m=−∞
[with θ = Arg(e iθ1 + e iθ2 ) ]
Binary collision term
Z
π
Icol [f ] = −f (r, θ, t)
dθ′ |e(θ′ ) − e(θ)|f (r, θ′ , t)
−π
Z ∞
Z π
Z π
dη p(η) |e(θ2 ) − e(θ1 )| f (r, θ1 , t) f (r, θ2 , t)
dθ2
dθ1
+
−π
−π
−∞
×
Eric Bertin
∞
X
m=−∞
HDR defence
δ(θ + η − θ + 2mπ)
Hydrodynamic equations
Hydrodynamic fields
Density field
ρ(r, t) =
Z
π
dθ f (r, θ, t)
−π
Velocity field u(r, t) and momentum field w(r, t)
Z π
dθ f (r, θ, t) e(θ)
w(r, t) = ρ(r, t) u(r, t) =
−π
Derivation of the hydrodynamic equations
Principle: take the moments of the Boltzmann equation
Integration over θ: continuity equation
∂ρ
+ ∇ · (ρu) = 0
∂t
Eric Bertin
HDR defence
Hydrodynamic equations
Velocity-field equation
Multiply by v = v0 e(θ) and integrate over θ: equation for the
velocity field (“Navier-Stokes”)
Not a closed equation in terms of ρ and u: need for an
approximation scheme
Fourier series expansion over the angle θ: f (r, θ, t) → fˆk (r, t)
Truncation and closure scheme, valid for small |w| = ρ|u|
∂w
1
+γ(w·∇)w = − ∇(ρ−κw2 )+(µ−ξw2 )w+ν∇2 w−κ(∇·w)w
∂t
2
Bertin, Droz, Grégoire, PRE 2006 & J. Phys. A 2009
Eric Bertin
HDR defence
Transport coefficients
ν=
γ=
κ=
µ=
ξ=
−1
4 14 2
1
−2σ 2
−2σ02
+ e
λ 1−e
+ ρ
4
π
15 3
8ν 16
2
2
+ 2e −2σ − e −σ /2
π 15
8ν 4
−2σ 2
−σ 2 /2
+ 2e
+e
π 15
4
2
2
−σ 2 /2
ρ e
−
− λ 1 − e −σ0 /2
π
3
2
1
64ν
−σ 2 /2
−2σ 2
e
−
+e
π2
5
3
Main result: explicit expression of the transport coefficients as
a function of microscopic parameters
Eric Bertin
HDR defence
Phase diagram in the noise-density plane
σ
1
u=0
0.8
0.6
u>0
0.4
0.2
0
ρ
0
1
2
3
4
5
Explicit phase diagram in terms of microscopic noise
(cannot be obtained from phenomenological
macroscopic equations)
Eric Bertin
HDR defence
OUTLINE
Main Results
Relevant macroscopic parameters
Fluctuations of global observables
Statistical physics of many-agent systems
Outlook
Research projects
A word on teaching
Eric Bertin
HDR defence
Ongoing projects
Applying statistical physics concepts to statistical signal
processing (and reciprocally)
Ph.D thesis of F. Angeletti (2009-2012),
co-supervised with P. Abry
Coupling experimental and theoretical approaches in the
physics of self-propelled particles (e.g., robots)
Ph.D thesis of M. Mathieu (2011-2014),
co-supervised with S. Ciliberto
Study of stochastic dissipative cascade models
Post-doc project of R. Lemoy (AP, 2011-2012)
Various ongoing collaborations with J.-C. Géminard and
P. Jensen (ENS Lyon), H. Chaté and O. Dauchot (Paris),
G. Györgyi (Budapest),...
Eric Bertin
HDR defence
Long-term project
Continue investigating the collective behavior of interacting
non-conservative “particles” or degrees of freedom
sand grains, bubbles in a foam
animals in a flock
robots, cars
models of social agents
Fourier modes in turbulent flows...
Common questions and methods? Are these systems too diverse?
Role of symmetries and conservation laws?
Relevant statistical framework or approximations?
Flat average over accessible configurations?
Eric Bertin
HDR defence
OUTLINE
Main Results
Relevant macroscopic parameters
Fluctuations of global observables
Statistical physics of many-agent systems
Outlook
Research projects
A word on teaching
Eric Bertin
HDR defence
Teaching statistical physics to non-physicists
Modelling of “complex systems” raises an increasing interest
in several disciplines beyond physics (biology, economics,
computer sciences, socials sciences,...)
Need for statistical physics teaching to students and
researchers from other disciplines
Responsible together with
P. Jensen for the second year
of Master Degree in
“Modelling of Complex
Systems”
(ENS Lyon, IXXI)
Eric Bertin
E. Bertin, Springer 2011
Introductory lecture on stat. phys.
HDR defence